{"id":1414,"date":"2023-07-28T10:23:19","date_gmt":"2023-07-28T10:23:19","guid":{"rendered":"https:\/\/williamcromar.com\/newmediawiki\/?p=1414"},"modified":"2024-03-16T23:27:55","modified_gmt":"2024-03-16T23:27:55","slug":"hidden-geometry","status":"publish","type":"post","link":"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/","title":{"rendered":"Hidden geometry"},"content":{"rendered":"\n<div class=\"wp-block-columns has-white-background-color has-background 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:25%\">\n<p class=\"has-text-align-center has-small-font-size\"><br><br><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:75%\">\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:33.33%\"><div class=\"wp-block-image is-style-rounded\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/fabbing\/\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" data-attachment-id=\"20\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/arthead_loopsm\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/arthead_loopsm.gif\" data-orig-size=\"300,300\" 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;0&quot;}\" data-image-title=\"arthead_loopsm\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/arthead_loopsm.gif\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/arthead_loopsm.gif\" alt=\"\" class=\"wp-image-20\" style=\"aspect-ratio:1;object-fit:cover;width:120px;height:undefinedpx\"\/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-center has-small-font-size\"><strong><a href=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/fabbing\/\">Fabbing<\/a><\/strong><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:33.33%\"><div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/williamcromar.com\/newmediawiki\/2d-3d-line-to-plane\/\"><img loading=\"lazy\" decoding=\"async\" width=\"579\" height=\"579\" data-attachment-id=\"331\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/fabbing\/linetoplane\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane.jpg\" data-orig-size=\"579,579\" 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;1282627376&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=\"lineToPlane\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane.jpg\" alt=\"\" class=\"wp-image-331\" style=\"aspect-ratio:1;object-fit:cover;width:120px;height:undefinedpx\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane.jpg 579w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane-300x300.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane-150x150.jpg 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/lineToPlane-100x100.jpg 100w\" sizes=\"auto, (max-width: 579px) 100vw, 579px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-center has-small-font-size\"><strong><a href=\"https:\/\/williamcromar.com\/newmediawiki\/2d-3d-line-to-plane\/\">2D-3D | line &gt; plane<\/a><\/strong><\/p>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:33.33%\"><div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"348\" height=\"348\" data-attachment-id=\"1382\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/2d-3d-line-to-plane\/geometry\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry.jpg\" data-orig-size=\"348,348\" 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=\"geometry\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry.jpg\" alt=\"\" class=\"wp-image-1382\" style=\"aspect-ratio:1;object-fit:cover;width:120px;height:undefinedpx\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry.jpg 348w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry-300x300.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry-150x150.jpg 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/geometry-100x100.jpg 100w\" sizes=\"auto, (max-width: 348px) 100vw, 348px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-center has-small-font-size\"><strong>hidden geo<\/strong><\/p>\n<\/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 is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:50%\"><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-69f13a7a5e3d3\" 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-69f13a7a5e3d3\"  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\/hidden-geometry\/#Hidden_geometry\" >Hidden geometry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Expressing_a_datum\" >Expressing a datum<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Expressing_axis_of_symmetry\" >Expressing axis of symmetry<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Bilateral_symmetry\" >Bilateral symmetry<\/a><\/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\/hidden-geometry\/#Radial_symmetry\" >Radial symmetry<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Self-similarity_symmetry_in_scale\" >Self-similarity: symmetry in scale<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-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-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Expressing_coordinate_systems\" >Expressing coordinate systems<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Orthographic_and_3-view_drawings\" >Orthographic and 3-view drawings<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Section_drawings\" >Section drawings<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Generating_irregular_form\" >Generating irregular form<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-geometry\/#Topographic_lines\" >Topographic lines<\/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\/hidden-geometry\/#Boat_lofting\" >Boat lofting<\/a><\/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\" 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\" \/><figcaption class=\"wp-element-caption\"><em>William Blake, The Ancient of Days, 1794<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\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>As you are now familiar with coordinate systems, structural geometries, and orthographic projection systems through your readings in <strong>parallelUniverses<\/strong>, let&#8217;s expand on those systems by seeing how they are expressed in a drawing. The samples below run across history, and are visual descriptions of such hidden geometries as the datum, various axes of symmetry, orthographic drawing, and systems for drawing irregular, curving forms.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expressing_a_datum\"><\/span>Expressing a datum<span class=\"ez-toc-section-end\"><\/span><\/h2>\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:&nbsp;Oronce Fine, <strong><a href=\"https:\/\/www.wga.hu\/html_m\/f\/fine\/1map.html\" target=\"_blank\" rel=\"noreferrer noopener\">Mappemonde en forme de&nbsp;c\u0153ur montrant la Terre australe.&nbsp;Recens et integra orbis descriptio<\/a><\/strong>,&nbsp;Paris, 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:&nbsp;Heinrich 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:&nbsp;Robert 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&nbsp;steam-powered warship, later renamed&nbsp;after its designer to<strong> USS Fulton<\/strong>, 1813.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expressing_axis_of_symmetry\"><\/span>Expressing axis of symmetry<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>One datum in particular that is often hidden but important to express is the <strong>axis of symmetry<\/strong>. The object you choose to model will likely contain an axis of symmetry or near-symmetry.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Bilateral_symmetry\"><\/span>Bilateral symmetry<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>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&nbsp;by 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&nbsp;<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>&nbsp;<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>,&nbsp;1570.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Radial_symmetry\"><\/span>Radial symmetry<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>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>,&nbsp;1570.<\/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<h3 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><\/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>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<h3 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><\/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>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<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expressing_coordinate_systems\"><\/span>Expressing coordinate systems<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 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><\/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 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>&nbsp; |&nbsp; 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>&nbsp;WWII 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<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Section_drawings\"><\/span>Section drawings<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>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,&nbsp;<strong>stacked-slice model of a shoe&nbsp;<\/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.&nbsp;Wilhelm L\u00fcbke, Max Semrau: <strong>Outlines of the History of Art<\/strong>.&nbsp;Paul 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&nbsp;Willson, <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<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Generating_irregular_form\"><\/span>Generating irregular form<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Topographic_lines\"><\/span>Topographic lines<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>A\u00a0<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<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Boat_lofting\"><\/span>Boat lofting<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Like topographic lines, the drafting technique of lofting can generate complex curving forms. Lofting is used in boat building to describe the streamlined shape of the hull. Usually, a profile was divided 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-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"662\" 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-1024x662.jpg\" alt=\"\" class=\"wp-image-1456\" srcset=\"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-300x194.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines-768x496.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/bjk-lines.jpg 1162w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><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>","protected":false},"excerpt":{"rendered":"<p>Fabbing 2D-3D | line &gt; plane hidden geo Hidden geometry As you are now familiar with coordinate systems, structural geometries, and orthographic projection systems through your readings in parallelUniverses, let&#8217;s expand on those systems by seeing how they are expressed in a drawing. The samples below run across history, and are visual descriptions of such&#8230; <a href=\"https:\/\/williamcromar.com\/newmediawiki\/hidden-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":[7,10],"tags":[],"class_list":["post-1414","post","type-post","status-publish","format-standard","hentry","category-fabbing","category-references"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Hidden geometry - newMediaWiki<\/title>\n<meta name=\"description\" content=\"Hidden geometry: everything has a datum, whether it&#039;s an origin, an axis of symmetry, or a coordinate system... find out how these work.\" \/>\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\/hidden-geometry\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hidden geometry - 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