{"id":13,"date":"2012-07-16T09:08:02","date_gmt":"2012-07-16T14:08:02","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=13"},"modified":"2012-07-16T22:29:38","modified_gmt":"2012-07-17T03:29:38","slug":"about-the-hedronometry-category","status":"publish","type":"post","link":"https:\/\/daylateanddollarshort.com\/bloog\/about-the-hedronometry-category\/","title":{"rendered":"About the “Hedronometry” Category"},"content":{"rendered":"

By the end of high school, most students should be familiar with the “three-dimensional” Pythagorean theorem (aka, “the distance formula”) that relates the length of the diagonal of a shoebox to the lengths of its edges. (Students should also know that this generalization extends to relate diagonals and edges in “any-dimensional” shoeboxes; always, the square of the diagonal equals the sum of the squares of the edges<\/em> at a corner.)<\/p>\n

However, view students seem to be aware of this result:<\/p>\n

<\/div>\n

Given a “right-corner tetrahedron”, the square\u00a0of the area<\/strong> of the face opposite the right corner is equal to the sum of the squares\u00a0of the areas<\/strong> of the other three faces.<\/p>\n

\"Pythagorean<\/a><\/p>\n

\\(X^2+Y^2+Z^2=R^2\\)<\/a><\/p><\/blockquote>\n

This result extends to “any-dimensional” space as simply as the distance formula: the square of the hypotenuse is equal to the sum of the squares of the legs<\/em>. However, unlike the distance formula —which only ever relates one-dimensional lengths— the values being squared here are\u00a0increasingly “dimensionally-enhanced” aspects of a figure: lengths in 2-d, areas in 3-d, volumes in 4-d, hyper-volumes in 5-d, and so forth.<\/p>\n

I was thrilled as a high school junior when I discovered this result, and I was devastated as a college freshman when I discovered that others had beaten me to it by at least a century. Nevertheless, in the years —decades!— since, I’ve worked off and on to find related, completely new, results in the field I call “(Tetra)Hedronometry”, the dimensionally-enhanced trigonometry of tetrahedra. I’ve had a bit of success —in particular, in the realm of non-Euclidean geometry, where (so far as I can tell) absolutely nothing was previously known— and I believe my focus on tetrahedral “pseudo-faces” is a significant contribution to tetrahedral lore. We’ll see.<\/p>\n","protected":false},"excerpt":{"rendered":"

By the end of high school, most students should be familiar with the “three-dimensional” Pythagorean theorem (aka, “the distance formula”) that relates the length of the diagonal of a shoebox to the lengths of its edges. (Students should also know that this generalization extends to relate diagonals and edges in “any-dimensional” shoeboxes; always, the square […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-13","post","type-post","status-publish","format-standard","hentry","category-hedronometry"],"_links":{"self":[{"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/13","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/comments?post=13"}],"version-history":[{"count":8,"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/13\/revisions"}],"predecessor-version":[{"id":30,"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/13\/revisions\/30"}],"wp:attachment":[{"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=13"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=13"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=13"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}