# Questions tagged [polytopes]

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### What's the length of the edge of a regular $d+1$ dimensional polytope with $K$ vertices on a $d$-sphere? [closed]

What's the length of the edge of a regular $d+1$ dimensional polytope with $K$ vertices that lie on a unit $d$-sphere ? Thank you for your help ! ----- Edited ---- I might add some more information to ...
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### Can every non-hemi uniform polytope tile hyperbolic space?

Can every uniform polytope which is not a hemipolytope tile hyperbolic space on its own?
1 vote
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### Are cells of 4-polytopes a convex polyhedron by definition?

I'm going by the Wikipedia definition for a 4-polytope. Do by definition, cells of 4-polytopes have to be a convex polyhedra? If not, then are there polyhedra with non-convex faces? If yes, is it the ...
42 views

### Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form $$M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\}$$ where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$. Goal I am looking for an (explicit) representation ...
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### Looking for clarification of C-H Sah's definition of abstract scissors congruence

In C-H Sah's book Hilbert's third problem: scissors congruence, the author defines the data for abstract scissors congruence in order to prove Zylev's theorem by combinatorial means in great ...
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### Realizing spherical complexes as convex polytope

A spherical polytope is the intersection of some closed hemispheres which is non-empty and does not contain a pair of antipodal points. A spherical complex is a tiling of the whole (d−1)-dimensional ...
51 views

### Realizability of abstract polytopes

What are the conditions that allow an abstract polytope to have a non-skew (but not necessarily convex) realization in Euclidean space?
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### Finding uniform polytopes in 3D versus in 4D

How was the exhaustive uniform polyhedron search (which found the great dirhombicosidodecahedron) conducted? What’s so difficult about doing the same thing but in four dimensions?
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### Simplicial polytope with regular cones

Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The ...
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### Possible new convex uniform polytope

Does there exist a convex uniform 9-polytope obtained by diminishing the 9-hypercube, removing 480 of its 512 vertices and turning each 8-hypercube facet into an 8-orthoplex?
197 views

### Triangulation of a simplex

I am looking for a triangulation of an $n$-dimensional simplex such that all sub-simplices are of comparable size, and are "as close as possible" to a regular simplex : the latter property ...
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### Dodecahedron deformation II

(Follow-up to this question) Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
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### Can a dodecahedron be deformed into a great stellated dodecahedron?

Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
1 vote
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### Graph diameter of the omnitruncated $E_8$ polytope

What is the graph diameter of the 1-skeleton of the omnitruncate of the $E_8$ family of uniform 8-polytopes?
86 views

### Expanded 24-simplices in the Leech polytope

The vertex coordinate set for the contact polytope of the Leech lattice listed on Wikipedia contains all permutations of: $\{4,-4,0^{22}\}$ $\{-3,1^{23}\}$ $\{3,-1^{23}\}$ The convex hull of these ...
168 views

### Textbooks/References for Solid Angles? [closed]

Are there any good textbooks that consider the properties of solid angles for polytopes? Being not the most well-versed in geometry, I am unsure of where to start looking. Thank you very much!
1 vote
41 views

### Lattice deformations of regular polytopes

It is trivial to see that the 24-cell, all hypercubes, and all polytopes with simplicial facets, can be deformed into lattice polytopes, and this blog post implies the same is true for the ...
1 vote
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### Diminishing of the $4_{21}$

One of the projections of the $4_{21}$ polytope (https://en.m.wikipedia.org/wiki/4_21_polytope) into four dimensions positions its vertices as those of two concentric 600-cells scaled by the golden ...
52 views

### Inverting "codimension matrix" for polytopes?

Let $P$ be an abstract polytope. Let's construct its square matrix $A$ as follows. Its lines and columns are labelled by all faces of $P$, of all dimensions. Put $A(F_1,F_2)=t^m$ if $F_1$ is a subface ...
104 views

### Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope? Here, I take a polytope to be a two-dimensional surface that could be ...
66 views

### Regiment map from Coxeter-Dynkin diagram

The following problem arises in the attempt to enumerate uniform polytopes: Given a Wythoffian polytope, what are all other Wythoffian polytopes which use maximal subsets of its vertices and edges, ...
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### If all 2-faces of a polytope are $2n$-gons, is the edge-graph bipartite?

This question on MSE has not received a satisfying answer. It can be summarized as follows: Question: Is is true that the edge-graph of a (convex) polytope is bipartite if and only if all 2-faces ...
86 views

### Uniquely describing a polytopal complex by prescribing the local structure around its vertices

Let $C$ be a $d$-dimensional (abstract) polytopal complex. Most of what I say below could be asked in this general setting, but for a start, let's further restrict to simple polytopal spheres, that is,...
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### Why are we interested in permutahedra, associahedra, cyclohedra, ...?

The following families of polytopes have received a lot of attention: permutahedra, associahedra, cyclohedra, ... My question is simple: Why? As I understand, at least the latter two were ...
1 vote
135 views

### The Fano plane, stericated 6-simplex, and pentallated 6-simplex

According to this link: https://en.wikipedia.org/wiki/Stericated_6-simplexes the stericated 5-simplex "scal" has 105 vertices defined as permutations of (0,0,1,1,1,1,2). In the course of my team's ...
Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope. Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in \$\mathrm{Aut}(P)\subset\mathrm O(\...