{"id":217,"date":"2012-11-24T05:54:24","date_gmt":"2012-11-24T11:54:24","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=217"},"modified":"2012-11-24T17:54:01","modified_gmt":"2012-11-24T23:54:01","slug":"a-hedronometric-theorem-of-menger","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/a-hedronometric-theorem-of-menger\/","title":{"rendered":"A Hedronometric Theorem of Menger"},"content":{"rendered":"\n

In 1928, Karl Menger outlined necessary and sufficient conditions for a set of edge lengths to determine an actual, non-degenerate, tetrahedron. The conditions amount to dead-simple sanity checks that the consequent face areas and volume have to be positive real numbers.<\/p>\n

In the short note “A Hedronometric Theorem of Menger”<\/a>, I derive (as the title suggests) a hedronometric counterpart of Menger’s result, giving necessary and sufficient conditions for a set of face and pseudoface areas to determine a tetrahedron. It’s really about time I did.<\/p>\n

The note came about when I observed a “Heronic duality” that I had never noticed (nor even suspected) before. Writing “\\([\\bullet]\\)” for what I’ll call the Heronic product,<\/p>\n

$$[x,y,z] := \\left(x+y+z\\right)\\left(-x+y+z\\right)\\left(x-y+z\\right)\\left(x+y-z\\right)$$<\/p>\n

we know that the area, \\(A\\), of a triangle with edge lengths \\(x, y, z\\) is given by<\/p>\n

$$16 A^2 = [x,y,z]$$<\/p>\n

That’s Heron’s formula (hence, the term “Heronic product”), and it gives us areas from edge lengths.<\/p>\n

As it turns out, Heron products give us tetrahedral edge lengths from face and pseudoface areas (and volume); for instance, if \\(a\\) is the edge between faces \\(Y\\) and \\(Z\\), and thus also a diagonal of pseudoface \\(H\\), then<\/p>\n

$$9V^2 a^2 = \\left[H,Y,Z\\right]$$<\/p>\n

Going back the other way, we can even express the areas of tetrahedral pseudofaces as Heronic products:<\/p>\n

$$16 H^2 = \\left[a,d,j\\right] \\qquad \\text{where} \\qquad j^2 = a^2-b^2+c^2+d^2-e^2+f^2$$<\/p>\n

The Heronic product is quite the workhorse!<\/p>\n

This note answers the an\u00a0Open Question<\/strong> from my “Heron-like Results …”<\/a> post, where I wrote that I had “other reasons to suspect Cayley-Menger-like ties” to the Pseudo-Heron volume formula. There, I wondered what matrix had that formula as its determinant; in the note, that matrix is \\(\\Delta^\\star\\). (There may be a better one, but that’s the best I have right now.)<\/p>\n

Update.<\/strong> I’ve added a Corollary that gives conditions for just six faces —three standard, three pseudo— to determine a tetrahedron.<\/p>\n","protected":false},"excerpt":{"rendered":"

In 1928, Karl Menger outlined necessary and sufficient conditions for a set of edge lengths to determine an actual, non-degenerate, tetrahedron. The conditions amount to dead-simple sanity checks that the consequent face areas and volume have to be positive real numbers. In the short note “A Hedronometric Theorem of Menger”, I derive (as the title […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4,11],"tags":[],"class_list":["post-217","post","type-post","status-publish","format-standard","hentry","category-hedronometry","category-open-question"],"_links":{"self":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/217","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=217"}],"version-history":[{"count":10,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/217\/revisions"}],"predecessor-version":[{"id":225,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/217\/revisions\/225"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=217"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=217"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=217"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}