{"id":2372,"date":"2021-09-11T08:25:34","date_gmt":"2021-09-11T12:25:34","guid":{"rendered":"https:\/\/josmfs.net\/?p=2372"},"modified":"2021-10-02T12:51:01","modified_gmt":"2021-10-02T16:51:01","slug":"existence-proofs","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2021\/09\/11\/existence-proofs\/","title":{"rendered":"Existence Proofs"},"content":{"rendered":"

\"\"Here is a seemingly simple problem from Futility Closet<\/em>.<\/p>\n

\u201cA quickie from Peter Winkler\u2019s Mathematical Puzzles<\/em>, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?\u201d<\/p>\n

Technically we might rephrase this as, can we inscribe a flat map of West Virginia in a square, since the boundary of most states is probably not differentiable everywhere, that is, has a tangent everywhere.<\/p>\n

But the real significance of the problem is that it is an example of an \u201cexistence proof\u201d, which in mathematics refers to a proof that asserts the existence<\/em> of a solution to a problem, but does not (or cannot) produce the solution itself.\u00a0 These proofs are second in delight only to the \u201cimpossible proofs\u201d which prove that something is impossible, such as trisecting an angle solely with ruler and compass.<\/p>\n

Here is another classic example (whose origin I don\u2019t recall).\u00a0 Consider the temperatures of the earth around the equator.\u00a0 At any given instant of time there must be at least two antipodal points that have the same temperature.\u00a0 (Antipodal points are the opposite ends of a diameter through the center of the earth.)<\/p>\n

See Existence Proofs<\/a> (revised)<\/p>\n

(Update 10\/2\/2021) <\/strong>I fixed a minor typo: “tail” should have been “head”<\/p>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here is a seemingly simple problem from Futility Closet. \u201cA quickie from Peter Winkler\u2019s Mathematical Puzzles, 2021: Can West Virginia be inscribed in a square? That is, is it possible to draw some square each of whose four sides is tangent to this shape?\u201d Technically we might rephrase this as, can we inscribe a flat […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[226,228,224,16,43,227,225],"class_list":["post-2372","post","type-post","status-publish","format-standard","hentry","category-math-inquiries","tag-brouwer","tag-differential-equations","tag-fixed-points","tag-futility-closet","tag-james-tanton","tag-mathologer","tag-winding-number"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2372","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=2372"}],"version-history":[{"count":3,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2372\/revisions"}],"predecessor-version":[{"id":2421,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/2372\/revisions\/2421"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=2372"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=2372"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=2372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}