All Questions
13 questions
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
8
votes
2
answers
278
views
Symmetries of contractable subsets of $\Bbb R^n$
Let $K\subset\Bbb R^n$ be a non-empty compact subset of $\Bbb R^n$. A symmetry of $K$ is an isometry of $\Bbb R^n$ that fixes $K$ set-wise. Since $K$ is compact, there is always a point $x\in\Bbb R^n$ ...
- 13.6k
3
votes
0
answers
40
views
Are there uniform compounds of 135 $BC_8$ polytopes?
The Coxeter group $D_8$ is an index-135 subgroup of $E_8$. One of the consequences of this is that the rectified 8-orthoplex, whose coordinates can be given as all permutations and sign changes of $\{...
- 2,773
3
votes
0
answers
103
views
Are there any other regular compounds?
Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected ...
- 2,773
1
vote
0
answers
45
views
How dense can a transitive sets of points be?
How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...
- 13.6k
4
votes
0
answers
114
views
Can we combine the symmetries of two polytopes to create a more symmetric polytope?
Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$.
The symmetry group $\mathrm{Aut}(P_i)\subset\...
- 13.6k
6
votes
1
answer
212
views
A polytope with congruent facets and an insphere that is not facet-transitive?
Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?
All facets are congruent,
it has an ...
- 13.6k
9
votes
0
answers
100
views
A characterization of root systems via their intersections with halfspaces
In a recent preprint I obtained a nice characterization of root systems as a side product.
I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
- 13.6k
4
votes
0
answers
49
views
Equiangular lines with symmetry requirements
Listing all possible arrangements of equiangular lines is non-trivial.
Does the problem become any easier when we additionally require that the symmetry group of that line arrangement acts ...
- 13.6k
2
votes
4
answers
997
views
Why does $\sqrt 5$ occur in manageable situations of these scenarios? [closed]
Banach-Mazur distance between $P_5$ and $P_3$ is $d(P_5,P_3)=1+\frac{\sqrt5}2$ where $P_n$ is regular polygon in $n$ sides https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7968198&tag=...
- 1,826
3
votes
2
answers
344
views
Is a vertex- and edge-transitive polytope already a uniform polytope?
I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive).
Question: Is every such ...
- 13.6k
3
votes
1
answer
236
views
Non-inherited symmetries of shadows of point sets
Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality).
This requires a ...
- 1,277