{"id":2488,"date":"2023-08-10T17:27:43","date_gmt":"2023-08-10T17:27:43","guid":{"rendered":"https:\/\/williamcromar.com\/newmediawiki\/?p=2488"},"modified":"2024-01-10T12:58:37","modified_gmt":"2024-01-10T12:58:37","slug":"euclid","status":"publish","type":"post","link":"https:\/\/williamcromar.com\/newmediawiki\/euclid\/","title":{"rendered":"Euclid"},"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-vertically-aligned-center is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:25%\">\n<p class=\"has-text-align-center has-small-font-size\"><\/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\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/2d-digital-art\/\"><img decoding=\"async\" data-attachment-id=\"15\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/campfire300\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/campfire300.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=\"campfire300\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/campfire300.gif\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/campfire300.gif\" alt=\"\" class=\"wp-image-15\" style=\"aspect-ratio:1;object-fit:cover;width:120px\" width=\"120\"\/><\/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\/2d-digital-art\/\">2D digital art<\/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\">\n<figure class=\"aligncenter size-full is-resized\"><a href=\"https:\/\/williamcromar.com\/newmediawiki\/mandala\/\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"73\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/nmw\/2d-digital-art\/mandala\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala.png\" data-orig-size=\"445,445\" 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=\"mandala\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala.png\" alt=\"\" class=\"wp-image-73\" style=\"width:120px;height:120px\" width=\"120\" height=\"120\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala.png 445w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala-300x300.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala-150x150.png 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/07\/mandala-100x100.png 100w\" sizes=\"auto, (max-width: 120px) 100vw, 120px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-center has-small-font-size\"><strong><a href=\"https:\/\/williamcromar.com\/newmediawiki\/mandala\/\">3 | mandala<\/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 decoding=\"async\" data-attachment-id=\"2162\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala\/web\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.jpg\" data-orig-size=\"2400,2400\" 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;1466635084&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;Web&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Web\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-1024x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.jpg\" alt=\"\" class=\"wp-image-2162\" style=\"aspect-ratio:1;object-fit:cover;width:120px\" width=\"120\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.jpg 2400w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-300x300.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-1024x1024.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-150x150.jpg 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-768x768.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-1536x1536.jpg 1536w\" sizes=\"(max-width: 2400px) 100vw, 2400px\" \/><\/figure>\n<\/div>\n\n\n<p class=\"has-text-align-center has-small-font-size\"><strong>euclid<\/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-69f125af076aa\" 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-69f125af076aa\"  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\/euclid\/#With_a_compass_and_straightedge_you_can_rule_the_world\" >With a compass and straightedge you can rule the world<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/#Compass_and_straightedge_the_first_computer\" >Compass and straightedge: the first computer?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/#Making_Illustrator_behave_like_a_compass_and_straightedge_the_rules\" >Making Illustrator behave like a compass and straightedge: the rules<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/#Other_helpful_hints\" >Other helpful hints<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/#Creating_simple_constructions\" >Creating simple constructions<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/#Perpendicular_bisector_of_a_given_line\" >Perpendicular bisector of a given line<\/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\/euclid\/#Tangent_to_a_given_circle\" >Tangent to a given circle<\/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\/euclid\/#Creating_simple_polygon_constructions\" >Creating simple polygon constructions<\/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\/euclid\/#Square\" >Square<\/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\/euclid\/#Hexagon\" >Hexagon<\/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\/euclid\/#Reverse-engineered_expanded_constructions_of_polygons\" >Reverse-engineered expanded constructions of polygons<\/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\/euclid\/#Expanding_the_square\" >Expanding the square<\/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\/euclid\/#Expanding_the_hexagon\" >Expanding the hexagon<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/#Save_and_share\" >Save and share<\/a><\/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%\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\" style=\"flex-basis:100%\">\n<p class=\"has-text-align-center\"><strong><a href=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.zip\">Download sample files here<\/a><\/strong><\/p>\n<\/div>\n<\/div>\n\n\n<div class=\"wp-block-image is-style-default\">\n<figure class=\"aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"2400\" height=\"2400\" data-attachment-id=\"2162\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/mandala\/web\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.jpg\" data-orig-size=\"2400,2400\" 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;1466635084&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;Web&quot;,&quot;orientation&quot;:&quot;1&quot;}\" data-image-title=\"Web\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-1024x1024.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.jpg\" alt=\"\" class=\"wp-image-2162\" style=\"aspect-ratio:1;object-fit:cover\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid.jpg 2400w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-300x300.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-1024x1024.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-150x150.jpg 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-768x768.jpg 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/euclid-1536x1536.jpg 1536w\" sizes=\"auto, (max-width: 2400px) 100vw, 2400px\" \/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"With_a_compass_and_straightedge_you_can_rule_the_world\"><\/span>With a compass and straightedge you can rule the world<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>In vector art drawing, powerful algorithms hide behind a graphical user interface. These algorithms are based on coordinate-generated geometries. In these exercises we will \u201coverride\u201d this computational logic by exploring another way of describing geometry: making Illustrator behave like a virtual compass and straightedge.<\/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=\"1024\" height=\"565\" data-attachment-id=\"2492\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/mesures\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/mesures.jpg\" data-orig-size=\"1024,565\" 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=\"mesures\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/mesures.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/mesures.jpg\" alt=\"\" class=\"wp-image-2492\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/mesures.jpg 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/mesures-300x166.jpg 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/mesures-768x424.jpg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">&nbsp;<em>Les Mesures, painting from drawing attributed to the school of Frans Floris, Flemish, late 16th C.<\/em><br><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Compass_and_straightedge_the_first_computer\"><\/span>Compass and straightedge: the first computer?<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>A strong relationship between art and computation is found in such constructed geometry. Evidence of this knowledge goes back to the Neolithic Age, but it was first codified into a system by <strong><a href=\"http:\/\/www.mathopenref.com\/euclid.html\" target=\"_blank\" rel=\"noreferrer noopener\">Euclid<\/a><\/strong>. <\/p>\n\n\n\n<p>Constructed geometry was critical to the Greeks: with it, they performed mathematical calculations their clumsy Roman-like numerical system couldn\u2019t handle; try long division in a numeral system that doesn&#8217;t have a zero and you\u2019ll see!<\/p>\n\n\n\n<p>Before means of measure and arithmetic, the only way to precisely define proof of geometric truth was through Euclidean geometric constructions. Even with the availability of marked rulers and protractors, it is ONLY geometric construction that, through an image, manifests the conceptual state of perfect precision geometry requires.<\/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=\"500\" height=\"767\" data-attachment-id=\"2493\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/davincicompass\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/davincicompass.jpg\" data-orig-size=\"500,767\" 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=\"davincicompass\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/davincicompass.jpg\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/davincicompass.jpg\" alt=\"\" class=\"wp-image-2493\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/davincicompass.jpg 500w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/davincicompass-196x300.jpg 196w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/davincicompass-300x460.jpg 300w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><figcaption class=\"wp-element-caption\"><em>Leonardo da Vinci<\/em>, <em>Two types of adjustable-opening compass<\/em>, <em>~1493 C.E.<\/em><\/figcaption><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<p>So for our project, knowledge of <strong><a href=\"http:\/\/www.mathopenref.com\/constructions.html\" target=\"_blank\" rel=\"noreferrer noopener\">compass and straightedge construction<\/a><\/strong> is necessary: read about it at the link. If your exposure to this was limited to high school, you might remember how to bisect a line, but you\u2019ll need more. Tutorials below&nbsp;will introduce the necessary knowledge, or act as a refresher if your compass and straightedge are a bit rusty. Of course, the compass and straightedge we\u2019ll be using are virtual: we\u2019re substituting straight lines and circles in Illustrator.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Making_Illustrator_behave_like_a_compass_and_straightedge_the_rules\"><\/span>Making Illustrator behave like a compass and straightedge: the rules<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>These rules for creating the constructions will persuade Illustrator to act like a compass and straightedge and override instructions where tutorials assume the use of physical tools:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Illustrator will draw ENTIRE circles, not just the arc as shown in some tutorials.<\/li>\n\n\n\n<li>Use ONLY a circle (<strong>Option+Shift<\/strong> constrained Ellipse Tool) and straight line (<strong>unconstrained<\/strong> Pen Tool).<\/li>\n\n\n\n<li>When using Pen Tool, only click to create points. Never click and drag.&nbsp;<\/li>\n\n\n\n<li>Use the <strong>Option-Drag<\/strong> shortcut to clone a circle when a tutorial asks you to move a compass to translate an arc. Never do this for a line.<\/li>\n\n\n\n<li>Scale a line or circle by dragging a <strong>Shift<\/strong> or <strong>Option+Shift<\/strong> constrained point. Scaling a line is like extending it with the straightedge. Scaling a circle is like changing the compass radius.&nbsp;<\/li>\n\n\n\n<li>Never measure. The first step in document setup below will guarantee you will not measure.\n<ul class=\"wp-block-list\">\n<li>Never use <em>Align<\/em> or <em>Transform<\/em> palette functions.<\/li>\n\n\n\n<li>Exception: Illustrator cannot draw a discrete point. To indicate a point, create a 0.05\u201d diameter circle with a black fill and no stroke, using its center as the actual point.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Other_helpful_hints\"><\/span>Other helpful hints<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>These are not rules per se, but are helpful suggestions for working with Illustrator as a compass-straightedge surrogate:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Toggle between <strong>Preview<\/strong> and <strong>Outline<\/strong> views using <strong>Command+Y<\/strong> to draw with precision.<\/li>\n\n\n\n<li>Practice using <strong>Shift-drag<\/strong> to pan, <strong>Command+Shift-click<\/strong> or <strong>drag<\/strong> to zoom in, <strong>Option+Shift-click<\/strong> or <strong>drag<\/strong> to zoom out, and <strong>Command+<\/strong> to recenter on the artboard if you get lost. Key command assists are much quicker than the menu or toolbar.<\/li>\n\n\n\n<li>Create layers to help control your drawing. Thee ability to lock, hide or change object order is essential on a complex construction.<\/li>\n\n\n\n<li>If you find it useful to label points, do so on a layer you can turn on and off. Final drawings should not express point labels (i.e. they should be turned off or the layer deleted).<\/li>\n\n\n\n<li>As you did with the <strong>thirteenWays<\/strong>&nbsp; project, you may find it useful to work the construction off to one side, moving and scaling the grouped result into a square when complete.<\/li>\n\n\n\n<li>Draw the construction with a compositional eye. How does the construction (a figure) relate to the designated square (the ground)? Does it crowd the edge? Too tiny? As you develop the figure, select and move, rotate or Shift-scale the entire construction to find a good figure-ground relationship inside its frame. Constructions are scale-less and have no correct orientation.<\/li>\n\n\n\n<li>You&#8217;ll be tempted to try to draw perfect horizontal or vertical lines at first: they don&#8217;t matter, don&#8217;t bother. Geometric proofs work at any angle, any orientation.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Creating_simple_constructions\"><\/span>Creating simple constructions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>As a warm-up, we&#8217;ll create two simple but highly useful constructions in the first column on the blank template file you can download above. There are a variety of tutorial styles, any of which you may use depending on your own learning style.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Perpendicular_bisector_of_a_given_line\"><\/span>Perpendicular bisector of a given line<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>The most basic and useful of all constructions, created in the upper left square of the downloadable sample file:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><a href=\"http:\/\/www.mathopenref.com\/constbisectline.html\" target=\"_blank\" rel=\"noreferrer noopener\">Interactive from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"http:\/\/www.mathopenref.com\/printbisectline.html\" target=\"_blank\" rel=\"noreferrer noopener\">Printable instructions from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"https:\/\/www.khanacademy.org\/math\/geometry\/hs-geo-congruence\/hs-geo-bisectors\/v\/constructing-a-perpendicular-bisector-using-a-compass-and-straightedge\" target=\"_blank\" rel=\"noreferrer noopener\">Video from Kahn Academy\u00bb<\/a><\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tangent_to_a_given_circle\"><\/span>Tangent to a given circle<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Applies the perpendicular bisector to a specific problem, and is found in the lower left square of the sample file:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><a href=\"http:\/\/www.mathopenref.com\/consttangent.html\" target=\"_blank\" rel=\"noreferrer noopener\">Interactive from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"http:\/\/www.mathopenref.com\/printtangent.html\" target=\"_blank\" rel=\"noreferrer noopener\">Printable instructions from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"https:\/\/www.khanacademy.org\/math\/geometry\/hs-geo-circles\/hs-geo-constructing-tangents\/v\/constructing-a-tangent-line-using-compass-and-straightedge\" target=\"_blank\" rel=\"noreferrer noopener\">Video from Kahn Academy\u00bb<\/a><\/strong><\/li>\n<\/ul>\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=\"200\" height=\"333\" data-attachment-id=\"2495\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/perpendicular_bisector\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Perpendicular_bisector.gif\" data-orig-size=\"200,333\" 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=\"Perpendicular_bisector\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Perpendicular_bisector.gif\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Perpendicular_bisector.gif\" alt=\"\" class=\"wp-image-2495\"\/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Creating_simple_polygon_constructions\"><\/span>Creating simple polygon constructions<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>In the next column of the template, we&#8217;ll create simple polygons: a square at center top, and a hexagon at center bottom.<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Square\"><\/span>Square<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>There are numerous solutions to the square construction problem. We will work with the simplest, which generates a diagonal square. You can see the usefulness of the perpendicular bisector in the animation.<\/p>\n\n\n\n<p><strong>SQUARE INSCRIBED IN A CIRCLE<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><a href=\"https:\/\/www.mathopenref.com\/constinsquare.html\" target=\"_blank\" rel=\"noreferrer noopener\">Interactive from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"https:\/\/www.mathopenref.com\/printinsquare.html\" target=\"_blank\" rel=\"noreferrer noopener\">Printable instructions from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"https:\/\/www.khanacademy.org\/math\/geometry\/hs-geo-circles\/hs-geo-inscribed-polygons\/v\/constructing-square-inscribed-in-circle\" target=\"_blank\" rel=\"noreferrer noopener\">Video from Kahn Academy\u00bb<\/a><\/strong><\/li>\n<\/ul>\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=\"480\" height=\"480\" data-attachment-id=\"2498\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/skewed_square_inscribed_in_a_circle\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Skewed_Square_Inscribed_in_a_Circle.gif\" data-orig-size=\"480,480\" 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=\"Skewed_Square_Inscribed_in_a_Circle\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Skewed_Square_Inscribed_in_a_Circle.gif\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Skewed_Square_Inscribed_in_a_Circle.gif\" alt=\"\" class=\"wp-image-2498\"\/><\/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\">\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Hexagon\"><\/span>Hexagon<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>A delightfully simple construction creates the hexagon, as the animation illustrates.<\/p>\n\n\n\n<p><strong>HEXAGON INSCRIBED IN A CIRCLE<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong><a href=\"http:\/\/www.mathopenref.com\/constinhexagon.html\" target=\"_blank\" rel=\"noreferrer noopener\">Interactive from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"http:\/\/www.mathopenref.com\/printinhexagon.html\" target=\"_blank\" rel=\"noreferrer noopener\">Printable instructions from Math Open Reference\u00bb<\/a><\/strong><\/li>\n\n\n\n<li><strong><a href=\"https:\/\/www.khanacademy.org\/math\/geometry\/hs-geo-circles\/hs-geo-inscribed-polygons\/v\/constructing-regular-hexagon-inscribed-in-circle\" target=\"_blank\" rel=\"noreferrer noopener\">Video from Kahn Academy\u00bb<\/a><\/strong><\/li>\n<\/ul>\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=\"480\" height=\"480\" data-attachment-id=\"2499\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/regular_hexagon_inscribed_in_a_circle\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Regular_Hexagon_Inscribed_in_a_Circle.gif\" data-orig-size=\"480,480\" 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=\"Regular_Hexagon_Inscribed_in_a_Circle\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Regular_Hexagon_Inscribed_in_a_Circle.gif\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/Regular_Hexagon_Inscribed_in_a_Circle.gif\" alt=\"\" class=\"wp-image-2499\"\/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<p>These are not the only regular polygons that can be constructed! <a href=\"http:\/\/en.wikipedia.org\/wiki\/User:Bender2k14\/Regular_Polygon_Constructions\" target=\"_blank\" rel=\"noreferrer noopener\">A full page of polygon constructions<\/a>, including a <strong>255-gon<\/strong>, has been collected by Wikipedia user Bender2k14! But, a square or hexagon is all we need for the next step.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Reverse-engineered_expanded_constructions_of_polygons\"><\/span>Reverse-engineered expanded constructions of polygons<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>The last two columns in the template are reserved for what we will call &#8220;expanded constructions&#8221; of a square and hexagon. <\/p>\n\n\n\n<p>This &#8220;construction&#8221; is a bit of a cheat. We will start by using Illustrator&#8217;s constrained Rectangle Tool and Polygon Tool to generate the basic shapes, then <strong>reverse-engineer<\/strong> the expanded construction into a complex geometric figure. So, while we are abandoning the strict rules of compass-and-straightedge, we will keep the sensibility.<\/p>\n\n\n\n<p>Why do this? While the pattern of circles and axes are not required to construct the polygons per se, they create a complex, interlocking web of arcs and intersections that can be quite useful as a structure to generate figures such as rosettes, stained glass windows, Islamic tile tessellations, Celtic knots&#8230; and, of course, mandalas!<\/p>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expanding_the_square\"><\/span>Expanding the square<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Create a 1&#8243; square using Rectangle Tool dialog box to enter the dimension.<\/li>\n\n\n\n<li>Using Pen Tool, draw two diagonal lines across vertices.<\/li>\n\n\n\n<li>Using Option-Shift constrained Ellipse Tool, create a circle whose center is at one vertex of the square, and whose radius anchor point touches a second vertex.<\/li>\n\n\n\n<li>Option-drag the circle to copy around until all four vertices have a circle centered on them.<\/li>\n\n\n\n<li>Scale the diagonals so that they intersect the outer perimeter of these circles.<\/li>\n\n\n\n<li>Create a horizontal and vertical axis with Pen Tool by creating endpoints at the outermost intersections between the circles.<\/li>\n\n\n\n<li>Finally, create an circle around the square. This one is a bit tough, as Illustrator&#8217;s &#8220;snap&#8221; functions operate orthogonally (that is, just in the x or y axis). So we trick it by duplicating the square on top of itself (Command+C, then Command+F), then rotating the duplicate square Shift-constrained to 45 degrees. Now, setting the center of the circle at the center of the square, we Option-Shift and drag until the circle&#8217;s anchor point intersects the diagonalized vertex of the copied square. When done, delete the copied square.<\/li>\n<\/ul>\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=\"1024\" data-attachment-id=\"2501\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/squareexpanded\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded.png\" data-orig-size=\"1507,1506\" 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=\"squareexpanded\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-1024x1024.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-1024x1024.png\" alt=\"\" class=\"wp-image-2501\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-1024x1024.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-300x300.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-150x150.png 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-768x767.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded-100x100.png 100w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/squareexpanded.png 1507w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/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\">\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Expanding_the_hexagon\"><\/span>Expanding the hexagon<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Using Polygon Tool, create a polygon with 6 sides, each side 1&#8243; long. This creates a 2&#8243; diameter hexagon.<\/li>\n\n\n\n<li>Draw a 2&#8243; diameter circle with&nbsp;Option-Shift constrained Ellipse Tool and snap center of circle to center of hexagon.&nbsp;<\/li>\n\n\n\n<li>Option-drag the circle to copy around the periphery of the hexagon, placing the center of a circle at each vertex.<\/li>\n\n\n\n<li>Using Pen Tool, draw 3 diameter lines between pairs of vertices such that they intersect at the center.<\/li>\n\n\n\n<li>Scale the diameter lines until they intersect with the outer periphery of the circles.<\/li>\n<\/ul>\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=\"1022\" data-attachment-id=\"2502\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/hexexpanded\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded.png\" data-orig-size=\"1508,1505\" 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=\"hexexpanded\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-1024x1022.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-1024x1022.png\" alt=\"\" class=\"wp-image-2502\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-1024x1022.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-300x299.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-150x150.png 150w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-768x766.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded-100x100.png 100w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/hexexpanded.png 1508w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div><\/div>\n<\/div>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Save_and_share\"><\/span>Save and share<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 has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column has-background is-layout-flow wp-block-column-is-layout-flow\" style=\"background-color:#eeeeee\">\n<p>Your finished exercise will look something like this:<\/p>\n<\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-columns has-background is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\" style=\"background-color:#eeeeee\">\n<div class=\"wp-block-column has-white-background-color has-background 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=\"663\" data-attachment-id=\"2504\" data-permalink=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/2018_cromar_w_geo-construction-1-01\/\" data-orig-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01.png\" data-orig-size=\"5100,3300\" 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=\"2018_cromar_w_geo-construction-1-01\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-1024x663.png\" src=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-1024x663.png\" alt=\"\" class=\"wp-image-2504\" style=\"object-fit:cover\" srcset=\"https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-1024x663.png 1024w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-300x194.png 300w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-768x497.png 768w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-1536x994.png 1536w, https:\/\/williamcromar.com\/newmediawiki\/wp-content\/uploads\/2023\/08\/2018_cromar_w_geo-construction-1-01-2048x1325.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div><\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n<p>Save the file as a native .ai file, and export a .jpg. The .ai file should be archived, and use the .jpg to illustrate your work as an embed in your process journal or blog.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>2D digital art 3 | mandala euclid Download sample files here With a compass and straightedge you can rule the world In vector art drawing, powerful algorithms hide behind a graphical user interface. These algorithms are based on coordinate-generated geometries. In these exercises we will \u201coverride\u201d this computational logic by exploring another way of describing&#8230; <a href=\"https:\/\/williamcromar.com\/newmediawiki\/euclid\/\">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":[3,11],"tags":[],"class_list":["post-2488","post","type-post","status-publish","format-standard","hentry","category-2d-digital-art","category-exercises"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Euclid - newMediaWiki<\/title>\n<meta name=\"description\" content=\"Euclid: Geometric construction in Illustrator for simple proofs, inscribed polygons, and expanded constructions used in mandala creation.\" \/>\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\/euclid\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Euclid - 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