Questions tagged [polytopes]
The polytopes tag has no usage guidance.
109
questions
1
vote
0
answers
33
views
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 ...
- 31
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 ...
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 ...
- 1,305
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?
- 2,415
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 $k$ ...
- 55
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 ...
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?
- 2,415
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 ...
user467453
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 ...
- 11.4k
4
votes
1
answer
104
views
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 ...
- 11.4k
2
votes
0
answers
39
views
Can every non-hemi uniform polytope tile hyperbolic space?
Can every uniform polytope which is not a hemipolytope tile hyperbolic space on its own?
- 2,415
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 ...
- 11
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 ...
- 181
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 ...
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 ...
- 223
0
votes
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?
- 2,415
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?
- 2,415
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 ...
- 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?
- 2,415
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 ...
- 43
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?
- 2,415
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?
- 2,415
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?
- 2,415
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 ...
- 2,415
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!
- 31
1
vote
0
answers
42
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 ...
- 2,415
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 ...
- 2,415
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 ...
- 755
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 ...
- 577
0
votes
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, ...
- 2,415
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
$$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...
- 11.4k
0
votes
0
answers
88
views
Number of vertices in a polyhedron
Consider polytopes
$$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$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...
- 13.2k
3
votes
1
answer
215
views
Intrinsic definition of a cone in a normal fan
Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities:
$$ \left<x,u_F\right> \geq -a_F$$
where $u_F\in \...
- 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 ...
- 191
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?
...
- 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\...
- 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). ...
- 2,415
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 $\{...
- 2,415
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 ...
- 2,415
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 ...
- 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 ...
- 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 ...
- 1,257
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$ ...
- 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 ...
- 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 ...
- 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 $$\...
- 11
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\...
- 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 ...
- 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 ...
- 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 ...
- 11.4k