{"id":11791,"date":"2026-01-16T20:31:07","date_gmt":"2026-01-16T16:31:07","guid":{"rendered":"https:\/\/beegraphy.com\/blog\/?p=11791"},"modified":"2026-02-02T10:24:33","modified_gmt":"2026-02-02T06:24:33","slug":"venn-diagrams-in-beegraphy","status":"publish","type":"post","link":"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/","title":{"rendered":"Venn Diagrams in BeeGraphy"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_75 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69e6ae77587e2\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #757575;color:#757575\" 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: #757575;color:#757575\" 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-69e6ae77587e2\" checked 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:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Overview\" >Overview<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Integrating_Venn_Diagrams_with_BeeGraphy\" >Integrating Venn Diagrams with BeeGraphy<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Extending_Venn_Diagrams_Beyond_Three_Sets\" >Extending Venn Diagrams Beyond Three Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#The_Geometric_Limitations_of_4-Circle_Venn_Diagrams\" >The Geometric Limitations of 4-Circle Venn Diagrams<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Why_does_this_happen\" >Why does this happen?<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Venns_Approach_to_the_4-Set_Diagram\" >Venn\u2019s Approach to the 4-Set Diagram<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Why_does_this_work\" >Why does this work?<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Constructing_a_4-Circle_Venn_Diagram_Using_BeeGraphy\" >Constructing a 4-Circle Venn Diagram Using BeeGraphy<\/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:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Why_BeeGraphy\" >Why BeeGraphy?<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Applying_Design_Thinking_to_Mathematical_Visualization\" >Applying Design Thinking to Mathematical Visualization<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Visualizing_Mathematics_Through_Parametric_Design\" >Visualizing Mathematics Through Parametric Design<\/a><\/li><\/ul><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#Step_by_Step_Tutorial_with_BeeGraphy\" >Step by Step Tutorial with BeeGraphy<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/beegraphy.com\/blog\/venn-diagrams-in-beegraphy\/#About_Jiyun_Ko_MathDesigner\" >About Jiyun Ko (MathDesigner)<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Overview\"><\/span><strong>Overview<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><em><span class=\"a_GcMg font-feature-liga-off font-feature-clig-off font-feature-calt-off text-decoration-none text-strikethrough-none\">This is a translated version of the <a href=\"https:\/\/beegraphy.com\/blog\/beegraphy%EC%97%90%EC%84%9C%EC%9D%98-%EB%B2%A4-%EB%8B%A4%EC%9D%B4%EC%96%B4%EA%B7%B8%EB%9E%A8\">Original Blog Written by Jiyun Ko<\/a><\/span><\/em><\/pre>\n<p><b>Cantor once said, \u201cThe essence of mathematics lies in its freedom.\u201d (Das Wesen der Mathematik liegt in ihrer Freiheit)<\/b><\/p>\n<p><i><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Picture1-229x300.jpg\" alt=\"\" width=\"229\" height=\"300\" \/><\/i><\/p>\n<p><i>This is the famous statement of Georg Cantor, the founder of set theory, which forms the foundation of modern mathematics.<\/i><\/p>\n<p><span style=\"font-weight: 400;\">But what does <\/span><i><span style=\"font-weight: 400;\">freedom<\/span><\/i><span style=\"font-weight: 400;\"> in mathematics actually mean? Does it mean mathematics itself is free? That sounds absurd altogether. So, are we freeing math from some chains? Not quite.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Even in mathematics which is often seen as the pinnacle of logical rigor, the standards we rely on aren\u2019t divine laws. These are frameworks that humans over time have collectively agreed to use. As long as the logic holds, mathematics becomes a space you can shape and explore freely.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">That\u2019s the spirit of the <a href=\"https:\/\/www.instagram.com\/mathdesigner_got\/\" target=\"_blank\" rel=\"noopener\">MathDesigner<\/a> classroom. Students are encouraged to imagine without constraints, play with ideas, and discover solutions with a sense of joy and curiosity. For us, it\u2019s not just about teaching math. It\u2019s about creating art through mathematics and equally discovering mathematics through art.<\/span><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Integrating_Venn_Diagrams_with_BeeGraphy\"><\/span><strong>Integrating Venn Diagrams with BeeGraphy<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><span style=\"font-weight: 400;\">In this lesson, we use <\/span><b>BeeGraphy<\/b><span style=\"font-weight: 400;\"> to understand the principles behind <\/span><b>Venn diagrams<\/b><span style=\"font-weight: 400;\">.<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">Venn diagrams form the fundamentals in set theory. Introduced by <\/span><b>John Venn in 1880, <\/b><span style=\"font-weight: 400;\">Venn diagrams were a new way to represent sets.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Venn diagrams consist of multiple overlapping circles. <\/span><b>Each circle represents a set<\/b><span style=\"font-weight: 400;\">. <\/span><i><span style=\"font-weight: 400;\">(A <\/span><\/i><b><i>set<\/i><\/b><i><span style=\"font-weight: 400;\"> is an unordered collection of distinct elements.)<\/span><\/i><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">For example, let<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Set <\/span><b>A = {1, 2, 3, 4}<\/b><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Set <\/span><b>B = {2, 4, 5}<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Here, 2 and 4 belong to both set A and set B.\u00a0<\/span><\/p>\n<p><b><\/b><img decoding=\"async\" class=\"alignright lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Picture2.jpg\" alt=\"\" width=\"236\" height=\"183\" \/>Let one circle be A, and B as another ,then the shared elements: 2 and 4, appear in the overlapping region. With just a couple of circles, anyone can grasp how sets relate to each other at a glance.<\/p>\n<p>If there is one circle, how many regions does it represent? The answer is<b> 2!<\/b><\/p>\n<ul>\n<li style=\"font-weight: 400;\">Inside the circle<\/li>\n<li style=\"font-weight: 400;\">Outside the circle<\/li>\n<\/ul>\n<p><img decoding=\"async\" class=\"wp-image-11822 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626-300x84.png\" alt=\"\" width=\"654\" height=\"183\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626-300x84.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626-1024x286.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626-768x215.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626-800x224.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626-20x6.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155626.png 1255w\" sizes=\"(max-width: 654px) 100vw, 654px\" \/><\/p>\n<p>Add another circle, so now we have two sets, and the number of regions jumps to <b>four. <\/b>Ding-dong-dang, exactly.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-11821 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643-300x105.png\" alt=\"\" width=\"651\" height=\"228\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643-300x105.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643-1024x357.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643-768x268.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643-800x279.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643-20x7.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155643.png 1253w\" sizes=\"(max-width: 651px) 100vw, 651px\" \/><\/p>\n<p>So, what happens as the number of sets (circles) increases?<br \/>\nThe number of regions created by <i>n<\/i> sets is given by = <b>2^<\/b><b>n<\/b> (where n is the number of sets).<\/p>\n<p><i>For example:<\/i><i><br \/>\n<\/i><span style=\"font-weight: 400;\">1 set \u2192 2^<\/span><span style=\"font-weight: 400;\">1<\/span><span style=\"font-weight: 400;\">=2 regions (inside the set, outside the set)<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">2 sets \u2192 2^<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> = 4 regions (all combinations of inside\/outside for the two sets)<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">3 sets \u2192 2^<\/span><span style=\"font-weight: 400;\">3<\/span><span style=\"font-weight: 400;\"> = 8 regions<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span><span style=\"font-weight: 400;\">4 sets \u2192 2^<\/span><span style=\"font-weight: 400;\">4<\/span><span style=\"font-weight: 400;\"> = 16 regions<\/span><span style=\"font-weight: 400;\"><br \/>\n<\/span>and so on\u2026<br \/>\nSo, as the number of sets increases, the required regions <b>grow exponentially<\/b>.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Extending_Venn_Diagrams_Beyond_Three_Sets\"><\/span>Extending Venn Diagrams Beyond Three Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>It is easy to draw a Venn diagram for <b>three<\/b> sets, but what happens when we increase the number of sets? How do we visualize those without the diagram turning into chaos? That\u2019s where BeeGraphy becomes useful.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-11820 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722-300x92.png\" alt=\"\" width=\"675\" height=\"207\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722-300x92.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722-1024x315.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722-768x237.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722-800x246.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722-20x6.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155722.png 1487w\" sizes=\"(max-width: 675px) 100vw, 675px\" \/><\/p>\n<p>The method is simple.<br \/>\nStart with a standard 3-set Venn diagram. Place one point in each region, then connect those points in a way that doesn\u2019t overlap or repeat any connections. The result looks a bit unusual at first\u2014but that strange-looking shape is exactly what becomes a 4-set Venn diagram.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Picture3.jpg\" alt=\"\" width=\"208\" height=\"209\" \/><\/p>\n<p><i>Pretty wild, right?<\/i><\/p>\n<h3><span class=\"ez-toc-section\" id=\"The_Geometric_Limitations_of_4-Circle_Venn_Diagrams\"><\/span>The Geometric Limitations of 4-Circle Venn Diagrams<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>With 2 or 3 circles, counting the regions in a Venn diagram is easy because students can see them. But with <b>4 circles<\/b>, there are 24 = 16 regions, some of which are hidden or irregularly shaped. Counting them by eye becomes tricky.<\/p>\n<p>When students try to draw a <b>4-circle Venn diagram<\/b>, it looks simple at first \u2014 \u201cjust add one more circle!\u201d<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-11985 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1-1024x548.png\" alt=\"\" width=\"1024\" height=\"548\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1-1024x548.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1-300x160.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1-768x411.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1-800x428.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1-20x11.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-155804-1.png 1322w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p>But this is where most students get stuck. A student may try many different arrangements of four circles, and some of them <i>look<\/i> correct. The diagram might look nicely symmetrical, or aesthetically pleasing, but when you actually <b>count the regions<\/b>, you only get <b>14<\/b> or <b>15<\/b> instead of the required <b>16<\/b>.<\/p>\n<h4><span class=\"ez-toc-section\" id=\"Why_does_this_happen\"><\/span>Why does this happen?<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><span style=\"font-weight: 400;\">This happens as four perfect circles cannot all intersect each other in the right way.<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Some circles do not intersect another circle twice.<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Some intersections overlap incorrectly.<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Some regions merge together without the student noticing.<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">This is why drawing 4 perfect circles <\/span><i><span style=\"font-weight: 400;\">simply does not work<\/span><\/i><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">For a true 4-set Venn diagram, every one of the 16 combinations (inside\/outside of A, B, C, and D) must exist as a separate region.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Venns_Approach_to_the_4-Set_Diagram\"><\/span><strong>Venn\u2019s Approach to the 4-Set Diagram<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Mathematicians realized long ago that\u00a0 it is impossible to get a proper 4-set Venn diagram using only four perfect circles. So the best way to go about it is:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignright lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Picture3.jpg\" alt=\"\" width=\"208\" height=\"209\" \/><\/p>\n<ul>\n<li><b>Start with a 3-circle Venn diagram:<\/b><i> It already has 8 regions.<\/i><\/li>\n<li><b>Place a centroid in each region: <\/b><i>Students put a little dot inside every region so none are missed.<\/i><\/li>\n<li><b>Now add the 4th set as a \u201cballoon-animal\u201d shape: <\/b><i>This shape moves around the 3-circle diagram and:<\/i>\n<ul>\n<li><i>passes through each centroid exactly once<\/i><\/li>\n<li><i>divides each of the 8 regions into two, resulting in <\/i><b><i>16 total regions<\/i><\/b><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">This shape is not a perfect circle, it is long and looping. But mathematically, it can result in a proper 4-set Venn diagram.<\/span><\/p>\n<h4><span class=\"ez-toc-section\" id=\"Why_does_this_work\"><\/span>Why does this work?<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p><span style=\"font-weight: 400;\">The curved shape ensures:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">every existing region gets split<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">no region is skipped<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">no regions merge incorrectly<\/span><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">So the student gets the correct <\/span><b>16-region Venn diagram<\/b><span style=\"font-weight: 400;\"> every time.<\/span><\/p>\n<p><i>Here are a few blogs to check out if you&#8217;re interested further in this theory:<\/i><i><br \/>\n<\/i><a href=\"https:\/\/medium.com\/@danyaltairoski\/why-you-cant-make-a-venn-diagram-with-4-circles-bea3c2dcbc5d\" target=\"_blank\" rel=\"noopener\"><i>Why you can\u2019t make a Venn Diagram with 4 Circles<\/i><i><br \/>\n<\/i><\/a><a href=\"https:\/\/www.quora.com\/Why-cant-we-draw-a-Venn-diagram-for-4-sets-with-circles-and-how-can-we-solve-it\" target=\"_blank\" rel=\"noopener\"><i>Why can&#8217;t we draw a Venn diagram for 4 sets with circles, and how can we solve it?<\/i><\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Constructing_a_4-Circle_Venn_Diagram_Using_BeeGraphy\"><\/span><strong>Constructing a 4-Circle Venn Diagram Using BeeGraphy<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Now, let\u2019s recreate this structure using BeeGraphy.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Why_BeeGraphy\"><\/span>Why BeeGraphy?<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><i>Because once students build it themselves, they actually feel the beauty of the idea. It turns a dry concept into a hands-on experience.<\/i><\/p>\n<h4><span class=\"ez-toc-section\" id=\"Applying_Design_Thinking_to_Mathematical_Visualization\"><\/span><i><span style=\"font-weight: 400;\">Applying Design Thinking to Mathematical Visualization<\/span><\/i><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>Let\u2019s organize this problem-solving process using the stages of design thinking. We\u2019ll begin with the first stage:<\/p>\n<ul>\n<li><b>Define: <\/b>We ask ourselves, \u201cHow can we effectively visualize four or more sets?\u201d This helps students to start exploring different approaches, thinking beyond textbooks.<\/li>\n<li><b>Ideate: <\/b>For three circles, they place a point in each region (using the Centroid nodeblock), index the points, and connect them in sequence. The key is making sure the lines don\u2019t overlap and that the curve passes through every region exactly once. By working through this, students naturally arrive at the general idea behind counting the maximum number of regions formed when a plane is divided. The discovery emerges through the process and not memorization.<\/li>\n<\/ul>\n<ul>\n<li><b>Prototype: <\/b>Using BeeGraphy\u2019s nodeblocks &#8211; Circle, Curve to Surface, Centroid, List Item, and others &#8211; students build their own versions of the diagram, test them, refine them, and iterate.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-11826 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-1024x520.png\" alt=\"\" width=\"1024\" height=\"520\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-1024x520.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-300x152.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-768x390.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-1536x779.png 1536w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-800x406.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445-20x10.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160445.png 1843w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<h4><span class=\"ez-toc-section\" id=\"Visualizing_Mathematics_Through_Parametric_Design\"><\/span><i><span style=\"font-weight: 400;\">Visualizing Mathematics Through Parametric Design<\/span><\/i><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p>By using parametric learning, learners can represent mathematics visually.\u00a0 That means, they are able to interact with a visual representation of a mathematical problem. When we integrate parametric design methodology, students are able to observe what perturbations occur in the mathematical formation while solving problems. As a result, they are better equipped to work with geometry, algebra, and above all to think creatively while using mathematical logic on their own. Their comprehension grows due to metacognitive awareness and visualization. This is BeeGraphy&#8217;s strength.<\/p>\n<ul>\n<li style=\"font-weight: 400;\"><b>Converting set structures into Venn diagrams in coordinate space: <\/b>Students transform abstract logical relationships into concrete, spatial forms. This \u201crepresentation shift\u201d makes logic feel tangible.<\/li>\n<li style=\"font-weight: 400;\"><b>Using height values to extend 2D regions into 3D forms: <\/b>Instead of relying on color, students give each region a vertical value\u2014making the diagram something they can perceive physically and intuitively.<\/li>\n<li style=\"font-weight: 400;\"><b>Applying parametric design to adjust the number of sets: <\/b>By manipulating the input values, they rebuild complex logical relationships as mathematical rules they can see and understand directly.<\/li>\n<\/ul>\n<p>Parametric design in math education allows students to instantly observe how a structure responds when inputs change. Concepts stop being static\u2014they become dynamic models. This encourages relational thinking over rote operations.<\/p>\n<p>In the process, students strengthen their ability to analyze conditions, visualize abstract ideas, and grasp structural relationships. The burden of processing complex information is reduced through visual clarity, making mathematical ideas easier to interpret at a glance.<\/p>\n<p>So really, isn\u2019t BeeGraphy a modern tool that brings the full freedom of mathematics to life?<\/p>\n<p><a href=\"https:\/\/beegraphy.com\/market\/product\/venn-diagram-d82?shareData=%7B%228882b93c-b411-4e8a-824f-719bb4b6e70f%22%3A%7B%22value%22%3A%7B%22type%22%3A%22Number%22%2C%22value%22%3A2%7D%7D%2C%2284cdcb7a-6313-4782-a84a-558d841e1f1c%22%3A%7B%22value%22%3A%7B%22type%22%3A%22Number%22%2C%22value%22%3A3.5%7D%7D%7D\"><b>You can try out the Demo for the above here.<\/b><\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Step_by_Step_Tutorial_with_BeeGraphy\"><\/span><b>Step by Step Tutorial with BeeGraphy<\/b><span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><i><span style=\"font-weight: 400;\">The step-by-step explanation for creating the structure is provided below for your reference.<\/span><\/i><\/p>\n<p><b><i>Step 1:<\/i><\/b><i><span style=\"font-weight: 400;\"> We begin making three circles and transforming them into surfaces so that we have shaded regions to work with.<\/span><\/i><\/p>\n<p><b><i>Step 2: <\/i><\/b><i><span style=\"font-weight: 400;\">Next, we use <\/span><\/i><b><i>Surface Split to divide each circle into distinct partitions. <\/i><\/b><i><span style=\"font-weight: 400;\">For example, if we want to distinguish the intersected region of circle A, we connect circle B and circle C to the Curve input, and circle A to the Surface input.<\/span><\/i><\/p>\n<p><i><span style=\"font-weight: 400;\">We repeat the steps above for circles B and C to get their distinctive intersection areas.<\/span><\/i><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-11827 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-1024x494.png\" alt=\"\" width=\"1024\" height=\"494\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-1024x494.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-300x145.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-768x370.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-1536x741.png 1536w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-800x386.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639-20x10.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160639.png 1860w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p><b><i>Step 3: <\/i><\/b><i><span style=\"font-weight: 400;\">To easily locate these sections, we use<\/span><\/i><b><i> List Item <\/i><\/b><i><span style=\"font-weight: 400;\">and assign indices to each region. This makes it easier to find all the partitions individually.<\/span><\/i><\/p>\n<p><b><i>Step 4: <\/i><\/b><i><span style=\"font-weight: 400;\">For every section, <\/span><\/i><b><i>we compute its centroid using the Centroid node, creating a one-to-one connection between each region and its centroid.<\/i><\/b><i><span style=\"font-weight: 400;\"> This gives us seven centroids for the seven regions formed by the three circles.<\/span><\/i><\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-11828 size-large lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-1024x532.png\" alt=\"venn diagram\" width=\"1024\" height=\"532\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-1024x532.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-300x156.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-768x399.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-1536x797.png 1536w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-800x415.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859-20x10.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-160859.png 1745w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p><b><i>Step 5: <\/i><\/b><i><span style=\"font-weight: 400;\">Finally, we place an additional point outside the circles to represent the eighth region. Make sure the point is placed in such a manner, it does not overlap any region. Next, we<\/span><\/i><b><i> interpolate all the points<\/i><\/b><i><span style=\"font-weight: 400;\"> such that there is cross intersection. Thus, creating a continuous path passing through all the points and regions.<\/span><\/i><\/p>\n<p><b><i>Step 6: <\/i><\/b><i><span style=\"font-weight: 400;\">To further interact with each region more easily, let us create a list of all the sections and connect them to the <\/span><\/i><b><i>input port<\/i><\/b><i><span style=\"font-weight: 400;\"> of the <\/span><\/i><b><i>List<\/i><\/b><i><span style=\"font-weight: 400;\"> node. To make multiple selections, <\/span><\/i><b><i>hold Shift<\/i><\/b><i><span style=\"font-weight: 400;\"> and drag a wire from the item output port into the List node.<\/span><\/i><\/p>\n<p><b><i>Step 7: <\/i><\/b><i><span style=\"font-weight: 400;\">Now, we create the index range. Add a <\/span><\/i><b><i>Range (Input)<\/i><\/b><i><span style=\"font-weight: 400;\"> node and set:<\/span><\/i><\/p>\n<ul>\n<li style=\"font-weight: 400;\"><b><i>Min:<\/i><\/b><i><span style=\"font-weight: 400;\"> 0<\/span><\/i><\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\"><b><i>Max:<\/i><\/b><i><span style=\"font-weight: 400;\"> 6 (since we have 7 surfaces, the index runs from 0 to 6)<\/span><\/i><\/li>\n<\/ul>\n<ul>\n<li style=\"font-weight: 400;\"><b><i>Step Count:<\/i><\/b><i><span style=\"font-weight: 400;\"> 1<\/span><\/i><i><span style=\"font-weight: 400;\"><br \/>\n<\/span><\/i><i><span style=\"font-weight: 400;\"> Rename the node to <\/span><\/i><b><i>Input<\/i><\/b><i><span style=\"font-weight: 400;\">, and connect it to the <\/span><\/i><b><i>Index<\/i><\/b><i><span style=\"font-weight: 400;\"> port.<\/span><\/i><\/li>\n<\/ul>\n<p><b><i>Step 8: <\/i><\/b><i><span style=\"font-weight: 400;\">Add an <\/span><\/i><b><i>Extrude Surface<\/i><\/b><i><span style=\"font-weight: 400;\"> node and connect the <\/span><\/i><b><i>List Item<\/i><\/b><i><span style=\"font-weight: 400;\"> output to its <\/span><\/i><b><i>Surface<\/i><\/b><i><span style=\"font-weight: 400;\"> input. You can now freely select any region using the index range.<\/span><\/i><\/p>\n<p><b><i>Step 9: <\/i><\/b><i><span style=\"font-weight: 400;\">To control the height of the section you want to extrude, add a <\/span><\/i><b><i>Vector Z<\/i><\/b><i><span style=\"font-weight: 400;\"> node and connect it to a <\/span><\/i><b><i>Range (Input)<\/i><\/b><i><span style=\"font-weight: 400;\"> node. This slider will allow you to choose your desired extrusion height.<\/span><\/i><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-large wp-image-11829 aligncenter lazyload\" src=\"data:image\/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==\" data-src=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-1024x431.png\" alt=\"\" width=\"1024\" height=\"431\" data-sizes=\"auto\" data-srcset=\"https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-1024x431.png 1024w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-300x126.png 300w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-768x323.png 768w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-1536x647.png 1536w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-800x337.png 800w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029-20x8.png 20w, https:\/\/beegraphy.com\/blog\/wp-content\/uploads\/2025\/12\/Screenshot-2025-12-08-161029.png 1905w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/p>\n<p><b><i>Step 10: <\/i><\/b><i><span style=\"font-weight: 400;\">Once everything is set up, head to <\/span><\/i><b><i>Demo<\/i><\/b><i><span style=\"font-weight: 400;\"> (top-right corner). You now have a fully interactive workspace where you can select any region and extrude it vertically to better understand its structure.<\/span><\/i><\/p>\n<p><a href=\"https:\/\/beegraphy.com\/market\/product\/venn-diagram-d82?shareData=%7B%228882b93c-b411-4e8a-824f-719bb4b6e70f%22%3A%7B%22value%22%3A%7B%22type%22%3A%22Number%22%2C%22value%22%3A2%7D%7D%2C%2284cdcb7a-6313-4782-a84a-558d841e1f1c%22%3A%7B%22value%22%3A%7B%22type%22%3A%22Number%22%2C%22value%22%3A3.5%7D%7D%7D\"><b>You can try out the Demo for the above here.<\/b><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><iframe id=\"model-692fc1f5413dfbd13ff86070\" style=\"border: none; border-radius: 12px;\" src=\"https:\/\/beegraphy.com\/embed\/692fc1f5413dfbd13ff86070\" width=\"1210\" height=\"860\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h3><span class=\"ez-toc-section\" id=\"About_Jiyun_Ko_MathDesigner\"><\/span>About Jiyun Ko (MathDesigner)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Jiyun Ko (MathDesigner) is a mathematics educator, researcher, and design-driven innovator whose work bridges mathematics, visualization, and educational technology.<\/p>\n<p>She is the CEO of GoMath Academy. Previously she has been worked as a developer of patented experimental math curricula and visual problem-solving systems, and a project manager for national R&amp;D initiatives on climate-change technologies.<\/p>\n<p>With an academic background spanning mathematics, industrial design, and educational technology, she received her PhD from from Korea University, Kookmin University where her research focuses on enhancing mathematical self-efficacy and developing visualization frameworks for math education.<\/p>\n<p>Her influence extends through academic research, admissions consulting, large-scale exam development, and frequent media features, including interviews across Korea and columns for Monthly Sisa News.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Overview This is a translated version of the Original Blog Written by Jiyun Ko Cantor once said, \u201cThe essence of mathematics lies in its freedom.\u201d (Das Wesen der Mathematik liegt in ihrer Freiheit) This is the famous statement of Georg Cantor, the founder of set theory, which forms the foundation of modern mathematics. But what [&hellip;]<\/p>\n","protected":false},"author":22,"featured_media":12024,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[99,417],"tags":[144,141,139],"class_list":["post-11791","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-parametric-education","category-tutorials","tag-mathprojects","tag-mathvisualization","tag-mathworkshop"],"_links":{"self":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts\/11791","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/comments?post=11791"}],"version-history":[{"count":62,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts\/11791\/revisions"}],"predecessor-version":[{"id":12153,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/posts\/11791\/revisions\/12153"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/media\/12024"}],"wp:attachment":[{"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/media?parent=11791"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/categories?post=11791"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/beegraphy.com\/blog\/wp-json\/wp\/v2\/tags?post=11791"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}