{"id":574,"date":"2012-09-16T18:05:10","date_gmt":"2012-09-16T23:05:10","guid":{"rendered":"http:\/\/daylateanddollarshort.com\/bloog\/?p=574"},"modified":"2012-09-21T20:31:57","modified_gmt":"2012-09-22T01:31:57","slug":"what-i-know-about-hyperbolic-tetrahedra","status":"publish","type":"post","link":"http:\/\/daylateanddollarshort.com\/bloog\/what-i-know-about-hyperbolic-tetrahedra\/","title":{"rendered":"What I know about hyperbolic tetrahedra"},"content":{"rendered":"\n

Inspired by Mednykh and Pashkevich’s “Elementary Formulas for a Hyperbolic Tetrahedron”<\/a>, I have compiled most of my disparate notes about hyperbolic hedronometry into one document: “Hedronometric Formulas for a Hyperbolic Tetrahedron”<\/a>. I consider this a “living document” that I will update as I learn more about the subject matter.<\/p>\n

It’s primarily a formula look-up list for myself, so I’ve left out a lot of exposition and proof. There may be a few typos, math-os, or outright errors. (There are at least two places where my results disagree with Mednykh and Pashkevich. One is a sign discrepancy, and the other an errant square root.) So, caveat reador<\/em>.<\/p>\n

As for content, “Hedronometric Formulas”<\/a>\u00a0covers familiar territory, from the foundational Laws of Cosines to the roadblock that is volume. There are also a few never-before-noted results, primarily involving “pseudo-altitudes” (each of which is a segment perpendicular to a pair of opposite edges); perhaps the coolest of these is the “Law of Side-Angle-Side-Altitude Sines”, which asserts that the product<\/p>\n

$$\\sinh{(\\text{edge})} \\;\\cdot\\; \\sinh{(\\text{edge})} \\;\\cdot\\; \\sin{(\\text{angle})} \\;\\cdot\\; \\sinh{(\\text{altitude})}$$<\/p>\n

is a metric invariant of a hyperbolic tetrahedron, having a constant value (related to the Gram determinant) for any choice of two distinct “edge”s and the “angle” and “altitude” they determine. (For adjacent edges, “angle” is the angle between them and “altitude” is the altitude to the face they bound; for opposite edges, “altitude” is the corresponding pseudo-altitude between them, and “angle” is the “twist” of those edges about the pseudo-altitude.) This is analogous to the Euclidean result in which “edge \\(\\cdot\\) edge \\(\\cdot\\) sin(angle) \\(\\cdot\\) altitude” always gives “\\(6 \\cdot \\text{volume}\\)”, which is also valid across standard and pseudo elements.<\/p>\n

I also provide an appendix describing a tetrahedron in coordinatized hyperbolic space. “Coordination” happens to have provided the route to proving key results about pseudo-altitudes and twists, but generally it seems even messier than Euclidean analytic geometry can often be.<\/p>\n

Some\u00a0Open Questions<\/strong>\u00a0raised (or repeated) here:<\/p>\n