{"id":210,"date":"2012-07-18T06:45:58","date_gmt":"2012-07-18T11:45:58","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=210"},"modified":"2014-02-10T23:33:49","modified_gmt":"2014-02-11T05:33:49","slug":"what-is-the-pythagorean-theorem-for-right-corner-simplices-in-hyperbolic-4-space","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/what-is-the-pythagorean-theorem-for-right-corner-simplices-in-hyperbolic-4-space\/","title":{"rendered":"What is the Pythagorean Theorem for Right-Corner Simplices in Hyperbolic 4-Space?"},"content":{"rendered":"\n

The Pythagorean theorem in hyperbolic 2-space is fundamental, and I discovered the hedronometric 3-space analogue years ago (see here<\/a>), but what about dimension 4 and beyond?<\/p>\n

As discussed in the Heron-like Strategies for Non-Euclidean Tetrahedral Volume<\/a> post, my investigations into the 4-space case stalls-out because hyperbolic volume is defined in terms of the thorny Derevnin-Mednykh integral.<\/p>\n

In the simplest (but hardly simple) case of an “isosceles” right-corner simplex with right-corner leg-faces and a regular hypotenuse-face, one arrives at these formulas for leg volume, \\(L\\), and hypotenuse volume, \\(H\\):<\/p>\n

$$\\begin{align}L &:= \\frac{3}{2}\\int_{1\/3}^{x} \\frac{1}{\\sqrt{u\\left(1-u\\right)}} \\; \\mathrm{atanh}\\sqrt{\\frac{3u-1}{1-u}} du \\\\[0.5em] H &:= 6 \\int_{1\/3}^{x} \\frac{1}{\\sqrt{1-u^2}} \\; \\mathrm{atanh}\\sqrt{\\frac{3u-1}{1-u}} du \\end{align}$$<\/p>\n

for \\( 1\/3 \\le x \\le 1\/2 \\). \u00a0Surely, a Pythagorean relation of some kind exists (right?), but I’m at a loss for what it might be. Given the specific extreme values at \\(x=1\/2\\)<\/p>\n

$$L^\\star := \\frac{1}{2} \\sum_{k=1}^{\\infty} \\frac{1}{k^2} \\sin \\frac{\\pi k}{2} = 0.45798\\dots \\qquad H^\\star := \\sum_{k=1}^{\\infty} \\frac{1}{k^2} \\sin \\frac{\\pi k}{3} = 1.01494 \\dots$$<\/p>\n

the relationship is all-but-guaranteed to be non-trivial. (Note: \\(L^{\\star}\\) is half of Catalan’s constant; and \\(H^\\star\\) is, among other things, the maximum value of the Clausen function, \\( \\mathrm{Cl}_2 \\). Maybe mentioning that here will make this blog post show up in web searches, and people who know a little about polylogarithms can give me some insights.)<\/p>\n

Here’s a TeX’d-up version of this blind alley: “What is the Pythagorean Theorem for Right-Corner Simplices in Hyperbolic 4-Space?”<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

The Pythagorean theorem in hyperbolic 2-space is fundamental, and I discovered the hedronometric 3-space analogue years ago (see here), but what about dimension 4 and beyond? As discussed in the Heron-like Strategies for Non-Euclidean Tetrahedral Volume post, my investigations into the 4-space case stalls-out because hyperbolic volume is defined in terms of the thorny Derevnin-Mednykh […]<\/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-210","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\/210","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=210"}],"version-history":[{"count":8,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/210\/revisions"}],"predecessor-version":[{"id":674,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/210\/revisions\/674"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=210"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=210"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=210"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}