All Questions
7 questions
18
votes
2
answers
573
views
Can the graph of a symmetric polytope have more symmetries than the polytope itself?
I consider convex polytopes $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-...
- 13.6k
10
votes
2
answers
255
views
Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
- 13.6k
10
votes
3
answers
500
views
Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?
Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$....
- 13.6k
5
votes
2
answers
1k
views
regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations
While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...
- 3,630
4
votes
2
answers
254
views
Does the edge-graph of a centrally symmetric polytope determine which vertices are antipodal?
Given two origin symmetric convex polytopes $P_1$ and $P_2$ (that is $P_i=-P_i$) with the same edge-graph, but potentially of different dimensions and combinatorial types.
Let $\phi: G_{P_1}\to G_{P_2}...
- 13.6k
4
votes
0
answers
132
views
Can a polytopal graph be "centrally symmetric" in more than one way?
Let $P,Q$ be two centrally symmetric convex polytopes, potentially of different dimensions and combinatorial type, but with the same edge-graph $G$.
The central symmetry of $P$ induces an involutory ...
- 13.6k
3
votes
3
answers
390
views
Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
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