{"id":611,"date":"2013-02-16T08:01:10","date_gmt":"2013-02-16T14:01:10","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=611"},"modified":"2013-03-01T07:52:20","modified_gmt":"2013-03-01T13:52:20","slug":"what-more-i-know-about-hyperbolic-tetrahedra","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/what-more-i-know-about-hyperbolic-tetrahedra\/","title":{"rendered":"What more I know about hyperbolic tetrahedra"},"content":{"rendered":"\n

I have updated my note, “Hedronometric Formulas for a Hyperbolic Tetrahedron” (PDF)<\/a>, with a brand new formula for the volume of an arbitrary tetrahedron in terms of its face and pseudo-face areas. (See Section 8.3.)<\/p>\n

The formula isn’t the monolithic and symmetric counterpart to Derevnin-Mednykh I’ve been seeking, but it’s a start. It’s complicated enough that I won’t attempt to render it here.<\/p>\n

The Open Question<\/strong>: As one might expect, the formula involves an integral. One of the limits of integration is the subject of a Conjecture. Again, the notion is too complicated to describe here, but the gist is that I *believe* that, by appropriately assigning names to the faces (and pseudo-faces), we guarantee that the lower limit is simply one-quarter of a particular pseudo-face area. (If I’m mistaken, then that limit is a less-obvious root of a trigonometric equation.) Numerical experiments in Mathematica suggest that the conjecture is true, but I don’t have even non-constructive proof. (Nevermind that the conjecture wouldn’t really be helpful without a practical way to determine what an “appropriate assignment” of names would be.) When (if?) a properly symmetric formula is finally discovered, the order of face names won’t matter at all; but for now, it makes for an irksome little wrinkle in the formula.<\/p>\n","protected":false},"excerpt":{"rendered":"

I have updated my note, “Hedronometric Formulas for a Hyperbolic Tetrahedron” (PDF), with a brand new formula for the volume of an arbitrary tetrahedron in terms of its face and pseudo-face areas. (See Section 8.3.) The formula isn’t the monolithic and symmetric counterpart to Derevnin-Mednykh I’ve been seeking, but it’s a start. It’s complicated enough […]<\/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-611","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\/611","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=611"}],"version-history":[{"count":4,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/611\/revisions"}],"predecessor-version":[{"id":615,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/posts\/611\/revisions\/615"}],"wp:attachment":[{"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/media?parent=611"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/categories?post=611"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/daylateanddollarshort.com\/bloog\/wp-json\/wp\/v2\/tags?post=611"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}