Timeline for A special tessellation
Current License: CC BY-SA 3.0
21 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| S Nov 22, 2013 at 11:22 | history | bounty ended | mathlove | ||
| S Nov 22, 2013 at 11:22 | history | notice removed | mathlove | ||
| Nov 22, 2013 at 11:22 | vote | accept | mathlove | ||
| Nov 22, 2013 at 9:57 | answer | added | user25199 | timeline score: 2 | |
| Nov 22, 2013 at 5:13 | comment | added | mathlove | @Carl: Interesting! I added the "billiards" tag. I think "billiards" is connected only with the condition 1. I'm very interested in the following questions : Does there exist a property such that the polygons which satisfy only condition 1 have as 'polygonal billard tables'? How about the polygons which satisfy both condition 1 and 2? Please let me know any relevant papers if you know some. | |
| Nov 22, 2013 at 4:47 | history | edited | mathlove |
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| Nov 22, 2013 at 0:05 | answer | added | Nathan Reading | timeline score: 7 | |
| Nov 20, 2013 at 17:34 | comment | added | user25199 | Your question has a nice connection to billiards (which you might include as a tag). Instead of reflecting the path of the billiard particle, it is often useful to reflect the table. Certainly in 2D and I think beyond, this question generates exactly the set of polygonal billiard tables with integrable ("regular") dynamics. | |
| Nov 16, 2013 at 15:27 | comment | added | Benoît Kloeckner | The group generated by the reflexions along faces of your polyhedron must be a finite group in order to get a tilling; you should therefore consider the classical classification of finite subgroups of $O(3)$. | |
| Nov 16, 2013 at 13:55 | comment | added | Benjamin Dickman | One difficulty is that there isn't a general formula like $\pi(n-2)$, which was nicely exploited in the answer for your first question, for the sum of the interior angles of a polyhedron. If you generalize the notion of an interior angle of a polygon to, say, a solid angle (see, e.g., Wikipedia), then you might find the following paper of interest: Barnette, D. (1972). The sum of the solid angles of a d-polytope. Geometriae dedicata, 1(1), 100-102. | |
| S Nov 16, 2013 at 11:44 | history | bounty started | mathlove | ||
| S Nov 16, 2013 at 11:44 | history | notice added | mathlove | Draw attention | |
| Nov 15, 2013 at 9:54 | history | edited | mathlove | CC BY-SA 3.0 |
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| Nov 14, 2013 at 10:00 | history | edited | mathlove | CC BY-SA 3.0 |
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| Nov 14, 2013 at 7:39 | answer | added | Gjergji Zaimi | timeline score: 17 | |
| Nov 14, 2013 at 5:33 | history | edited | mathlove | CC BY-SA 3.0 |
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| Nov 13, 2013 at 14:44 | comment | added | mathlove | @j.c. Thank you very much for introducing the book. I'm going to edit. | |
| Nov 13, 2013 at 12:53 | comment | added | j.c. | It seems what you are after is related to tilings generated by reflection symmetries, so I would suggest reading about Poincaré's polyhedron theorem. There is an accessible account in Bonahon's book "Low-Dimensional Geometry". | |
| Nov 13, 2013 at 12:52 | comment | added | j.c. | Your operations are quite hard to follow - can you please edit to make it more precise? In operation 1, I take it that $P_0$ is a convex $n$-gon - you did not state this explicitly? Then in operation 2, are you constructing the reflections of $P_0$ in each of its edges? And in what follows I think you should use a better notation or set of names than "Let these $P$s be $P_1$" which seems at odds with your earlier conventions. | |
| Nov 13, 2013 at 11:31 | history | asked | mathlove | CC BY-SA 3.0 |