{"id":6843,"date":"2024-02-17T22:17:22","date_gmt":"2024-02-17T22:17:22","guid":{"rendered":"https:\/\/williamcromar.com\/newmediawiki\/?p=6843"},"modified":"2024-03-16T23:27:30","modified_gmt":"2024-03-16T23:27:30","slug":"chapter-10-expressions-of-geometry","status":"publish","type":"post","link":"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/","title":{"rendered":"CHAPTER 10 | Expressions  of Geometry"},"content":{"rendered":"\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title\" style=\"cursor:inherit\">Contents<\/div>\n<label for=\"ez-toc-cssicon-toggle-item-69f10d382c1f0\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"ez-toc-cssicon\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69f10d382c1f0\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#The_Idea_of_a_Model_Forms_and_Primitives\" >The Idea of a Model: Forms and Primitives<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#The_Theory_of_Forms\" >The Theory of Forms<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Platonic_solids\" >Platonic solids<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Pre-platonic_polyhedra\" >Pre-platonic polyhedra<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Primitives\" >Primitives<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Digital_primitives\" >Digital primitives<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Coordinates\" >Coordinates<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Cartesian_coordinate_system\" >Cartesian coordinate system<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Axis_Mundi\" >Axis Mundi<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#GeoMATER_Joyce_Frascari_and_three_kinds_of_geometry\" >GeoMATER: Joyce, Frascari, and three kinds of geometry<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Mantic_geometry\" >Mantic geometry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Structure_geometry\" >Structure geometry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Body_geometry\" >Body&nbsp;geometry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Mantic_geometry_in_creating_a_proportioning_system\" >Mantic geometry in creating a proportioning system<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Format\" >Format<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Proportion_grids_in_the_format\" >Proportion grids in the format<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Structure_geometry_for_internal_proportioning\" >Structure geometry for internal proportioning<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Body_geometry_and_the_visible_world\" >Body geometry and the visible world<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Hidden_geometry\" >Hidden geometry<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Expressing_a_datum\" >Expressing a datum<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Expressing_axes_of_symmetry\" >Expressing axes of symmetry<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Bilateral_symmetry\" >Bilateral symmetry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Radial_symmetry\" >Radial symmetry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Self-similarity_symmetry_in_scale\" >Self-similarity: symmetry in scale<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#%E2%80%9CNear%E2%80%9D_symmetry\" >&#8220;Near&#8221; symmetry<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Expressing_coordinate_systems\" >Expressing coordinate systems<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Orthographic_and_3-view_drawings\" >Orthographic and 3-view drawings<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Section_drawings\" >Section drawings<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Generating_irregular_form\" >Generating irregular form<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Topographic_lines\" >Topographic lines<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/#Boat_lofting\" >Boat lofting<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p class=\"has-text-align-right has-small-font-size\"><em>18-minute read<\/em><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"694\" height=\"694\" data-attachment-id=\"30\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/paralleluniverses_01\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01.png\" data-orig-size=\"694,694\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"parallelUniverses_01\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01.png\" src=\"https:\/\/i0.wp.com\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01.png?resize=694%2C694&amp;ssl=1\" alt=\"\" class=\"wp-image-30\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01.png 694w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01-300x300.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01-150x150.png 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/parallelUniverses_01-100x100.png 100w\" sizes=\"auto, (max-width: 694px) 100vw, 694px\" \/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"402\" data-attachment-id=\"6844\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs.png\" data-orig-size=\"6392,2507\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-1024x402.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-1024x402.png\" alt=\"\" class=\"wp-image-6844\" style=\"width:1200px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-1024x402.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-300x118.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-768x301.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-1536x602.png 1536w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/67cfb2e0-d85e-476b-a131-f408b7951db82aba3d7085203a1b9d_orbs-2048x803.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Neolithic&nbsp;<a href=\"http:\/\/www.ashmolean.org\/ash\/britarch\/highlights\/stone-balls.html\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Carved Stone Balls<\/strong><\/a>, c. 2000 BCE, Ashmolean Museum, University of Oxford, UK. These balls contain anywhere from 4 to 6 to 7 to 14 knobs, but some in other collections contain more. They don&#8217;t indicate a systematic representation of the Platonic Solids \u2014 one would expect 12 knobs for a dodecahedron, for example. However, they indicate an intuitive understanding of geometry at least a thousand years before any written record describing geometric properties of polyhedra.<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"The_Idea_of_a_Model_Forms_and_Primitives\"><\/span>The Idea of a Model: Forms and Primitives<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Picasso\u2019s &#8220;art is the lie that makes us realize the truth&#8221; statement is based on a fairly ancient idea. Making art, we have argued elsewhere, is fundamentally an act of modeling. <a href=\"https:\/\/www.merriam-webster.com\/dictionary\/model\">Merriam-Webster<\/a>&nbsp;defines modeling as an act of designing or imitating forms, an act of shaping, molding, or planning. It defines a model as a material representation or embodiment of an ideal. <\/p>\n\n\n\n<p>Picasso&#8217;s art\/lie statement is cagey for effect. He&#8217;s arguing that art is an embodiment of ideals, a giving form to things that should, but don&#8217;t, exist. What he&#8217;s saying is that the artist sees the world of the ideal, the suitable model for that which ought to be in the world and which it is the artist\u2019s task to show, through the imperfect material lens of art. This is another way of saying that artists give form to <em>Forms<\/em>, a basic and ancient idea that permeates Western philosophy.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"The_Theory_of_Forms\"><\/span>The Theory of Forms<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Plato first postulated the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Theory_of_Forms\"><strong>Theory of Forms<\/strong><\/a>. He argues that immaterial abstractions known as <strong>Forms<\/strong>, which some authors translate tellingly as <strong>Ideas<\/strong>, possess a more fundamental reality than the objects that we see and touch. Tangible objects, in other words, are imperfect, mimetic copies of the Forms. The Forms are true, and the objects we see are a dull kind of mirror, a <em>partial<\/em>&nbsp;reflection of the truth embodied in the Forms. A partial truth is not the whole truth, and it is in that sense that Picasso&#8217;s &#8220;lie&#8221; should be understood.<\/p>\n\n\n\n<p>Plato&#8217;s Theory of Forms provides an early definition of modeling and can be considered a rigorous formulation of the philosophy of modeling. Among the tools Hellenistic philosophers had for describing Forms included words and images, and images are governed among many other things by \u2014 what else? \u2014 <strong>Geometry<\/strong>. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Platonic_solids\"><\/span>Platonic solids<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>Beloved by the Greeks because it could describe abstractions and perform mathematical calculations, Geometry was a subject intertwined with the Forms. Although Plato <a href=\"https:\/\/en.wikipedia.org\/wiki\/Platonic_solid\" target=\"_blank\" rel=\"noreferrer noopener\">did not discover them<\/a>, he features the&nbsp;Platonic Solids prominently in his writings on the classical elements. He associates fire, earth, air, and water with the tetrahedron, cube (or hexahedron), octahedron, and icosahedron respectively. He links the fifth solid, the dodecahedron, to the construction of the heavens. There are only 5 Platonic Solids by definition: a closed, regular polyhedron consisting of the same polygon for each face. Any polygon larger than a pentagon creates a flat plane or an impossible tiling.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"400\" height=\"600\" data-attachment-id=\"6846\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/8535989_orig\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/8535989_orig.jpg\" data-orig-size=\"400,600\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"8535989_orig\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/8535989_orig.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/8535989_orig.jpg\" alt=\"\" class=\"wp-image-6846\" style=\"object-fit:cover\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/8535989_orig.jpg 400w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/8535989_orig-200x300.jpg 200w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/8535989_orig-300x450.jpg 300w\" sizes=\"auto, (max-width: 400px) 100vw, 400px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\">\n<p><em>Leonardo da Vinci,&nbsp;<strong>Platonic Solids<\/strong>, illustration for Luciano Pacioli&#8217;s text <a href=\"https:\/\/archive.org\/details\/divinaproportion00paci\/page\/n185\/mode\/2up\" target=\"_blank\" rel=\"noreferrer noopener\">De Divina Proportione, 1509<\/a>. Top: tetrahedron, cube (hexahedron). Middle: octahedron, icosahedron. Bottom: dodecahedron. Da Vinci represented the solids in both open form (seen here) and solid form. The open form suggests a computer modeling view known as&nbsp;wireframe modeling, wherein one can see all sides of a solid simultaneously, although here da Vinci gives the &#8220;wire&#8221; some thickness to avoid spatial ambiguity \u2014 that is, he gives the mathematical construct a tactile geometric reading. For more on Platonic Solids, visit the demonstration videos at the <a href=\"http:\/\/demonstrations.wolfram.com\/PlatonicSolids\/\">Wolfram MathWorld<\/a> website.<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Pre-platonic_polyhedra\"><\/span>Pre-platonic polyhedra<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>Some attribute the discovery of the Platonic Solids to the Greek mathematician <a href=\"http:\/\/en.wikipedia.org\/wiki\/Pythagoras\">Pythagoras<\/a>, but they turn out to have been known in some form since prehistory. Hundreds of intricately <a href=\"http:\/\/en.wikipedia.org\/wiki\/Carved_Stone_Balls\">carved stone balls<\/a> suggesting knowledge of polyhedra have been found in Scotland and dated to the Neolithic period, a millennium before Plato&#8217;s birth. We can only speculate about their use absent any kind of written record. However, we can be certain the Neolithic Pictish people who occupied Scotland understood these volumes to have exceptional properties and therefore manifested the presence of something exceptional in the world, even if they did not possess the mathematical understanding of the later Greeks. Modeling is indeed an ancient enterprise.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Primitives\"><\/span>Primitives<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>3D modeling is the closest means we have yet devised to peek through our dark glass of material existence into Plato&#8217;s rarefied world of pure Form, and one can see homage paid to the Platonic Solids in certain modeling user interfaces. Another homage often exists in the UI: the world of what we call <a href=\"http:\/\/en.wikipedia.org\/wiki\/Geometric_primitive\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Primitives<\/strong><\/a>. <\/p>\n\n\n\n<p>Primitives overlap slightly with the special-case convex polyhedra that constitute Platonic Solids. We see one example in the <strong>cube<\/strong> and another in the 3-sided pyramid known as the <strong>tetrahedron<\/strong>. We see shapes \u2014 the <strong>sphere<\/strong>, <strong>cylinder<\/strong>, <strong>cone<\/strong>, <strong>torus<\/strong>, and others \u2014 as the fundamental building blocks of more complex geometries. The painter Paul C\u00e9zanne&nbsp;explained in a conversation with Emile Bernard that a painter &#8220;must first of all study geometric forms: the cone, the cube, the cylinder, the sphere. When one knows how to render these things in their forms and their planes, one ought to know how to paint.&#8221; <sup><a href=\"#footnote_1_6843\" id=\"identifier_1_6843\" class=\"footnote-link footnote-identifier-link\" title=\"Loran, Erle. &quot;C&eacute;zanne&#039;s Composition.&quot; University of California Press. 2006. p. 9.\">1<\/a><\/sup><\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"2250\" height=\"1695\" data-attachment-id=\"6851\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/r674\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890.jpg\" data-orig-size=\"2250,1695\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;R674&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"R674\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-1024x771.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890.jpg\" alt=\"\" class=\"wp-image-6851\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890.jpg 2250w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-300x226.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-1024x771.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-768x579.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-1536x1157.jpg 1536w\" sizes=\"auto, (max-width: 2250px) 100vw, 2250px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p><em>Paul C\u00e9zanne,&nbsp;<strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Paul_C%C3%A9zanne,_Still_Life_With_Apples,_c._1890.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Still Life With Apples<\/a><\/strong>, 1890. One level of geometric structure exists in what you see \u2014 apples being structured by the sphere, for example. The next level of structure exists in what&nbsp;controls what you see \u2014 a tetrahedral organization for the apples. We discover an even deeper level in the brushwork \u2014&nbsp;note the V-shaped shadow below the far right apple. It visually aligns with shadows on the wall to suggest a purely compositional\/spatial cone.<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"300\" data-attachment-id=\"6852\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/4895819323_0ee8c37246_o\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4895819323_0ee8c37246_o.jpg\" data-orig-size=\"600,300\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"4895819323_0ee8c37246_o\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4895819323_0ee8c37246_o.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4895819323_0ee8c37246_o.jpg\" alt=\"\" class=\"wp-image-6852\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4895819323_0ee8c37246_o.jpg 600w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4895819323_0ee8c37246_o-300x150.jpg 300w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p><em><strong>Geometric Primitives<\/strong> in digital modeling. C\u00e9zanne&#8217;s Primitives \u2014 cone, cube, cylinder, sphere \u2014 are at right in the image. Other Primitives include the pyramid, plane (technically, a 2D object), torus, and pipe. Modeling often employs distortions and combinations based on these simple volumes.<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Digital_primitives\"><\/span>Digital primitives<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>C\u00e9zanne did not invent the Primitives. However, his sophisticated empirical study of these building-block geometries, which simultaneously structure and are structured by our perception of the natural world, led to the first complete reinvention of space in painting since the invention of Renaissance perspective, ultimately inspiring Picasso to develop Cubism. Today, a 3D modeler can manipulate C\u00e9zannesque geometric primitives into complex forms in the same analytical way. This analysis is not something software will do, any more than a brush makes a painting for a painter.<\/p>\n\n\n\n<p>Digital modeling allows us to see and manipulate Forms and Primitives with unheard-of precision. Yet, we usually view them filtered through an imperfect mechanism: the two-dimensional computer monitor representing the 3D world through a fairly clumsy window. Regardless, we are privileged to witness an unprecedented development in the history of visualizing geometry.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Coordinates\"><\/span>Coordinates<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Plato and C\u00e9zanne share the view that a precise, pure, invisible geometry controls our perception of the less pure reflections of geometry in the material world. The abstract geometry of the Forms and Primitives is a mathematical construct. As such, we have devised several mathematical means to make that geometry tangible. <\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Cartesian_coordinate_system\"><\/span>Cartesian coordinate system<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>In Enlightenment Europe, another philosopher, <a href=\"http:\/\/en.wikipedia.org\/wiki\/Ren%C3%A9_Descartes\">Ren\u00e9 Descartes<\/a>, developed a system for linking geometry with algebra that bears his name. The <a href=\"http:\/\/en.wikipedia.org\/wiki\/Cartesian_coordinate_system\"><strong>Cartesian coordinate system<\/strong><\/a> allows a given point to be specified relative to unit measure distances from a point of origin. We prescribe this origin by two perpendicular lines (in the case of a 2D plane) or three mutually perpendicular and intersecting lines (in the case of a 3D space). Digital modeling uses this mathematical means to define the volumetric elements you create.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"600\" data-attachment-id=\"6862\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/4897466994_239a901481_o\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897466994_239a901481_o.png\" data-orig-size=\"620,600\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"4897466994_239a901481_o\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897466994_239a901481_o.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897466994_239a901481_o.png\" alt=\"\" class=\"wp-image-6862\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897466994_239a901481_o.png 620w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897466994_239a901481_o-300x290.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p><em>&nbsp;<a href=\"http:\/\/en.wikipedia.org\/wiki\/File:Coord_system_CA_0.svg\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Cartesian coordinate system<\/strong><\/a>. In 3D, we define the origin (O) as the intersection of 3 mutually perpendicular lines X, Y, and Z, from which each reference plane \u2014 XY, XZ, and YZ \u2014 gets its name. The XY plane is traditionally the horizontal reference plane, but&#8230;<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"460\" height=\"460\" data-attachment-id=\"6863\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/4897467050_4fede1c445_o\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o.jpg\" data-orig-size=\"460,460\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"4897467050_4fede1c445_o\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o.jpg\" alt=\"\" class=\"wp-image-6863\" style=\"width:799px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o.jpg 460w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o-300x300.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o-150x150.jpg 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/4897467050_4fede1c445_o-100x100.jpg 100w\" sizes=\"auto, (max-width: 460px) 100vw, 460px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\">\n<p><em>&nbsp;&#8230; it is an eccentricity of Maya that it uses Y to measure height and Z to measure depth. If you are coming to Maya from another 3D program or are already familiar with the conventions of Cartesian orientation, this can cause minor confusion.<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n<p>This system for quantifying and systematizing geometry seems fairly dry and straightforward, but it contains powerful metaphorical implications. Of one kind of tactile expression of Cartesian geometry, <a href=\"http:\/\/faculty.frostburg.edu\/phil\/forum\/Descartes.htm\">Jorn K. Bramann<\/a> writes:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><em>The French garden architecture of the 17th and 18th centuries is a striking demonstration of the spirit of Cartesian thinking. This type of landscaping, which was widely copied throughout Europe, is characterized by the demonstrative superimposition of geometric shapes and figures on nature. The natural terrain of a garden is not allowed to remain as it is found, but is carefully leveled, and then sectioned into regular parts until it resembles a mathematician&#8217;s blueprint. Plants are placed in such a way that they form straight lines, circles, ellipses, or artfully designed mazes. Individual trees and bushes are clipped until they represent perfect spheres, cones, squares, or other geometrical figures. It was the most deep-seated passion of the age to press nature into designs that are not natural. Pure geometry is a human creation, a creation of the abstract mind. To superimpose geometry on what otherwise grows in irregular forms was the lustful demonstration of the detached sovereign mind&#8217;s power over the external world.<\/em> <sup><a href=\"#footnote_2_6843\" id=\"identifier_2_6843\" class=\"footnote-link footnote-identifier-link\" title=\"Bramann, Jorn K. &quot;Descartes: The Solitary Self.&quot; Philosophical Forum at Frostburg State University. 2004. http:\/\/faculty.frostburg.edu\/phil\/forum\/Descartes.htm\">2<\/a><\/sup><\/p>\n<\/blockquote>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1200\" height=\"873\" data-attachment-id=\"6870\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/labyrinth_gallery_05\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05.jpg\" data-orig-size=\"1200,873\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"labyrinth_gallery_05\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05-1024x745.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05.jpg\" alt=\"\" class=\"wp-image-6870\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05.jpg 1200w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05-300x218.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05-1024x745.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/labyrinth_gallery_05-768x559.jpg 768w\" sizes=\"auto, (max-width: 1200px) 100vw, 1200px\" \/><figcaption class=\"wp-element-caption\"><em>Hans Vredeman de Vries, from&nbsp;<a href=\"http:\/\/www.vilt.net\/nkdee\/sisforstanzas.jsp\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Elegant and Multifarious Depictions of Gardens and Parks<\/strong><\/a>, late 16th C. The garden engravings of the Flemish architect de Vries are designs prescient of the development of the Cartesian coordinate system in French formal gardening.<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Axis_Mundi\"><\/span>Axis Mundi<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Note especially the tree in the center of the garden, acting as a kind of vertical axis to the horizontal plane. Even the axial system underpinning the Cartesian grid itself contains symbolic power. The concept of the <a href=\"http:\/\/en.wikipedia.org\/wiki\/Axis_mundi\"><strong>Axis Mundi<\/strong><\/a>, a mythological center of the world, is another example of convergence between mathematical and tactile expressions of geometry. A powerful transcultural metaphor, the&nbsp;<em>axis mundi<\/em>&nbsp;is a vertical axis or column expressing both an umbilical and phallic origin of the world, a link between earth and sky. Sculptor Constantin Brancusi used this symbolism in <em><strong>The Endless Column<\/strong><\/em>, a monument to Romanian war dead.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" data-attachment-id=\"6875\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/endless-column\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column.jpg\" data-orig-size=\"1150,1150\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"endless-column\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-1024x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-1024x1024.jpg\" alt=\"\" class=\"wp-image-6875\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-1024x1024.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-300x300.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-150x150.jpg 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-768x768.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column-100x100.jpg 100w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/endless-column.jpg 1150w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Constantin Brancusi,&nbsp;<a href=\"http:\/\/www.panoramio.com\/photo\/12771269\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>The Endless Column<\/strong><\/a>, 1938, restored 2000. An expression of the <strong>axis mundi<\/strong>, this and other towers become a tactile expression of the vertical Cartesian axis, giving it symbolic power. In its rhythmic modulation, the column seems to invite the viewer to ascend.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"1004\" data-attachment-id=\"6876\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/f794c-brancusi-endless-column\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/f794c-brancusi-endless-column.jpg\" data-orig-size=\"670,1004\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"f794c-brancusi-endless-column\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/f794c-brancusi-endless-column.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/f794c-brancusi-endless-column.jpg\" alt=\"\" class=\"wp-image-6876\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/f794c-brancusi-endless-column.jpg 670w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/f794c-brancusi-endless-column-200x300.jpg 200w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/f794c-brancusi-endless-column-300x450.jpg 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/figure>\n<\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"GeoMATER_Joyce_Frascari_and_three_kinds_of_geometry\"><\/span><strong>GeoMATER<\/strong>: Joyce, Frascari, and three kinds of geometry<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p><em>Now<\/em>, aqua in buccat<em>. I\u2019ll make you to see figuratleavely the whome of your eternal&nbsp;<\/em><strong><em>geomater<\/em><\/strong><em>. And if you flung her headdress on her from under her highlows you\u2019d wheeze whyse Salmonson set his seel on a hexen-gown.<\/em><\/p>\n<cite>\u2014&nbsp;James Joyce, Finnegans Wake&nbsp;[<strong>emphasis<\/strong>&nbsp;by the author]<sup><a href=\"#footnote_3_6843\" id=\"identifier_3_6843\" class=\"footnote-link footnote-identifier-link\" title=\"Joyce, James. Finnegans Wake. Penguin Books. 1999. pp. 296-7.\">3<\/a><\/sup><\/cite><\/blockquote>\n\n\n\n<p>Geometry is a very specific way of creating order, and modeling is a very specific way of working with geometry. But there is more than meets the eye \u2014 literally \u2014 when we discuss the use of geometry to create a model.<\/p>\n\n\n\n<p><a href=\"http:\/\/en.wikipedia.org\/wiki\/Marco_Frascari\" target=\"_blank\" rel=\"noreferrer noopener\">Marco Frascari<\/a>, an Italian architect and design theorist, formulated three specific ways geometry is used in creative work, channeling the Irish author James Joyce to do so.&nbsp;Frascari appropriates the pun word <strong>geomater<\/strong>, a rich and metaphorical term coined by Joyce, who was famous (or infamous) for mashing up words to create new meanings. <\/p>\n\n\n\n<p>Frascari reveals how Joyce combined the words&nbsp;<em>matrix<\/em>,&nbsp;<em>mater<\/em>, and <em>meter<\/em> with&nbsp;<em>geology<\/em>&nbsp;and&nbsp;<em>geometry<\/em>&nbsp;in a pregnant (literally and figuratively) metaphor describing geometry as a discipline of prediction (the <strong>Mantic<\/strong>), creation (the <strong>Structure<\/strong>), and observation (the <strong>Body<\/strong>). He defines artists and designers as image makers constructing geomaternal figures in, about, and on this world. These figures are paintings, drawings, buildings, sculptures, and designed objects of every kind. They are, in his words, &#8220;material manifestations of the geometric science of visualization.&#8221; Frascari\u2019s useful concept of geomater connotes that human-made things are developed in three coincidental and co-equal parts:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"752\" height=\"1024\" data-attachment-id=\"1416\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/blake_ancient\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient.jpg\" data-orig-size=\"800,1089\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"blake_ancient\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-752x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-752x1024.jpg\" alt=\"\" class=\"wp-image-1416\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-752x1024.jpg 752w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-220x300.jpg 220w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-768x1045.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-300x408.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient.jpg 800w\" sizes=\"auto, (max-width: 752px) 100vw, 752px\" \/><figcaption class=\"wp-element-caption\"><em>William Blake<br><a href=\"https:\/\/en.wikipedia.org\/wiki\/The_Ancient_of_Days\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>The Ancient of Days<\/strong><\/a><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h3 class=\"wp-block-heading has-text-align-center\"><span class=\"ez-toc-section\" id=\"Mantic_geometry\"><\/span><strong><a href=\"http:\/\/en.wikipedia.org\/wiki\/Mantic\" target=\"_blank\" rel=\"noreferrer noopener\">Mantic<\/a><\/strong><br><strong>geometry<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p><em><strong>Mantic<\/strong><\/em> is a term associated with prophecy and divination. We see it as a root for words such as <em>ro<\/em><strong><em>mantic<\/em><\/strong> (as in visionary).<\/p>\n\n\n\n<p>Mantic geometry projects a possible future, making thinkable the yet-to-be-imagined, associated with layout and format.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-white-color has-alpha-channel-opacity has-white-background-color has-background\"\/>\n\n\n\n<p>An example will be how we use a layout device, such as the rule-of-thirds grid, for composition:<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"664\" height=\"1024\" data-attachment-id=\"2383\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/manticgeometry\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry.png\" data-orig-size=\"829,1279\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"manticgeometry\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry-664x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry-664x1024.png\" alt=\"\" class=\"wp-image-2383\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry-664x1024.png 664w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry-194x300.png 194w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry-768x1185.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry-300x463.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/manticgeometry.png 829w\" sizes=\"auto, (max-width: 664px) 100vw, 664px\" \/><figcaption class=\"wp-element-caption\"><em>Rule-of-thirds mantic grid<br>in a vertical format<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"365\" height=\"500\" data-attachment-id=\"2387\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/vitruvian\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg\" data-orig-size=\"365,500\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"vitruvian\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg\" alt=\"\" class=\"wp-image-2387\" style=\"width:600px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg 365w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian-219x300.jpg 219w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian-300x411.jpg 300w\" sizes=\"auto, (max-width: 365px) 100vw, 365px\" \/><figcaption class=\"wp-element-caption\"><em>Leonardo da Vinci<br><a href=\"https:\/\/en.wikipedia.org\/wiki\/Vitruvian_Man\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Vitruvian Man<\/strong><\/a><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h3 class=\"wp-block-heading has-text-align-center\"><span class=\"ez-toc-section\" id=\"Structure_geometry\"><\/span><strong><a href=\"http:\/\/en.wikipedia.org\/wiki\/Structure\" target=\"_blank\" rel=\"noreferrer noopener\">Structure<\/a><\/strong><br><strong>geometry<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-text-align-left\"><strong><em>Structure<\/em><\/strong> is an organization of interrelated elements in an object or a system.<\/p>\n\n\n\n<p>Structural geometry controls pattern and hierarchy, making invisible relationships visible, associated with creation through ratio and proportion.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-white-color has-alpha-channel-opacity has-white-background-color has-background\"\/>\n\n\n\n<p>An example will be how we use polygon construction in developing a drawing structure:<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"663\" height=\"1024\" data-attachment-id=\"2388\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/web-2\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-scaled.jpg\" data-orig-size=\"1656,2560\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1466598017&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;Web&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Web\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-663x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-663x1024.jpg\" alt=\"\" class=\"wp-image-2388\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-663x1024.jpg 663w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-194x300.jpg 194w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-768x1187.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-994x1536.jpg 994w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-1325x2048.jpg 1325w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-300x464.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/structuregeometry-scaled.jpg 1656w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><figcaption class=\"wp-element-caption\"><em>Top: hexagon construction<br>Bottom: square construction<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"360\" height=\"500\" data-attachment-id=\"2390\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/cezannepaul\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg\" data-orig-size=\"360,500\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"cezannepaul\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg\" alt=\"\" class=\"wp-image-2390\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg 360w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul-216x300.jpg 216w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul-300x417.jpg 300w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><figcaption class=\"wp-element-caption\"><em>Paul Cezanne<br><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Paul_C%C3%A9zanne,_La_vase_paill%C3%A9,_c._1895.jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>La Vase Paill\u00e9<\/strong><\/a><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h3 class=\"wp-block-heading has-text-align-center\"><span class=\"ez-toc-section\" id=\"Body_geometry\"><\/span><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Physical_body\" target=\"_blank\" rel=\"noreferrer noopener\">Body<\/a>&nbsp;geometry<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>In physics, a <strong><em>body<\/em><\/strong> is a discrete unit of material that is movable in 3-dimensional space. A body is also one&#8217;s physical self.<\/p>\n\n\n\n<p>Body geometry determines tangible volumes, spaces, and surfaces, making material immaterial, through the application of geometric primitives: polygons, polyhedra, cones, cylinders, and spheres.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-text-color has-white-color has-alpha-channel-opacity has-white-background-color has-background\"\/>\n\n\n\n<p>An example will be how we develop shapes and color into meaningful visual elements:<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"663\" height=\"1024\" data-attachment-id=\"2391\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/goldenspiral\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-scaled.jpg\" data-orig-size=\"1656,2560\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1466774816&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;goldenspiral&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"goldenspiral\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-663x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-663x1024.jpg\" alt=\"\" class=\"wp-image-2391\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-663x1024.jpg 663w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-194x300.jpg 194w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-768x1187.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-994x1536.jpg 994w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-1325x2048.jpg 1325w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-300x464.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/bodygeometry-scaled.jpg 1656w\" sizes=\"auto, (max-width: 663px) 100vw, 663px\" \/><figcaption class=\"wp-element-caption\"><em>Color applied to<br>geometric construction<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<p>Frascari formulates these types of geometry to describe and control the image-making potential of visualization. They are not differences in kind, but rather different co-existing and co-reinforcing aspects of geomater, sharing the process of geometric construction and the logic of&nbsp;<a href=\"http:\/\/en.wikipedia.org\/wiki\/Poetics\" target=\"_blank\" rel=\"noreferrer noopener\">poetics<\/a>, in the mechanical sense of an extended visual image or metaphor. Digital modelers may use these concepts of geometry to guide the creation of work.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mantic_geometry_in_creating_a_proportioning_system\"><\/span>Mantic geometry in creating a proportioning system<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Format\"><\/span>Format<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"752\" height=\"1024\" data-attachment-id=\"1416\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/blake_ancient\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient.jpg\" data-orig-size=\"800,1089\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"blake_ancient\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-752x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-752x1024.jpg\" alt=\"\" class=\"wp-image-1416\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-752x1024.jpg 752w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-220x300.jpg 220w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-768x1045.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient-300x408.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/blake_ancient.jpg 800w\" sizes=\"auto, (max-width: 752px) 100vw, 752px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\">\n<p>Frascari&#8217;s <strong>Mantic<\/strong> geometry establishes the format and orientation of a composition. In painting, even the orientation of a canvas presents an authorial choice: horizontal or vertical \u2014  or omnidirectional? This creates what is known as the <strong>format<\/strong>: the basic ratio relationship of height to width of the visual field, or the volume of a mass or void space. While there are unusual formats to be found in the world \u2014 square, diamond, oval, and circles \u2014 the common choices in basic drawing are these:<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:39%\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"664\" height=\"1024\" data-attachment-id=\"2604\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala-of-the-elements\/standard_grid_vert\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert.png\" data-orig-size=\"829,1279\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"standard_grid_vert\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert-664x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert-664x1024.png\" alt=\"\" class=\"wp-image-2604\" style=\"object-fit:contain;width:220px;height:340px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert-664x1024.png 664w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert-194x300.png 194w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert-768x1185.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert-300x463.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid_vert.png 829w\" sizes=\"auto, (max-width: 664px) 100vw, 664px\" \/><figcaption class=\"wp-element-caption\"><em>Vertical format, or <strong>portrait<\/strong><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:61%\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"664\" data-attachment-id=\"2605\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala-of-the-elements\/standard_grid\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid.png\" data-orig-size=\"1279,829\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"standard_grid\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid-1024x664.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid-1024x664.png\" alt=\"\" class=\"wp-image-2605\" style=\"object-fit:contain;width:340px;height:220px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid-1024x664.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid-300x194.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid-768x498.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/standard_grid.png 1279w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Horizontal format, or<strong> landscape<\/strong><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<p>Recall that Bauhaus Master Wassily Kandinsky called the visual field of the paper or canvas the <strong>Basic Plane<\/strong>. He described vertical and horizontal as having an innate character he terms <strong>tonality<\/strong>. He considers the vertical basic plane to be <strong>warm<\/strong> and therefore active, while the horizontal is <strong>cool<\/strong> and therefore passive. So we make our first decision, the format, based on whether we wish to develop an energetic or calming presence in our work. We often see a blank piece of paper or a void space and we freeze, not knowing where to make the first move. Once we realize a blank is not blank, but instead contains innate Mantic clues, it becomes less difficult to create a composition.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Proportion_grids_in_the_format\"><\/span>Proportion grids in the format<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>Any visual field format contains intrinsic geometric relationships that can be expressed by drawing controlling grids or geometries. Just a few of the major types, compared to the ratio of an 11 x 17 piece of paper:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<p class=\"has-text-align-center\"><strong>Dynamic Symmetry Grid<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"664\" data-attachment-id=\"2619\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala-of-the-elements\/dynamic_symmetry\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry.png\" data-orig-size=\"1279,829\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"dynamic_symmetry\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry-1024x664.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry-1024x664.png\" alt=\"\" class=\"wp-image-2619\" style=\"width:340px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry-1024x664.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry-300x194.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry-768x498.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/dynamic_symmetry.png 1279w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>45\u00b0 diagonals from each corner combined with corner-to-corner or perpendiculars to corners<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<p class=\"has-text-align-center\"><strong>Rule of Thirds Grid<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"664\" data-attachment-id=\"2620\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala-of-the-elements\/ruleof3\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3.png\" data-orig-size=\"1279,829\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"ruleof3\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3-1024x664.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3-1024x664.png\" alt=\"\" class=\"wp-image-2620\" style=\"width:340px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3-1024x664.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3-300x194.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3-768x498.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/ruleof3.png 1279w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Common in photography<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<p class=\"has-text-align-center\"><strong>Golden Ratio ( \u0278 )<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"664\" data-attachment-id=\"2621\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala-of-the-elements\/phi\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi.png\" data-orig-size=\"1279,829\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"phi\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi-1024x664.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi-1024x664.png\" alt=\"\" class=\"wp-image-2621\" style=\"width:340px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi-1024x664.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi-300x194.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi-768x498.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/phi.png 1279w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>1:0.61803&#8230; <\/em><br><em>equals the <strong>Golden Ratio<\/strong>,&nbsp;<strong>\u0278<\/strong> (phi).<br><\/em><br><em>The inverse of the Golden Ratio:<\/em><br><em>1\/0.61803&#8230; = 1.61803&#8230;<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<p class=\"has-text-align-center\"><strong>Fibonacci Sequence (&nbsp;<em>Fn<\/em>&nbsp;)<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"664\" data-attachment-id=\"2622\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala-of-the-elements\/fibonacci1\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1.png\" data-orig-size=\"1279,829\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"fibonacci1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1-1024x664.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1-1024x664.png\" alt=\"\" class=\"wp-image-2622\" style=\"width:340px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1-1024x664.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1-300x194.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1-768x498.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/fibonacci1.png 1279w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>F<sub>n<\/sub> = F<sub>n-1<\/sub> + F<sub>n-2&nbsp;<\/sub><\/em><br><em>gives the sequence:<\/em><br><em>0,&nbsp;1, 1, 2, 3, 5, 8, 13, 21, 34,&#8230; \u221e.<\/em><br><em> <\/em><br><em>As&nbsp;F<sub>n <\/sub>rises, ratios approach <strong>\u0278<\/strong>!<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<p>Some smartphone photo apps contain these types of grids, and we see a rule-of-thirds grid in Photoshop&#8217;s Crop tool. A modeled primitive solid divided into faces and edges can be modulated in the manner of these grids. The various geometries help to predict symmetry or asymmetry, stable or dynamic compositions, focal points, and hierarchies. They can be flipped horizontally or vertically, and can even be combined.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"921\" height=\"1024\" data-attachment-id=\"6889\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/ch10_013\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013.jpg\" data-orig-size=\"951,1057\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"ch10_013\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013-921x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013-921x1024.jpg\" alt=\"\" class=\"wp-image-6889\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013-921x1024.jpg 921w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013-270x300.jpg 270w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013-768x854.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013-300x333.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_013.jpg 951w\" sizes=\"auto, (max-width: 921px) 100vw, 921px\" \/><figcaption class=\"wp-element-caption\"><em>Le Corbusier,\u00a0<strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Modulor\" target=\"_blank\" rel=\"noreferrer noopener\">Modulor<\/a><\/strong>,\u00a0circa 1948.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>There is no set recipe for using any proportioning system, but one painter-architect of the 20th century, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Le_Corbusier\" target=\"_blank\" rel=\"noreferrer noopener\">Le Corbusier<\/a>, found a way to reconcile competing units of measure (the English and Metric systems) with human proportion and three-dimensional space. He called this system the<strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Modulor\" target=\"_blank\" rel=\"noreferrer noopener\">  Modulor<\/a><\/strong>. Notice how the values in the system relate to the Golden Ratio \u0278, and how some sequences behave in a Fibonnaci-like manner.<\/p>\n\n\n\n<p>Of Le Corbusier&#8217;s system, Albert Einstein said: &#8220;It is a scale of proportions which makes the bad difficult and the good easy.&#8221; <sup><a href=\"#footnote_4_6843\" id=\"identifier_4_6843\" class=\"footnote-link footnote-identifier-link\" title=\"Quoted by Le Corbusier in &quot;The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics, Volume 2.&quot; Birkh&auml;user. Basel, Switzerland. Reprint 2004. p.58.&nbsp;\">4<\/a><\/sup><\/p>\n<\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Structure_geometry_for_internal_proportioning\"><\/span>Structure geometry for internal proportioning<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"365\" height=\"500\" data-attachment-id=\"2387\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/vitruvian\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg\" data-orig-size=\"365,500\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"vitruvian\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg\" alt=\"\" class=\"wp-image-2387\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian.jpg 365w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian-219x300.jpg 219w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/vitruvian-300x411.jpg 300w\" sizes=\"auto, (max-width: 365px) 100vw, 365px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\">\n<p>The Modulor Man relates the human form to the built environment. As such, it builds a bridge between Frascari&#8217;s <strong>Mantic<\/strong> state of proportioning and his <strong>Structure<\/strong> state of geometry. Where Mantic grids control compositional relationships <em>among<\/em> objects, Structure is a geometry that controls what you see <em>within<\/em> an object.\u00a0The easiest expression of structural geometry to understand is that of an axis of symmetry. Recalling da Vinci&#8217;s Vitruvian Man, we understand there is a line, an axis, that runs down the center of the human form \u2014 it&#8217;s not <em>there<\/em>, but we all see it inferred by left-right correlations.<\/p>\n<\/div>\n<\/div>\n\n\n\n<p>Branching off of that geometric axis, several other lines of geometric proportioning can be derived, as we see in da Vinci&#8217;s sketchbooks:<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1469\" height=\"2000\" data-attachment-id=\"6891\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/912601-tif\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490.jpg\" data-orig-size=\"1469,2000\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1312385820&quot;,&quot;copyright&quot;:&quot;Royal Collection Trust\/\\u00c2\\u00a9 Her Majesty Queen Elizabeth II 2016&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;912601.tif&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"912601.tif\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490-752x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490.jpg\" alt=\"\" class=\"wp-image-6891\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490.jpg 1469w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490-220x300.jpg 220w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490-752x1024.jpg 752w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490-768x1046.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490-1128x1536.jpg 1128w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_RCIN_912601_The_proportions_of_the_head_and_a_standing_nude_c.1490-300x408.jpg 300w\" sizes=\"auto, (max-width: 1469px) 100vw, 1469px\" \/><figcaption class=\"wp-element-caption\"><em>Leonardo da Vinci,\u00a0<a href=\"https:\/\/en.m.wikipedia.org\/wiki\/File:Leonardo_da_Vinci_-_RCIN_912601,_The_proportions_of_the_head,_and_a_standing_nude_c.1490.jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Study of Proportion<\/strong><\/a>,\u00a0silverpoint on paper, circa 1490.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1372\" height=\"975\" data-attachment-id=\"6892\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/leonardo_da_vinci_-_studi_di_proporzioni_del_volto_e_dellocchio_con_note_d-c-_1557415576_recto\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto.png\" data-orig-size=\"1372,975\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto-1024x728.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto.png\" alt=\"\" class=\"wp-image-6892\" style=\"width:799px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto.png 1372w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto-300x213.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto-1024x728.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Leonardo_da_Vinci_-_Studi_di_proporzioni_del_volto_e_dellocchio_con_note_D.C._1557415576_recto-768x546.png 768w\" sizes=\"auto, (max-width: 1372px) 100vw, 1372px\" \/><figcaption class=\"wp-element-caption\"><em>Leonardo da Vinci,\u00a0Study on the<a href=\"https:\/\/www.finestresullarte.info\/en\/works-and-artists\/all-the-drawings-of-leonardo-da-vinci-at-the-royal-library-of-turin\" target=\"_blank\" rel=\"noreferrer noopener\"> <strong>Proportion of Head and Eyes<\/strong><\/a>,\u00a0pen and ink, circa 1500.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<p>Da Vinci&#8217;s method of geometric analysis was also employed by Albrecht D\u00fcrer, whose volumetric analysis presages wireframe modeling, as Structure geometry approaches the condition of Body geometry:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"940\" height=\"1257\" data-attachment-id=\"6895\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/375-figure3-18-1\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1.png\" data-orig-size=\"940,1257\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"375-Figure3.18-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-766x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1.png\" alt=\"\" class=\"wp-image-6895\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1.png 940w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-224x300.png 224w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-766x1024.png 766w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-768x1027.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-300x401.png 300w\" sizes=\"auto, (max-width: 940px) 100vw, 940px\" \/><figcaption class=\"wp-element-caption\"><em>Albrecht D\u00fcrer,<\/em>&nbsp;Stereometric Diagrams and Planar Transformations of the Human Head,&nbsp;<em>circa 1500.<\/em><\/figcaption><\/figure>\n<\/div>\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Body_geometry_and_the_visible_world\"><\/span>Body geometry and the visible world<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"360\" height=\"500\" data-attachment-id=\"2390\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/geometry-color-meaning\/cezannepaul\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg\" data-orig-size=\"360,500\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"cezannepaul\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg\" alt=\"\" class=\"wp-image-2390\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul.jpg 360w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul-216x300.jpg 216w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/cezannepaul-300x417.jpg 300w\" sizes=\"auto, (max-width: 360px) 100vw, 360px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\">\n<p><strong>Mantic<\/strong> geometry predicts a compositional logic and <strong>Structure<\/strong> analyzes found proportion within individual composed elements, but both are <strong>hidden geometries<\/strong> \u2014 a topic we&#8217;ll explore in greater depth. It is Frascari&#8217;s <strong>Body<\/strong> geometry that articulates the things one sees.  We&#8217;ll explore this by returning to C\u00e9zanne and the Primitives, and observing how he works with Body geometry.<\/p>\n<\/div>\n<\/div>\n\n\n\n<p>Here, we&#8217;ll analyze his <em><strong>Still Life With Apples<\/strong><\/em> that we introduced the Primitives with: <\/p>\n\n\n\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"2250\" height=\"1695\" data-attachment-id=\"6851\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/r674\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890.jpg\" data-orig-size=\"2250,1695\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;R674&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"R674\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-1024x771.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890.jpg\" alt=\"\" class=\"wp-image-6851\" style=\"width:600px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890.jpg 2250w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-300x226.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-1024x771.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-768x579.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/Paul_Cezanne_Still_Life_With_Apples_c._1890-1536x1157.jpg 1536w\" sizes=\"auto, (max-width: 2250px) 100vw, 2250px\" \/><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"998\" height=\"751\" data-attachment-id=\"6897\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/ch10_018\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_018.png\" data-orig-size=\"998,751\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"ch10_018\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_018.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_018.png\" alt=\"\" class=\"wp-image-6897\" style=\"width:600px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_018.png 998w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_018-300x226.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/ch10_018-768x578.png 768w\" sizes=\"auto, (max-width: 998px) 100vw, 998px\" \/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<p>Notice it&#8217;s not enough to say &#8220;OK, the apples are round.&#8221; For example, he incorporates fruit based on elliptical solids that create a transition between more spherical fruit and perceived oval states of plate and cup. Rationally, we know the plate and cup to be based on cylindrical and conic primitives, but visually they generate ellipses. Groups of elements, such as the stack of apples, create a larger hierarchic element that is in turn controlled structurally by a tetrahedron. Even without color and contrast, the shapes themselves begin to develop a tension of pairings and groupings controlled by geometry.<\/p>\n\n\n\n<p>Now, we understand that C\u00e9zanne creates an abstraction of these forms, not a highly realistic image. He distills and presents shape and color. When we further isolate shape, we begin to perceive the Primitives, but they are still peeking out from underneath the veneer of color. So we can say that Body geometry, while more apparent to the eye, is still hiding a bit under the surface of the object we see \u2014 an apple, after all, is not perfectly round like a primitive sphere.<\/p>\n\n\n\n<p>So, if all of Frascari&#8217;s three kinds of geometry are hiding to one degree or another, how can we use them in modeling?<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Hidden_geometry\"><\/span>Hidden geometry<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Let&#8217;s expand on those geometries by seeing how we might express them visually. The samples are visual descriptions of such hidden geometries as the <strong>datum<\/strong>, various <strong>axes of symmetry<\/strong>, <strong>orthographic<\/strong> drawing, and systems for drawing irregular, curving forms.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expressing_a_datum\"><\/span>Expressing a datum<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>The <strong>Axis Mundi<\/strong> we learned about above is the first example of a fanciful, metaphorical datum. A datum is an arbitrary but conventionally agreed-upon point of reference, something used as the basis for measure or calculation. Among many others, some historical expressions of a datum include:<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"857\" height=\"1200\" data-attachment-id=\"7522\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/oluf_olufsen_bagge_-_yggdrasil_the_mundane_tree_1847_-_full_page\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page.jpg\" data-orig-size=\"857,1200\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page-731x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page.jpg\" alt=\"\" class=\"wp-image-7522\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page.jpg 857w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page-214x300.jpg 214w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page-731x1024.jpg 731w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page-768x1075.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Oluf_Olufsen_Bagge_-_Yggdrasil_The_Mundane_Tree_1847_-_full_page-300x420.jpg 300w\" sizes=\"auto, (max-width: 857px) 100vw, 857px\" \/><figcaption class=\"wp-element-caption\"><em>The<strong> Axis Mundi<\/strong> expressed as the <strong>World Tree<\/strong>: Oluf Olufsen Bagge, <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Oluf_Olufsen_Bagge_-_Yggdrasil,_The_Mundane_Tree_1847_-_full_page.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Yggdrasil, the World Tree<\/a> <\/strong>of Norse mythology, from<strong> Northern Antiquities<\/strong>, an English translation of the Prose Edda, 1847.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"974\" data-attachment-id=\"7525\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/batterypoint_usgs_marker2\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/BatteryPoint_USGS_Marker2.jpg\" data-orig-size=\"1024,974\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;8&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;DMC-FX12&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1332084948&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;5.8&quot;,&quot;iso&quot;:&quot;100&quot;,&quot;shutter_speed&quot;:&quot;0.005&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"BatteryPoint_USGS_Marker2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/BatteryPoint_USGS_Marker2.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/BatteryPoint_USGS_Marker2.jpg\" alt=\"\" class=\"wp-image-7525\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/BatteryPoint_USGS_Marker2.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/BatteryPoint_USGS_Marker2-300x285.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/BatteryPoint_USGS_Marker2-768x731.jpg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em><strong>Sea level<\/strong> expressing height elevation on Earth: <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:BatteryPoint_USGS_Marker.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">USGS Bench Mark<\/a><\/strong>, installed at Battery Point Light, Crescent City, California, 1915.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1022\" height=\"900\" data-attachment-id=\"7527\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/map-heart-0542\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Map-heart-0542.jpg\" data-orig-size=\"1022,900\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Map-heart-0542\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Map-heart-0542.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Map-heart-0542.jpg\" alt=\"\" class=\"wp-image-7527\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Map-heart-0542.jpg 1022w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Map-heart-0542-300x264.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Map-heart-0542-768x676.jpg 768w\" sizes=\"auto, (max-width: 1022px) 100vw, 1022px\" \/><figcaption class=\"wp-element-caption\"><em>The <strong>Equator<\/strong> and <strong>Poles<\/strong> expressing division of hemispheres and the axis of rotation for Earth:\u00a0Oronce Fine, <strong><a href=\"https:\/\/www.wga.hu\/html_m\/f\/fine\/1map.html\" target=\"_blank\" rel=\"noreferrer noopener\">Mappemonde en forme de\u00a0c\u0153ur montrant la Terre australe.\u00a0Recens et integra orbis descriptio<\/a><\/strong>,\u00a0Paris, 1536.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"899\" height=\"913\" data-attachment-id=\"7529\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/pentagram_and_human_body_agrippa-2\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa.png\" data-orig-size=\"899,913\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Pentagram_and_human_body_Agrippa\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa.png\" alt=\"\" class=\"wp-image-7529\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa.png 899w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa-295x300.png 295w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa-768x780.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Pentagram_and_human_body_Agrippa-300x305.png 300w\" sizes=\"auto, (max-width: 899px) 100vw, 899px\" \/><figcaption class=\"wp-element-caption\"><em>The<strong> center line<\/strong> of a human form expressing biological mirror symmetry:\u00a0Heinrich Cornelius Agrippa, illustration from<strong> <a href=\"https:\/\/en.m.wikipedia.org\/wiki\/File:Pentagram_and_human_body_(Agrippa).jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Libri tres de occulta philosophia<\/a><\/strong>, 1533.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"800\" height=\"1000\" data-attachment-id=\"1421\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/800px-fulton_sections\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections.jpg\" data-orig-size=\"800,1000\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"800px-Fulton_Sections\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections.jpg\" alt=\"\" class=\"wp-image-1421\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections.jpg 800w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections-240x300.jpg 240w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections-768x960.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Fulton_Sections-300x375.jpg 300w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><figcaption class=\"wp-element-caption\"><em>The<strong> water line<\/strong> expressing the line of displacement of water by a vessel:\u00a0Robert Fulton drawings for <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Fulton_Sections.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">USS Demilogos<\/a><\/strong>, the first\u00a0steam-powered warship, later renamed\u00a0after its designer to<strong> USS Fulton<\/strong>, 1813.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expressing_axes_of_symmetry\"><\/span>Expressing axes of symmetry<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>One datum in particular that is often hidden but important to express is the <strong>axis of symmetry<\/strong>.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Bilateral_symmetry\"><\/span>Bilateral symmetry<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>As we see da Vinci&#8217;s drawing of the <strong>Vitruvian Man<\/strong> above, humans along with many other living beings are biologically organized along an axis of bilateral symmetry. Many machines and architectural constructions also harbor bilateral axes of symmetry.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1575\" height=\"2560\" data-attachment-id=\"1424\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/v0001948-titlepage-of-robert-fludds-utriusque-cosmi-maioris-scilice\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Titlepage_of_Robert_Fludds__Utriusque_cosmi_maioris_scilice_Wellcome_V0001948-scaled.jpg\" data-orig-size=\"1575,2560\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;Wellcome Library, London&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;V0001948 Titlepage of Robert Fludd&#039;s \\&quot;Utriusque cosmi maioris scilice\\nCredit: Wellcome Library, London. Wellcome Images\\nimages@wellcome.ac.uk\\nhttp:\/\/wellcomeimages.org\\nTitlepage of Robert Fludd&#039;s \\&quot;Utriusque cosmi maioris scilicet et minoris ... historia\\&quot;. Line engraving by T. de Bry, 1617.\\n1617 after: Theodore de BryPublished:  - \\n\\nCopyrighted work available under Creative Commons Attribution only licence CC BY 4.0 http:\/\/creativecommons.org\/licenses\/by\/4.0\/&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;Copyrighted work available under Creative Commons Attribution only licence CC BY 4.0 http:\/\/creativecommons.org\/licenses\/by\/4.0\/&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;V0001948 Titlepage of Robert Fludd&#039;s \\&quot;Utriusque cosmi maioris scilice&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"V0001948 Titlepage of Robert Fludd&amp;#8217;s &amp;#8220;Utriusque cosmi maioris scilice\" data-image-description=\"\" data-image-caption=\"&lt;p&gt;V0001948 Titlepage of Robert Fludd&amp;#8217;s &amp;#8220;Utriusque cosmi maioris scilice&lt;br \/&gt;\nCredit: Wellcome Library, London. Wellcome Images&lt;br \/&gt;\nimages@wellcome.ac.uk&lt;br \/&gt;\nhttp:\/\/wellcomeimages.org&lt;br \/&gt;\nTitlepage of Robert Fludd&amp;#8217;s &amp;#8220;Utriusque cosmi maioris scilicet et minoris &amp;#8230; historia&amp;#8221;. Line engraving by T. de Bry, 1617.&lt;br \/&gt;\n1617 after: Theodore de BryPublished:  &amp;#8211; &lt;\/p&gt;\n&lt;p&gt;Copyrighted work available under Creative Commons Attribution only licence CC BY 4.0 http:\/\/creativecommons.org\/licenses\/by\/4.0\/&lt;\/p&gt;\n\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Titlepage_of_Robert_Fludds__Utriusque_cosmi_maioris_scilice_Wellcome_V0001948-630x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Titlepage_of_Robert_Fludds__Utriusque_cosmi_maioris_scilice_Wellcome_V0001948-scaled.jpg\" alt=\"\" class=\"wp-image-1424\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Titlepage_of_Robert_Fludds__Utriusque_cosmi_maioris_scilice_Wellcome_V0001948-scaled.jpg 1575w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Titlepage_of_Robert_Fludds__Utriusque_cosmi_maioris_scilice_Wellcome_V0001948-185x300.jpg 185w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Titlepage_of_Robert_Fludds__Utriusque_cosmi_maioris_scilice_Wellcome_V0001948-630x1024.jpg 630w\" sizes=\"auto, (max-width: 1575px) 100vw, 1575px\" \/><figcaption class=\"wp-element-caption\"><em>T. de Bry, line engraving frontispiece featuring symmetrical human similar to Agrippa and da Vinci, <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Titlepage_of_Robert_Fludd's_%22Utriusque_cosmi_maioris_scilice_Wellcome_V0001948.jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Utriusque cosmi maioris scilicet et minoris &#8230; <\/strong><\/a>history\u00a0by Robert Fludd, 1617.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"654\" height=\"863\" data-attachment-id=\"1425\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/albrecht_durer_-_studies_on_the_proportions_of_the_female_body\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body.jpg\" data-orig-size=\"654,863\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body.jpg\" alt=\"\" class=\"wp-image-1425\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body.jpg 654w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body-227x300.jpg 227w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Albrecht_Durer_-_Studies_on_the_Proportions_of_the_Female_Body-300x396.jpg 300w\" sizes=\"auto, (max-width: 654px) 100vw, 654px\" \/><figcaption class=\"wp-element-caption\"><em>Albrecht D\u00fcrer, <a href=\"https:\/\/en.m.wikipedia.org\/wiki\/File:Albrecht_D%C3%BCrer_-_Studies_on_the_Proportions_of_the_Female_Body_-_WGA07263.jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Studies on the Proportions of the Female Body<\/strong><\/a>, 1528.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"654\" height=\"863\" data-attachment-id=\"1426\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/anatomical_and_geometrical_proportions_-_albrecht_durer\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Anatomical_and_geometrical_proportions_-_Albrecht_Durer.png\" data-orig-size=\"654,863\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Anatomical_and_geometrical_proportions_-_Albrecht_Durer\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Anatomical_and_geometrical_proportions_-_Albrecht_Durer.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Anatomical_and_geometrical_proportions_-_Albrecht_Durer.png\" alt=\"\" class=\"wp-image-1426\" style=\"width:798px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Anatomical_and_geometrical_proportions_-_Albrecht_Durer.png 654w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Anatomical_and_geometrical_proportions_-_Albrecht_Durer-227x300.png 227w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Anatomical_and_geometrical_proportions_-_Albrecht_Durer-300x396.png 300w\" sizes=\"auto, (max-width: 654px) 100vw, 654px\" \/><figcaption class=\"wp-element-caption\">Albrecht D\u00fcrer, <em><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Anatomical_and_geometrical_proportions_-_Albrecht_D%C3%BCrer.png\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Anatomical and Geometrical Proportions<\/strong><\/a><\/em>, ca 1528.<\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1000\" height=\"1203\" data-attachment-id=\"7541\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/modelt-2\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt.png\" data-orig-size=\"1000,1203\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"modelt\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt-851x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt.png\" alt=\"\" class=\"wp-image-7541\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt.png 1000w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt-249x300.png 249w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt-851x1024.png 851w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt-768x924.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/modelt-300x361.png 300w\" sizes=\"auto, (max-width: 1000px) 100vw, 1000px\" \/><figcaption class=\"wp-element-caption\"><em>Henry Ford, front view of\u00a0<strong><a href=\"https:\/\/drawingdatabase.com\/ford-model-t\/\" target=\"_blank\" rel=\"noreferrer noopener\">Ford Model T<\/a><\/strong>, 1928.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"890\" height=\"600\" data-attachment-id=\"1428\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/villa_rotonda_front\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Villa_Rotonda_front.jpg\" data-orig-size=\"890,600\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Villa_Rotonda_front\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Villa_Rotonda_front.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Villa_Rotonda_front.jpg\" alt=\"\" class=\"wp-image-1428\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Villa_Rotonda_front.jpg 890w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Villa_Rotonda_front-300x202.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Villa_Rotonda_front-768x518.jpg 768w\" sizes=\"auto, (max-width: 890px) 100vw, 890px\" \/><figcaption class=\"wp-element-caption\"><em>Andrea Palladio,<strong>\u00a0<a href=\"https:\/\/en.m.wikipedia.org\/wiki\/File:Villa_Rotonda_front.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Villa Capra<\/a><\/strong>, also known as <strong>La Rotonda<\/strong>,\u00a01570.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Radial_symmetry\"><\/span>Radial symmetry<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>In the Villa Rotonda facade we see bilateral symmetry. In its floor plan, we see <strong>axes<\/strong> of symmetry in multiple directions, creating the condition of <strong>radial symmetry<\/strong>. Many types exist:<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1280\" height=\"899\" data-attachment-id=\"7547\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/andrea_palladio_his_life_and_works_1902_14577899720\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720.jpg\" data-orig-size=\"1280,899\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Andrea_Palladio_his_life_and_works_1902_14577899720\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720-1024x719.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720.jpg\" alt=\"\" class=\"wp-image-7547\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720.jpg 1280w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720-300x211.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720-1024x719.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Andrea_Palladio_his_life_and_works_1902_14577899720-768x539.jpg 768w\" sizes=\"auto, (max-width: 1280px) 100vw, 1280px\" \/><figcaption class=\"wp-element-caption\">Andrea Palladio, plan showing 4-fold symmetry for <em><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Andrea_Palladio,_his_life_and_works_(1902)_(14577899720).jpg\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>La Rotonda<\/strong><\/a><\/em>,\u00a01570.<\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"864\" height=\"624\" data-attachment-id=\"1431\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/starfish_1920s\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/starfish_1920s.jpg\" data-orig-size=\"864,624\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"starfish_1920s\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/starfish_1920s.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/starfish_1920s.jpg\" alt=\"\" class=\"wp-image-1431\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/starfish_1920s.jpg 864w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/starfish_1920s-300x217.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/starfish_1920s-768x555.jpg 768w\" sizes=\"auto, (max-width: 864px) 100vw, 864px\" \/><figcaption class=\"wp-element-caption\"><em>Walter Coleman, illustration for starfish showing both bilateral symmetry and 5-fold radial symmetry, for <a href=\"https:\/\/archive.org\/details\/beginnerszoowest00cole\/page\/34\/mode\/2up\"><strong>Beginner&#8217;s Zoology<\/strong><\/a>, 1922.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"491\" height=\"449\" data-attachment-id=\"1432\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/bentley_snowflake20\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Bentley_Snowflake20.jpg\" data-orig-size=\"491,449\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Bentley_Snowflake20\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Bentley_Snowflake20.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Bentley_Snowflake20.jpg\" alt=\"\" class=\"wp-image-1432\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Bentley_Snowflake20.jpg 491w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Bentley_Snowflake20-300x274.jpg 300w\" sizes=\"auto, (max-width: 491px) 100vw, 491px\" \/><figcaption class=\"wp-element-caption\"><em>6-fold radial symmetry in William &#8220;Snowflake&#8221; Bentley&#8217;s <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Bentley_Snowflake20.jpg\"><strong>Snow Flake No. 20<\/strong><\/a>, ca. late 19th or early 20th century.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"702\" height=\"600\" data-attachment-id=\"1433\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/702px-7_cactus\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/702px-7_cactus.jpg\" data-orig-size=\"702,600\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"702px-7_cactus\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/702px-7_cactus.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/702px-7_cactus.jpg\" alt=\"\" class=\"wp-image-1433\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/702px-7_cactus.jpg 702w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/702px-7_cactus-300x256.jpg 300w\" sizes=\"auto, (max-width: 702px) 100vw, 702px\" \/><figcaption class=\"wp-element-caption\"><em><strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Get_the_point_(301733819).jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Cactus<\/a><\/strong> with 7-fold symmetry.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"768\" data-attachment-id=\"1434\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/1024px-roof_hafez_tomb\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/1024px-Roof_hafez_tomb.jpg\" data-orig-size=\"1024,768\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"1024px-Roof_hafez_tomb\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/1024px-Roof_hafez_tomb.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/1024px-Roof_hafez_tomb.jpg\" alt=\"\" class=\"wp-image-1434\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/1024px-Roof_hafez_tomb.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/1024px-Roof_hafez_tomb-300x225.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/1024px-Roof_hafez_tomb-768x576.jpg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>16-fold symmetry in the domed ceiling of Persian poet <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Roof_hafez_tomb.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Hafez&#8217;s Tomb<\/a><\/strong>, 1935.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Self-similarity_symmetry_in_scale\"><\/span>Self-similarity: symmetry in scale<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>When a part is similar to its whole, such an object exhibits<strong> self-similarity<\/strong>. It is a scalar complement to symmetry.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"560\" height=\"202\" data-attachment-id=\"1436\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/van_ecyk\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/van_ecyk.jpg\" data-orig-size=\"560,202\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"van_ecyk\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/van_ecyk.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/van_ecyk.jpg\" alt=\"\" class=\"wp-image-1436\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/van_ecyk.jpg 560w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/van_ecyk-300x108.jpg 300w\" sizes=\"auto, (max-width: 560px) 100vw, 560px\" \/><figcaption class=\"wp-element-caption\"><em><a href=\"https:\/\/web-app.usc.edu\/soc\/syllabus\/20173\/11230.pdf\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Diagram<\/strong><\/a> by architect Aldo van Eyck, stating &#8220;tree is leaf and leaf is tree &#8211; house is city and city is house,&#8221; 1962.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"800\" height=\"500\" data-attachment-id=\"1437\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/kochsnowgif16_800x500\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/KochSnowGif16_800x500.gif\" data-orig-size=\"800,500\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"KochSnowGif16_800x500\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/KochSnowGif16_800x500.gif\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/KochSnowGif16_800x500.gif\" alt=\"\" class=\"wp-image-1437\"\/><figcaption class=\"wp-element-caption\"><em><strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/File:KochSnowGif16_800x500.gif\" target=\"_blank\" rel=\"noreferrer noopener\">Koch snowflake fractal<\/a><\/strong>, work by Wikimedia contributor Leofun01.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"602\" height=\"1024\" data-attachment-id=\"1438\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/20170623165214barnsley_fern\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern.png\" data-orig-size=\"720,1224\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"20170623165214Barnsley_fern\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern-602x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern-602x1024.png\" alt=\"\" class=\"wp-image-1438\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern-602x1024.png 602w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern-176x300.png 176w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern-300x510.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/20170623165214Barnsley_fern.png 720w\" sizes=\"auto, (max-width: 602px) 100vw, 602px\" \/><figcaption class=\"wp-element-caption\"><em><strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Barnsley_fern.png\">Barnsley fern<\/a><\/strong> \u2014 not a real fern, but an expression of a fractal algorithm by Wikimedia contributor Surt91.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E2%80%9CNear%E2%80%9D_symmetry\"><\/span>&#8220;Near&#8221; symmetry<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>Certain objects display a tendency toward symmetry. We may even recognize an axis, although it (or the object controlled by it) exhibits distortion.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"726\" height=\"560\" data-attachment-id=\"1440\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/hannah_cohoon_tree_of_life_or_blazaing_tree_1845\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hannah_Cohoon_Tree_of_Life_or_Blazaing_Tree_1845.jpg\" data-orig-size=\"726,560\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Hannah_Cohoon_Tree_of_Life_or_Blazaing_Tree_1845\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hannah_Cohoon_Tree_of_Life_or_Blazaing_Tree_1845.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hannah_Cohoon_Tree_of_Life_or_Blazaing_Tree_1845.jpg\" alt=\"\" class=\"wp-image-1440\" style=\"width:797px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hannah_Cohoon_Tree_of_Life_or_Blazaing_Tree_1845.jpg 726w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hannah_Cohoon_Tree_of_Life_or_Blazaing_Tree_1845-300x231.jpg 300w\" sizes=\"auto, (max-width: 726px) 100vw, 726px\" \/><figcaption class=\"wp-element-caption\"><em>Hannah Cohoon, <strong><a href=\"https:\/\/en.m.wikipedia.org\/wiki\/File:Hannah_Cohoon,_Tree_of_Life_or_Blazaing_Tree,_1845.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Tree of Life<\/a><\/strong>, 1845, displays more symmetry as a drawing than an actual tree, but see if you can spot the anomalies.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"914\" height=\"1024\" data-attachment-id=\"7559\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/shoeprint_comparison\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison.jpg\" data-orig-size=\"914,1024\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Shoeprint_comparison\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison.jpg\" alt=\"\" class=\"wp-image-7559\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison.jpg 914w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison-268x300.jpg 268w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison-768x860.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/Shoeprint_comparison-300x336.jpg 300w\" sizes=\"auto, (max-width: 914px) 100vw, 914px\" \/><figcaption class=\"wp-element-caption\"><em>A forensic study of a <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Shoeprint_comparison.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">shoe print<\/a><\/strong> with an S-shape axial distortion.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"510\" height=\"599\" data-attachment-id=\"1442\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/01_16_x-ray_of_hand\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/01_16_X-ray_of_Hand.jpg\" data-orig-size=\"510,599\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"01_16_X-ray_of_Hand\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/01_16_X-ray_of_Hand.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/01_16_X-ray_of_Hand.jpg\" alt=\"\" class=\"wp-image-1442\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/01_16_X-ray_of_Hand.jpg 510w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/01_16_X-ray_of_Hand-255x300.jpg 255w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/01_16_X-ray_of_Hand-300x352.jpg 300w\" sizes=\"auto, (max-width: 510px) 100vw, 510px\" \/><figcaption class=\"wp-element-caption\"><em><strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:01_16_X-ray_of_Hand.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">An x-ray of a hand<\/a><\/strong> exhibits a symmetry of number, but not of form.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expressing_coordinate_systems\"><\/span>Expressing coordinate systems<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Orthographic_and_3-view_drawings\"><\/span>Orthographic and 3-view drawings<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>The orthographic 3-view drawing remains an important tool to visualize coordinate systems, even in a post-mechanical-drafting workflow. We&#8217;ll drill much deeper into orthographic projection in another chapter. <\/p>\n\n\n\n<p>As a practical tool, it verifies that information seen in one coordinate view correlates with information seen in others, guaranteeing the precision of data.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"731\" data-attachment-id=\"7563\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/alfaromeo\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo.webp\" data-orig-size=\"1588,1133\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"alfaromeo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo-1024x731.webp\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo-1024x731.webp\" alt=\"\" class=\"wp-image-7563\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo-1024x731.webp 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo-300x214.webp 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo-768x548.webp 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo-1536x1096.webp 1536w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/alfaromeo.webp 1588w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Scuderia Ferrari <strong><a href=\"https:\/\/www.etsy.com\/listing\/1175402180\/original-alfa-romeo-16c-bimotore-1935\" target=\"_blank\" rel=\"noreferrer noopener\">Bimotore<\/a><\/strong> racing car, 1935. What is the <strong>datum<\/strong> or <strong>data<\/strong> in this drawing? Can you relate these drawings to one another? These are in a proper 3-view relationship.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1363\" height=\"1706\" data-attachment-id=\"7565\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/spads7\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7.png\" data-orig-size=\"1363,1706\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"spads7\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7-818x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7.png\" alt=\"\" class=\"wp-image-7565\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7.png 1363w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7-240x300.png 240w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7-818x1024.png 818w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7-768x961.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7-1227x1536.png 1227w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/spads7-300x375.png 300w\" sizes=\"auto, (max-width: 1363px) 100vw, 1363px\" \/><figcaption class=\"wp-element-caption\"><em>J. D. Carrick or F. Yeoman, <strong>SPAD S.7<\/strong>, illustration in <strong><a href=\"https:\/\/archive.org\/details\/fighteraircrafto0000unse\/page\/104\/mode\/2up?view=theater\" target=\"_blank\" rel=\"noreferrer noopener\">Fighter Aircraft of the 1914-1918 War<\/a><\/strong> compiled by W. M. Lamberton, published by Harleyford Publications Ltd., 1960. What is the <strong>datum<\/strong> or <strong>data<\/strong>? Can you find <strong>centerlines<\/strong>? <strong>Sections<\/strong>? Only two drawings are in the correct 3-view relationship \u2014 which ones<\/em>?<\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"846\" height=\"1024\" data-attachment-id=\"1446\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/mitsubishi_a6m_zero_drawing-01\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01.png\" data-orig-size=\"3918,4744\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Mitsubishi_A6M_Zero_drawing-01\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-846x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-846x1024.png\" alt=\"\" class=\"wp-image-1446\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-846x1024.png 846w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-248x300.png 248w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-768x930.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-1269x1536.png 1269w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-1691x2048.png 1691w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Mitsubishi_A6M_Zero_drawing-01-300x363.png 300w\" sizes=\"auto, (max-width: 846px) 100vw, 846px\" \/><figcaption class=\"wp-element-caption\"><em><strong>3<\/strong>\u00a0 |\u00a0 Peter Kaboldy, <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Mitsubishi_A6M_Zero_drawing.svg\" target=\"_blank\" rel=\"noreferrer noopener\">Mitsubishi A6M2 Zero<\/a><\/strong>\u00a0WWII fighter plane. This is a classic, correct <strong>3-view<\/strong>.<\/em><\/figcaption><\/figure>\n<\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"800\" height=\"483\" data-attachment-id=\"1447\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/800px-durer_foot\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Durer_foot.jpg\" data-orig-size=\"800,483\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"800px-Durer_foot\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Durer_foot.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Durer_foot.jpg\" alt=\"\" class=\"wp-image-1447\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Durer_foot.jpg 800w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Durer_foot-300x181.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/800px-Durer_foot-768x464.jpg 768w\" sizes=\"auto, (max-width: 800px) 100vw, 800px\" \/><figcaption class=\"wp-element-caption\"><em>Albrecht D\u00fcrer,<strong> <a href=\"https:\/\/en.m.wikipedia.org\/wiki\/File:Durer_foot.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Orthographic projection of a Foot<\/a><\/strong>, 1528. Another classically correct <strong>3-view<\/strong>, including a pair of <strong>sections<\/strong>.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Section_drawings\"><\/span>Section drawings<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<p>If orthographic drawings show us the surface, sections oriented to particular coordinate planes will reveal the interior of the object: the topological division between the mass of the object and the void that surrounds it. Let&#8217;s explore D\u00fcrer&#8217;s head analysis visualizing formal slices, along with other examples.<\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"940\" height=\"1257\" data-attachment-id=\"6895\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/375-figure3-18-1\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1.png\" data-orig-size=\"940,1257\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"375-Figure3.18-1\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-766x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1.png\" alt=\"\" class=\"wp-image-6895\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1.png 940w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-224x300.png 224w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-766x1024.png 766w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-768x1027.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/02\/375-Figure3.18-1-300x401.png 300w\" sizes=\"auto, (max-width: 940px) 100vw, 940px\" \/><figcaption class=\"wp-element-caption\"><em>Albrecht D\u00fcrer, <a href=\"https:\/\/www.semanticscholar.org\/paper\/Learning-to-draw%2C-drawing-to-learn%3A-Theory-and-in-Greist\/dd62d0b8225dda1f8fb40bcd7f0ed20b0bb18235\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>Sections through the human head<\/strong><\/a>, ca. 1520s. Showing the relationship between <strong>orthogonal drawing<\/strong> and spatial form.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"630\" height=\"484\" data-attachment-id=\"1451\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/cross-contour-line-drawings\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/cross-contour-line-drawings.jpg\" data-orig-size=\"630,484\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"cross-contour-line-drawings\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/cross-contour-line-drawings.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/cross-contour-line-drawings.jpg\" alt=\"\" class=\"wp-image-1451\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/cross-contour-line-drawings.jpg 630w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/cross-contour-line-drawings-300x230.jpg 300w\" sizes=\"auto, (max-width: 630px) 100vw, 630px\" \/><figcaption class=\"wp-element-caption\"><em>Anonymous student, <strong><a href=\"https:\/\/www.studentartguide.com\/articles\/line-drawings\" target=\"_blank\" rel=\"noreferrer noopener\">cross contour drawing<\/a><\/strong>, date unknown. <strong>Cross contour<\/strong> is a kind of analytical drawing that suggests multiple slicing planes through a mass.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"757\" data-attachment-id=\"7572\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/2018_shoe_final_model\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-scaled.jpg\" data-orig-size=\"2560,1892\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;2.2&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;iPhone 6&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;1538290829&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;4.15&quot;,&quot;iso&quot;:&quot;320&quot;,&quot;shutter_speed&quot;:&quot;0.066666666666667&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"2018_shoe_final_model\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-1024x757.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-1024x757.jpg\" alt=\"\" class=\"wp-image-7572\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-1024x757.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-300x222.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-768x568.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2024\/03\/2018_shoe_final_model-1536x1135.jpg 1536w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Anonymous student,\u00a0<strong>stacked-slice model of a shoe\u00a0<\/strong>using laser-cut corrugated cardboard sectioning. Where the drawing at 2 is sectioned horizontally, this model is sectioned at a 30\u02da tilt. From a studio by the author.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"735\" height=\"599\" data-attachment-id=\"1450\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/735px-pantheon-drawing\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/735px-Pantheon.drawing.jpg\" data-orig-size=\"735,599\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"735px-Pantheon.drawing\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/735px-Pantheon.drawing.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/735px-Pantheon.drawing.jpg\" alt=\"\" class=\"wp-image-1450\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/735px-Pantheon.drawing.jpg 735w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/735px-Pantheon.drawing-300x244.jpg 300w\" sizes=\"auto, (max-width: 735px) 100vw, 735px\" \/><figcaption class=\"wp-element-caption\"><em><strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Pantheon.drawing.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Pantheon, Rome<\/a><\/strong>, cross-section.\u00a0Wilhelm L\u00fcbke, Max Semrau: <strong>Outlines of the History of Art<\/strong>.\u00a0Paul Neff Verlag, Esslingen, 14th edition, 1908. The section in gray contains a rich rendering of the interior <strong>elevation<\/strong>.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"600\" data-attachment-id=\"1452\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/600px-prr_rail_section_drawing\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing.png\" data-orig-size=\"600,600\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"600px-PRR_rail_section_drawing\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing.png\" alt=\"\" class=\"wp-image-1452\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing.png 600w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing-300x300.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing-150x150.png 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/600px-PRR_rail_section_drawing-100x100.png 100w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><figcaption class=\"wp-element-caption\"><em>Frederick Newton\u00a0Willson, <strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:PRR_rail_section_drawing.jpg\" target=\"_blank\" rel=\"noreferrer noopener\">Cross-Section of Standard Rail<\/a><\/strong>, one hundred pounds to the yard, as used on the Pennsylvania Railroad, 1898. The material is extruded, much like an extrusion operation can create form in a model.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Generating_irregular_form\"><\/span>Generating irregular form<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Topographic_lines\"><\/span>Topographic lines<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>A&nbsp;<strong>topographic map<\/strong> uses a stack of horizontal sections: a regular division of elevation to describe the height of land above sea level. Think of the topographic levels as slicing planes parallel to the plane of sea level. This modulates an irregular surface, making it easy (ok, <strong>easier<\/strong>) to describe. Here are three topographic descriptions of the island of Hawaii for comparison, the bottom one interactive.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"413\" height=\"413\" data-attachment-id=\"1454\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/hawaii_simple_topo\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo.png\" data-orig-size=\"413,413\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"hawaii_simple_topo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo.png\" alt=\"\" class=\"wp-image-1454\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo.png 413w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo-300x300.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo-150x150.png 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/hawaii_simple_topo-100x100.png 100w\" sizes=\"auto, (max-width: 413px) 100vw, 413px\" \/><figcaption class=\"wp-element-caption\"><em>Topographic map of <strong><a href=\"http:\/\/facstaff.gpc.edu\/~pgore\/images\/hawaiitopo.gif\" target=\"_blank\" rel=\"noreferrer noopener\">Hawaii<\/a><\/strong><\/em><\/figcaption><\/figure>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-column is-vertically-aligned-top is-layout-flow wp-block-column-is-layout-flow\"><div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"844\" data-attachment-id=\"1455\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/hawaii_island_topographic_map-en-svg\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hawaii_Island_topographic_map-en.svg.png\" data-orig-size=\"1024,844\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;0&quot;}\" data-image-title=\"Hawaii_Island_topographic_map-en.svg\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hawaii_Island_topographic_map-en.svg.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hawaii_Island_topographic_map-en.svg.png\" alt=\"\" class=\"wp-image-1455\" style=\"width:798px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hawaii_Island_topographic_map-en.svg.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hawaii_Island_topographic_map-en.svg-300x247.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/Hawaii_Island_topographic_map-en.svg-768x633.png 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><em>Topographic map with <a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Hawaii_Island_topographic_map-fr.svg\"><strong>color gradient<\/strong><\/a><\/em><\/figcaption><\/figure>\n<\/div>\n\n\n<iframe src=\"https:\/\/williamcromar.com\/code_samples\/Hawaii_Island_topographic_map_SMIL.svg.html\" style=\"border:none; width:100%; height:400px;\" scrolling=\"no\"><\/iframe>\n\n\n\n<p><em><strong><a href=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/2\/29\/Hawaii_Island_topographic_map_SMIL.svg\" target=\"_blank\" rel=\"noreferrer noopener\">Animated SVG<\/a><\/strong>: drag or tap to turn<\/em><\/p>\n<\/div>\n<\/div>\n\n\n\n<h4 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Boat_lofting\"><\/span>Boat lofting<span class=\"ez-toc-section-end\"><\/span><\/h4>\n\n\n\n<p>Like topographic lines, the drafting technique of lofting can generate complex curving forms. We use <strong>lofting<\/strong> in boat building to describe the streamlined shape of the hull. Usually, we divide a profile into 10 or so equal parts, creating a series of parallel sections that describe the shape of the hull. Because the boat is bilaterally symmetrical from the front coordinate orientation, you can see how the drawing is &#8220;halved&#8221; to describe the rear, then the front, of the boat. The deck plan is also a half-drawing, cut at the centerline as a datum.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1162\" height=\"751\" data-attachment-id=\"1456\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/bjk-lines\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines.jpg\" data-orig-size=\"1162,751\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"bjk-lines\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines-1024x662.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines.jpg\" alt=\"\" class=\"wp-image-1456\" style=\"width:800px\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines.jpg 1162w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines-300x194.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines-1024x662.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines-768x496.jpg 768w\" sizes=\"auto, (max-width: 1162px) 100vw, 1162px\" \/><figcaption class=\"wp-element-caption\"><em><strong><a href=\"https:\/\/commons.wikimedia.org\/wiki\/File:Blanchard_Jr._Knockabout,_Hull_No._7,_Lines_-_Blanchard_Junior_Knockabout,_Hull_No._7,_The_Center_for_Wooden_Boats,_Seattle,_King_County,_WA_HAER_WA-181_(sheet_1_of_2).png\" target=\"_blank\" rel=\"noreferrer noopener\">Blanchard Junior Knockabout<\/a><\/strong>, Historic American Engineering Record, Library of Congress.<\/em><\/figcaption><\/figure>\n<\/div><ol class=\"footnotes\"><li id=\"footnote_1_6843\" class=\"footnote\">Loran, Erle. &#8220;C\u00e9zanne&#8217;s Composition.&#8221; University of California Press. 2006. p. 9.<span class=\"footnote-back-link-wrapper\">[<a href=\"#identifier_1_6843\" class=\"footnote-link footnote-back-link\">&#8617;<\/a>]<\/span><\/li><li id=\"footnote_2_6843\" class=\"footnote\">Bramann, Jorn K. &#8220;Descartes: The Solitary Self.&#8221; Philosophical Forum at Frostburg State University. 2004. http:\/\/faculty.frostburg.edu\/phil\/forum\/Descartes.htm<span class=\"footnote-back-link-wrapper\">[<a href=\"#identifier_2_6843\" class=\"footnote-link footnote-back-link\">&#8617;<\/a>]<\/span><\/li><li id=\"footnote_3_6843\" class=\"footnote\">Joyce, James. Finnegans Wake. Penguin Books. 1999. pp. 296-7.<span class=\"footnote-back-link-wrapper\">[<a href=\"#identifier_3_6843\" class=\"footnote-link footnote-back-link\">&#8617;<\/a>]<\/span><\/li><li id=\"footnote_4_6843\" class=\"footnote\">Quoted by Le Corbusier in &#8220;The Modulor: A Harmonious Measure to the Human Scale, Universally Applicable to Architecture and Mechanics, Volume 2.&#8221; Birkh\u00e4user. Basel, Switzerland. Reprint 2004. p.58.&nbsp;<span class=\"footnote-back-link-wrapper\">[<a href=\"#identifier_4_6843\" class=\"footnote-link footnote-back-link\">&#8617;<\/a>]<\/span><\/li><\/ol>","protected":false},"excerpt":{"rendered":"<p>18-minute read The Idea of a Model: Forms and Primitives Picasso\u2019s &#8220;art is the lie that makes us realize the truth&#8221; statement is based on a fairly ancient idea. Making art, we have argued elsewhere, is fundamentally an act of modeling. Merriam-Webster&nbsp;defines modeling as an act of designing or imitating forms, an act of shaping,&#8230; <a href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/\">read more &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_eb_attr":"","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[8],"tags":[],"class_list":["post-6843","post","type-post","status-publish","format-standard","hentry","category-paralleluniverses"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>CHAPTER 10 | Expressions of Geometry - newMediaWiki<\/title>\n<meta name=\"description\" content=\"Chapter 10 Expressions of Geometry: From Forms through Primitives to Cartesian coordinates, an exploration of geometries hidden &amp; visible.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/williamcromar.com\/newmediawiki\/chapter-10-expressions-of-geometry\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"CHAPTER 10 | Expressions of Geometry - 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