{"id":1058,"date":"2015-10-25T11:42:37","date_gmt":"2015-10-25T03:42:37","guid":{"rendered":"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/?p=1058"},"modified":"2024-02-16T21:09:13","modified_gmt":"2024-02-16T13:09:13","slug":"circles-wherever-you-look","status":"publish","type":"post","link":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/2015\/10\/circles-wherever-you-look\/","title":{"rendered":"Circles wherever you look."},"content":{"rendered":"<p>You see a silhouette of a three-dimensional shape. The silhouette is a circle. Then, you turn to get another view. The new silhouette is still a circle, so you turn again. And again. Every time you look at the shape, you see a circular outline.<\/p>\n<p>Is it a sphere?<\/p>\n<p><!--more--><\/p>\n<p>In my son&#8217;s math book, in the geometry section, there was one question like this. He was given the top, side and front view of an object, and asked &#8220;what shape is this?&#8221;<\/p>\n<p>The front, top and side views were all circles.<\/p>\n<p>Clearly, the correct answer &#8211; for elementary school level math &#8211; is that the shape is a sphere. However, like many things in elementary school math, the correct answer is a lie, a simplification made to make the subject simpler to teach.<\/p>\n<p>To show him this, I cooked up the shape shown below.<\/p>\n<p><a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-montage.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1059\" src=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-montage.png\" alt=\"NonSphere-montage\" width=\"384\" height=\"126\" srcset=\"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-montage.png 384w, https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-montage-300x98.png 300w\" sizes=\"(max-width: 384px) 100vw, 384px\" \/><\/a><\/p>\n<p>I&#8217;ve shown, here, the front view, top view and side view of the shape. All three views are circles. Is the shape a sphere?<\/p>\n<p>Probably, the shading gave the game away. Here&#8217;s the same shape, viewed from a different angle:<\/p>\n<p><a href=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-Angled.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-1060\" src=\"http:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-Angled.png\" alt=\"Non-Sphere - Angled\" width=\"642\" height=\"720\" srcset=\"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-Angled.png 642w, https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-content\/files\/2015\/10\/NonSphere-Angled-268x300.png 268w\" sizes=\"(max-width: 642px) 100vw, 642px\" \/><\/a><\/p>\n<p>Clearly, this is not a sphere. The elementary-school math book has simplified\u00a0things, as, perhaps, it must.<\/p>\n<p>So, clearly three views is not enough to be sure the shape is a sphere. Would four views help? Five? How many more do we need?<\/p>\n<p>Imagine the shape <em>was<\/em> a sphere. Each view gives me information about the sphere&#8217;s outline, as seen from one direction. The edge of the silhouette carves a great circle around the ball.<\/p>\n<p>Alas, no matter how many circles are drawn, there will still be points on the surface of the sphere that I have not seen. You can then gouge the surface of the sphere\u00a0&#8211; or even, to an extent,\u00a0fill it out, as long as you don&#8217;t touch the disks I&#8217;ve seen.<\/p>\n<p>So, to be truly sure you&#8217;re looking at a sphere, you need to see it from an infinite number of angles.<\/p>\n<p>Think about that next time you&#8217;re looking at some architectural drawings. Will this house be as it seems, or are there hidden Hogwarts-like passages and rooms hidden out of view?<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>You see a silhouette of a three-dimensional shape. The silhouette is a circle. Then, you turn to get another view. The new silhouette is still a circle, so you turn again. And again. Every time you look at the shape, you see a circular outline. Is it a sphere?<\/p>\n","protected":false},"author":1,"featured_media":1059,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[81],"tags":[282,549,547,63,548,546,544,545],"_links":{"self":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/1058"}],"collection":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/comments?post=1058"}],"version-history":[{"count":1,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/1058\/revisions"}],"predecessor-version":[{"id":1061,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/posts\/1058\/revisions\/1061"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media\/1059"}],"wp:attachment":[{"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/media?parent=1058"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/categories?post=1058"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.dr-mikes-math-games-for-kids.com\/blog\/wp-json\/wp\/v2\/tags?post=1058"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}