{"id":678,"date":"2020-05-12T16:23:14","date_gmt":"2020-05-12T16:23:14","guid":{"rendered":"http:\/\/blog.atlant.is\/?p=678"},"modified":"2021-02-21T12:30:41","modified_gmt":"2021-02-21T10:30:41","slug":"a-sense-of-closure","status":"publish","type":"post","link":"https:\/\/blog.douzeb.is\/?p=678","title":{"rendered":"Mathematical Cover-Up"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

Great mathematician John H. Conway<\/a> passed away Saturday April 11th, as a result of complications from COVID-19.<\/p>\n\n\n\n\n\n\n\n

He had a slyly bent sense of humour. Together, he and Princeton University colleague Alexander Soifer managed to get the world record for the shortest paper ever published in a serious Math Journal.<\/p>\n\n\n\n

The paper was published in the January 2005 issue of the American Mathematical Monthly<\/a><\/em>. Numberphile recalls the event<\/a>.<\/p>\n\n\n\n

The paper<\/a> asks the question: Can $n^2+ 1$ unit equilateral triangles cover an equilateral triangle of side $> n$, say $n + \\epsilon$?<\/em> then proceeds to give the answer: $n^2+ 2$ can<\/em>, together with figures showing two different ways of achieving the result<\/span>1<\/sup><\/a><\/span><\/span>2<\/sup><\/a><\/span>.<\/p>\n\n\n\n

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Figure 1:<\/figcaption><\/figure><\/div>\n\n\n\n
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Figure 2:<\/figcaption><\/figure><\/div>\n\n\n\n

The folks at numberphile worry that Conway and Soifer have written nothing about the possibility of an $n^2 + 1$ unit triangles cover-up: Tony Padilla (at 4 min 40): \u00ab\u00a0Yes, so that’s what I think about it. I don’t think they even answered the question.\u00a0\u00bb<\/p>\n\n\n\n

Actually, in his book How Does One Cut a Triangle?<\/a>, Soifer recollects the $n^2 + 1$ case was the easy part (\u00ab\u00a0Area considerations alone show the need for at least $n^2 + 1$ of them\u00a0\u00bb). So they needed not bother to encumber their proof with it.<\/p>\n\n\n\n

It could have gone like this:<\/p>\n\n\n\n

Consider the \u00ab\u00a0canonic\u00a0\u00bb cover-up of the triangle of side $n + \\epsilon$ with $n^2$ small triangles of side $1 + {\\epsilon \\over n}$.
Consider the combined $n^2 + 2$ vertices of these small triangles.
The distance between any two vertices is greater than or equal to $1 + {\\epsilon \\over n}$, so no two vertices belong to the same unit triangle.
Consequently a cover-up of the $n + \\epsilon$<\/em> triangle must be comprised of at least $n^2 + 2$ unit triangles.<\/em> $\\Box$<\/p>\n\n\n\n

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

Great mathematician John H. Conway passed away Saturday April 11th, as a result of complications from COVID-19.<\/p>\n","protected":false},"author":1,"featured_media":691,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[16],"tags":[],"ppma_author":[76],"authors":[{"term_id":76,"user_id":1,"is_guest":0,"slug":"fred","display_name":"Fred","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/ec0326d654fdf9f66e9eb42bb34a9bc4?s=96&d=mm&r=g","description":"","first_name":"","last_name":"","user_url":""}],"_links":{"self":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/678"}],"collection":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=678"}],"version-history":[{"count":25,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/678\/revisions"}],"predecessor-version":[{"id":3469,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/posts\/678\/revisions\/3469"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=\/wp\/v2\/media\/691"}],"wp:attachment":[{"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=678"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=678"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=678"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/blog.douzeb.is\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=678"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}