Timeline for Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 29, 2021 at 17:03 | vote | accept | Giulio Belletti | ||
| Jan 29, 2021 at 16:22 | comment | added | Joseph O'Rourke | @GiulioBelletti: I think the discrepancy is that all is "up to projective equivalence," and all tetrahedra are projectively equivalent. | |
| Jan 29, 2021 at 15:15 | comment | added | Giulio Belletti | I'm still confused about the dimension. If the space of realizations of a graph is $\mathbb{R}^{E-6}$ then for the tetrahedron it would be a single point; however I expect there to be 12 degrees of freedom for tetrahedra in $\mathbb{R}^3$ (6 if we consider them up to isometry) | |
| Jan 29, 2021 at 13:48 | comment | added | Joseph O'Rourke | @GiulioBelletti: 2nd try on your followup. One proof approach for Steinitz's theorem is via framework equilibrium stresses. Planar graphs with a self-stress with positive weights are Schlegel diagrams, which when lifted form a polyhedron. Varying the weights results in different realizations. The space of realizations is $\mathbb{R}^{E-6}$ where $E$ is the number of edges. | |
| Jan 29, 2021 at 13:36 | history | edited | Joseph O'Rourke | CC BY-SA 4.0 |
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| Jan 29, 2021 at 12:46 | comment | added | Joseph O'Rourke | @GiulioBelletti: I would have to look up the original Satz to be certain, but I believe Steinitz also proved the realization space is $\mathbb{R}^k$ where $k$ depends on the number of edges. (I realize this does not fully answer your followup question.) | |
| Jan 29, 2021 at 12:31 | comment | added | Giulio Belletti | Thank you for your answer! I have a follow-up: I have seen Steinitz's Theorem stated roughly as "A graph is the $1$-skeleton of a $3$-polytope if and only if it is planar and $3$-connected". How does this relate to the Isotopy property? | |
| Jan 29, 2021 at 12:24 | history | edited | Joseph O'Rourke | CC BY-SA 4.0 |
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| Jan 29, 2021 at 12:17 | history | answered | Joseph O'Rourke | CC BY-SA 4.0 |