{"id":529,"date":"2019-06-29T22:10:10","date_gmt":"2019-06-29T22:10:10","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=529"},"modified":"2019-11-19T00:24:39","modified_gmt":"2019-11-19T00:24:39","slug":"equidecomposability","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/equidecomposability\/","title":{"rendered":"Equidecomposability"},"content":{"rendered":"\n<p>Two sets A and B are said to be\u00a0<em>equidecomposable<\/em>\u00a0if you can partition set A into a finite number of subsets and reassemble them (by rigid motions only) to form set B.<\/p>\n\n\n\n<p>Let A be a unit circle, and let B be a unit circle with one point X missing (called a &#8220;deleted circle&#8221;). Are sets A and B equidecomposable?<\/p>\n\n\n\n<p>Believe it or not, yes! In fact you can do it using just 2 subsets. Can you figure out how?<\/p>\n\n\n\n<p><strong>Presentation&nbsp;Suggestions:<\/strong><br>Some students will be tempted to &#8220;push together&#8221; the ends of the deleted circle, but this is not a rigid motion, and because of the openness of the endpoints, the ends will never &#8220;meet&#8221; unless they intersect.<\/p>\n\n\n\n<p><strong>The&nbsp;Math&nbsp;Behind&nbsp;the&nbsp;Fact:<\/strong><br>Consider set B and let U be the subset consisting of all points that are a positive integer number of radians clockwise from X along the circle. This is a countably infinite set (the irrationality of Pi prevents two such points from coinciding). Let set V be everything else.<\/p>\n\n\n\n<p>If you pick set U up and rotate it&nbsp;<em>counterclockwise<\/em>&nbsp;by one radian, something very interesting happens. The deleted hole at X gets filled by the point 1 radian away, and the point at the (n-1)-th radian gets filled by the point at the n-th radian. Every point vacated gets filled, and in addition, the empty point at X gets filled too!<\/p>\n\n\n\n<p>Thus, B may be decomposed into sets U and V, which after this reassembling, form set A, a complete circle!<\/p>\n\n\n\n<p>This elementary example forms the beginnings of the idea of how to accomplish the\u00a0Banach-Tarski paradox.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Equidecomposability.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:<\/strong>   <br>Francis Su <\/p>\n","protected":false},"excerpt":{"rendered":"<p>Two sets A and B are said to be\u00a0equidecomposable\u00a0if you can partition set A into a finite number of subsets&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[57,12,63,75],"class_list":["post-529","page","type-page","status-publish","hentry","tag-advanced","tag-other","tag-set-theory","tag-unit-circle"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/529","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=529"}],"version-history":[{"count":3,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/529\/revisions"}],"predecessor-version":[{"id":1422,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/529\/revisions\/1422"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=529"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=529"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}