Timeline for Pythagorean theorem for right-corner hyperbolic simplices?
Current License: CC BY-SA 4.0
22 events
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| May 7, 2020 at 20:09 | history | edited | YCor | CC BY-SA 4.0 |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| S Aug 8, 2013 at 2:47 | history | suggested | Michael Albanese | CC BY-SA 3.0 |
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| Dec 23, 2011 at 11:51 | history | edited | Blue | CC BY-SA 3.0 |
Added range of parameter $x$ in volume formulas
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| Dec 23, 2011 at 11:20 | history | edited | Blue | CC BY-SA 3.0 |
Added strikingly-similar leg/hypotenuse volume formulas; made minor edits to discussion.
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| Jan 1, 2011 at 3:13 | answer | added | Igor Rivin | timeline score: 4 | |
| Nov 15, 2010 at 6:31 | comment | added | Blue | @Joseph: Yup. As I mention in a previous comment, I'm aware of the D-M formula. In fact, the explicit expressions for $L$ and $H$, given in my Edit2, arise from that formula. I appreciate the pointer. | |
| Nov 15, 2010 at 0:08 | comment | added | Joseph O'Rourke | I am not sure if this will be of interest to you: "A formula for the volume of a hyperbolic tetrahedon," D. A. Derevnin and A. D. Mednykh, 2005 Russ. Math. Surv. 60 346. iopscience.iop.org/0036-0279/60/2/L07 | |
| Nov 14, 2010 at 7:32 | history | edited | Blue | CC BY-SA 2.5 |
added discussion of special case
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| Jul 19, 2010 at 1:47 | comment | added | Blue | @Victor: If this suggests that the 4-simplex case potentially gets (comparatively) easy again, I'm all for it. :) So far, though, I'm not seeing it. :( | |
| Jul 19, 2010 at 1:17 | comment | added | Victor Protsak | Re "dramatic divergence": I am not at all sure that this is relevant for your question, but there seems to be a dichotomy between even-dimensional spaces and odd-dimensional spaces (both in Euclidean and non-Euclidean setting). A classical example is the distinction between the arc length of circle and the area of a spherical belt, which was generalized by V.I.Arnold and V.Vasiliev, see e.g. ams.org/mathscinet-getitem?mr=1024727 | |
| Jul 18, 2010 at 23:13 | comment | added | Blue | Also, I'm aware of the Derevnin-Mednykh formula for hyperbolic tetrahedral volume. (It's what I've toyed with the past few days, but I've also poked at it off and on for far longer.) I can't even seem to tease out a particularly insight-providing formula for the right-corner case, let alone determine how "leg-volumes" might relate to hypotenuse-volume. | |
| Jul 18, 2010 at 23:05 | comment | added | Blue | @Willie: "Right-corner" tetrahedra (your "N-rectangular"s) aren't orthoschemes. @Agol: I'm not after the most natural class of simplices study. I'm after a Pythagorean theorem; my class of simplices is pretty much chosen for me. :) @Willie again: I plowed through a similar matrix-based derivation of the generalized Euclidean Pythagorean Theorem in the 80s; I was told then that the result was already well-known. :( | |
| Jul 18, 2010 at 22:52 | comment | added | Willie Wong | Google also brought up this presentation math.sci.hiroshima-u.ac.jp/~shimada/branched10/Mednykh/… | |
| Jul 18, 2010 at 22:32 | comment | added | Willie Wong | @Agol: I think the OP was already talking about orthoschemes? @Don: For the orthoscheme's in arbitrary dimension Euclidean space I found the formula around 10 years ago (5 years too late to be novel) dpmms.cam.ac.uk/~ww278/papers/gp.pdf I'll think about the homogeneous space case for a bit. | |
| Jul 18, 2010 at 22:14 | comment | added | Ian Agol | A more natural class of simplices is the orthoschemes. I think you can obtain such formulae by induction on dimension. These are natural because they arise from geometric versions of barycentric subdivision. en.wikipedia.org/wiki/Orthoscheme | |
| Jul 18, 2010 at 22:14 | comment | added | Blue | @Willie: (1) I've added a link to my notes deriving the formulas; it's all basic trig. (2) Curvature ±1: Yes. (That always gives the cleanest formulas, right? :) (3) Geodesically convex hull: Yes. | |
| Jul 18, 2010 at 22:07 | history | edited | Blue | CC BY-SA 2.5 |
added references
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| Jul 18, 2010 at 18:00 | history | edited | Willie Wong |
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| Jul 18, 2010 at 17:59 | comment | added | Willie Wong | Do you have a reference for the spherical and hyperbolic formulae you gave? Also, am I correct in assuming that you are fixing the representative spherical/hyperbolic manifolds to have curvature $\pm 1$, since area deviation general depends on the curvature. And for the simplices, I assume you take them to be the geodesically convex hull of the vertices? | |
| Jul 18, 2010 at 16:36 | history | asked | Blue | CC BY-SA 2.5 |