# Questions tagged [polytopes]

The polytopes tag has no usage guidance.

100
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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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32
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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?

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33
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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 ...

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42
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### 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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1
answer

104
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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 ...

5
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1
answer

123
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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 ...

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51
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### 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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29
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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?

4
votes

1
answer

52
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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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72
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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?

4
votes

1
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197
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### 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 ...

2
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77
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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?

6
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1
answer

227
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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
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1
answer

89
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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?

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86
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### 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 ...

3
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0
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168
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### 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!

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### 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 ...

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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 ...

2
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52
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### 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 ...

2
votes

1
answer

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 ...

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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, ...

2
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54
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### An easy way to recognize the edges of an orbit polytope?

Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is
$$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...

0
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79
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### Number of vertices in a polyhedron

Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...

3
votes

1
answer

146
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### Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...

3
votes

0
answers

48
views

### Classifying/enumerating vertex-transitive simplicial polytopes

I'm interested in understanding the class of simplicial polytopes in $\mathbb R^n$ whose Euclidean isometry group $G$ acts transitively on the vertices. These are examples that I know of:
simplicial ...

5
votes

3
answers

488
views

### Convex lattice polygons with equal area and perimeter

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon.
Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter?
...

4
votes

1
answer

110
views

### A combinatorial characterization of the central inversion of a polytope?

Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...

1
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0
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50
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### Classification of pseudoregular polyhedra

In contrast to a regular polyhedron, which has one orbit of flags, I’ve been studying what I call pseudoregular polyhedra, which have two orbits of flags interchanged by conjugation (explained here). ...

3
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### 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 $\{...

3
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0
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### 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 ...

6
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1
answer

418
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### What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?

Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...

5
votes

1
answer

193
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### Is there a polytope with an essentially unique shape?

More percisely:
Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations?
I suppose I have to assume that it has more than $d+2$ vertices/facets if ...

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2
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336
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### Do Bernoulli polynomials know about face vectors?

This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the ...

3
votes

1
answer

91
views

### Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...

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1
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699
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### Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that
all vertices end up on a common sphere, and
the ...

4
votes

3
answers

270
views

### Minimal data required to determine a convex polytope

Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...

1
vote

0
answers

47
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### What are the corners of this polytope?

Let $f$ be a non-negative function on the positive integers such that $f(s+t)\geq f(s) + f(t)$ for all $s,t\in\mathbb{Z}^+$. Consider the polytope consisting of all $x\in \mathbb{R}^n$ such that $$\...

4
votes

0
answers

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### 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\...

6
votes

1
answer

143
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### 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 ...

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### Is a polytope that has in-spheres for faces of all dimensions already regular?

Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points).
A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...

5
votes

1
answer

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### What is known about the duals of cyclic polytopes?

What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...

3
votes

1
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### Reference for "every 5-dimensional polytope has a 3-gonal or 4-gonal face"

It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.
Alternatively, I would be ...

5
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2
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### Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?

The question is in the title:
Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?
I consider only convex polytopes (convex ...

2
votes

0
answers

80
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### Polytopes with large dihedral angles

The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$.
The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...

4
votes

0
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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 ...

2
votes

1
answer

86
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### 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,...

29
votes

7
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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

1
answer

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### 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 ...

2
votes

1
answer

83
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### A matrix that commutes with all symmetries of a vertex-transitive polytope

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(\...

0
votes

0
answers

40
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### Every point in a regular polytope has its own antipodal point or antipodal face

I apologize for using non-common language. When this problem comes to my mind, it seems quite easy but It's not.
Maybe It can be rewritten as,
There exists a unique facet containing the most far ...