Questions tagged [polytopes]

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About the number of faces of the conification of a polytope

Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
ElliptCg's user avatar
2 votes
0 answers
68 views

Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance: https://math.stackexchange.com/a/14469/454016 Poincaré duality is explained through a duality of ...
Alexander Praehauser's user avatar
1 vote
0 answers
39 views

Parametrize regions of positivity of a polynomial

I realize that this problem is extremely generic, so I am pessimistic that there may be concrete solutions, but let me try... Consider a multi-variate polynomial $P(x)$, is it possible to find ...
giulio bullsaver's user avatar
3 votes
0 answers
85 views

Is there a 5-cell-600-cell honeycomb?

Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
Daniel Sebald's user avatar
2 votes
1 answer
52 views

Counting the number of pair of d-uplets with upper bounded distance

Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
Ludwich's user avatar
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4 votes
0 answers
186 views

Characterization of curves contained in the boundary of convex bodies

Given a continuous closed curve $\gamma$ in $\mathbb R^n$ does there exist a convex body $K$ (convex set with non-empty interior) such that $\gamma\subset \partial K$? I am looking for a reference to ...
Vadim Semenov's user avatar
1 vote
0 answers
35 views

Barnes-Wall lattices’ contact polytopes

The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?
Daniel Sebald's user avatar
1 vote
1 answer
89 views

Are simplicial polytopes a dense set?

Consider the space of non-empty, compact, and convex subsets of $\mathbb{R}^d$ equipped with the Hausdorff metric. Are simplicial polytopes a dense subset of that space? Probably this is just a ...
user avatar
4 votes
2 answers
244 views

A rational polytope that is not a 01-polytope?

A 01-polytope is the convex hull of some points $S\subseteq\{0,1\}^n$. I wonder, which polytopes can be represented (combinatorially) as 01-polytopes? There are polytopes that cannot have rational ...
M. Winter's user avatar
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4 votes
1 answer
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Given a polytope $P$ with bipartite edge-graph, if the bipartition classes are equal in size and lie on spheres, is $P$ inscribed?

Suppose that $P\subset\Bbb R^n, n\ge 3$ is a (full-dimensional) convex polytope with a bipartite edge-graph $G=(V_1\cup V_2,E)$ (for example, a zonotope). Suppose further that there are concentric ...
M. Winter's user avatar
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2 votes
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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?
Daniel Sebald's user avatar
1 vote
0 answers
50 views

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 ...
Ron Michal's user avatar
0 votes
0 answers
71 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 ...
Felix B.'s user avatar
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5 votes
1 answer
142 views

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 ...
Roselyn van Waalen's user avatar
6 votes
1 answer
161 views

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 ...
hyyyyy's user avatar
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0 answers
66 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?
Daniel Sebald's user avatar
2 votes
0 answers
31 views

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?
Daniel Sebald's user avatar
4 votes
1 answer
65 views

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 ...
user73577's user avatar
  • 395
0 votes
0 answers
79 views

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?
Daniel Sebald's user avatar
4 votes
1 answer
303 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 ...
Bruno's user avatar
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2 votes
0 answers
82 views

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?
Daniel Sebald's user avatar
6 votes
1 answer
239 views

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?
Daniel Sebald's user avatar
1 vote
1 answer
97 views

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?
Daniel Sebald's user avatar
3 votes
0 answers
101 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 ...
Daniel Sebald's user avatar
3 votes
0 answers
199 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!
rianko's user avatar
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1 vote
0 answers
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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 ...
Daniel Sebald's user avatar
1 vote
0 answers
69 views

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 ...
Daniel Sebald's user avatar
2 votes
0 answers
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 ...
Dasha Poliakova's user avatar
2 votes
1 answer
109 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 ...
Talmsmen's user avatar
  • 577
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0 answers
68 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, ...
Daniel Sebald's user avatar
2 votes
0 answers
71 views

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 \...
M. Winter's user avatar
  • 11.4k
0 votes
0 answers
88 views

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 ...
Turbo's user avatar
  • 13.2k
3 votes
1 answer
215 views

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 \...
Luis Ferroni's user avatar
  • 1,816
3 votes
0 answers
54 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 ...
Brent Kerby's user avatar
5 votes
3 answers
532 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? ...
Nandakumar R's user avatar
  • 4,577
4 votes
1 answer
138 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\...
M. Winter's user avatar
  • 11.4k
1 vote
0 answers
55 views

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). ...
Daniel Sebald's user avatar
3 votes
0 answers
37 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 $\{...
Daniel Sebald's user avatar
3 votes
0 answers
100 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 ...
Daniel Sebald's user avatar
6 votes
1 answer
472 views

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 ...
OzoneNerd's user avatar
  • 179
5 votes
1 answer
211 views

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 ...
M. Winter's user avatar
  • 11.4k
10 votes
2 answers
348 views

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 ...
David Richter's user avatar
3 votes
1 answer
98 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 $\epsilon$ ...
James Propp's user avatar
  • 18.8k
19 votes
1 answer
809 views

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 ...
M. Winter's user avatar
  • 11.4k
4 votes
3 answers
290 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 ...
M. Winter's user avatar
  • 11.4k
1 vote
0 answers
56 views

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 $$\...
Bob Mullins's user avatar
4 votes
0 answers
108 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\...
M. Winter's user avatar
  • 11.4k
6 votes
1 answer
176 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 ...
M. Winter's user avatar
  • 11.4k
9 votes
2 answers
287 views

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 ...
M. Winter's user avatar
  • 11.4k
5 votes
1 answer
255 views

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 ...
M. Winter's user avatar
  • 11.4k