Timeline for Which polytopes can be deformed while keeping their edge-lengths?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 12, 2020 at 3:15 | comment | added | Manfred Weis | maybe looking structural rigidity is helpful. In the linked article it is stated that "In any dimension, the rigidity of rod-and-hinge linkages is described by a matroid" | |
| Oct 11, 2020 at 0:23 | comment | added | M. Winter | What I take a away from this discussion in the comments is, that there are many more flexible polytopes than what I have naively assumed. So I suppose I have to rethink this question. I still think it would be interesting to ask whether there are any "unexpected" flexible polytopes, that are not polygons, zonotopes, or are created by products, Minkowski sums, gluing, or any other (not yet considered) "trivial" process. But this should certainly be made more precise. In any way, thanks to all of you for your input. | |
| Oct 10, 2020 at 20:50 | comment | added | Ilya Bogdanov | Perhaps, it is also better to search for “minimal” ones. E.g., you may glue anything to a facet of a cube, and it remains flexible (although with less freedom). Perhaps, you are not interested in such examples? | |
| Oct 10, 2020 at 17:29 | comment | added | M. Winter | @MattF. Right. A prism over any flexible polytope is flexible. Probably the same holds for the cartesian product of a flexible polytope and an arbitrary polytope. What about Minkowsi sums? It might be useful to find the "elementary" flexible polytopes. | |
| Oct 10, 2020 at 16:53 | comment | added | user44143 | What about a pentagonal cylinder, i.e. $\{(\cos 2\pi n/5, \sin 2\pi n/5, \pm 1)\}$? Or a cylinder of any other polygon? | |
| Oct 10, 2020 at 13:55 | comment | added | Joseph O'Rourke | Sure, please edit as you see fit. | |
| Oct 10, 2020 at 12:22 | history | edited | M. Winter | CC BY-SA 4.0 |
added 67 characters in body
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| Oct 10, 2020 at 12:18 | comment | added | M. Winter | @JosephO'Rourke Oh I see. I suspect you mean to generate a polytope $$Z=\sum\mathrm{conv}\{-r_i,r_i\}$$ for some (generic) unit vectors $r_i$, and then you change these unit vectors continuously. So the question might be whether there are any that are not zonohedra/zonotopes (or polygons)? Is it still okay for me to edit the question? | |
| Oct 10, 2020 at 12:15 | comment | added | Joseph O'Rourke | Possibly: All zonohedra? | |
| Oct 10, 2020 at 12:09 | history | asked | M. Winter | CC BY-SA 4.0 |