Questions tagged [polyhedra]
The polyhedra tag has no usage guidance.
199
questions
1
vote
0answers
85 views
Integrality of polyhedra
Given two polyhedra in $H$ representation $P_1:Ax\leq b$ and $P_2:Bx\leq c$ which are integral are bounded when is their intersection also integral?
Given two polyhedra in $H$ representation $P_1:Ax\...
0
votes
0answers
56 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 ...
1
vote
1answer
104 views
Exactly counting number of vertices of a polyhedron
Suppose $Ax\leq b$ is a polyhedron, where the number of rows in $A$ is $r$, the vector $x$ lies in $\mathbb R^n$ and the rank of $A$ is $t$. Assume minimal number of hyperplane inequalities to define ...
0
votes
1answer
60 views
Number of linear inequalities describing a polyhedron with prescribed number of vertices
If a polytope has $d$ vertices in $k$ dimensions how many linear inequalities is required to describe it?
5
votes
1answer
209 views
Embedding an icosahedron
A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.
Let me start with the example of a ...
2
votes
0answers
95 views
Distinguishable knots (with constraints) over polyhedra
I'm trying to find the number of distinguishable knot projections over certain convex regular polyhedra according to the following constraints. On each face on the polyhedron the knot will have a ...
4
votes
1answer
113 views
Finding Motzkin's original paper on copositive quadratic forms
I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...
1
vote
0answers
43 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). ...
1
vote
0answers
45 views
Decomposition of Polyhedral - An example
There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system
$$ \...
4
votes
1answer
79 views
Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected?
Let $\mathcal{A}_\Gamma$ be the space of convex (non-degenerate) Euclidean polyhedra with $1$-skeleton a certain polyhedral graph $\Gamma$. This space can be seen as a subset of $\mathcal{Gr}_2(\...
3
votes
1answer
117 views
Duality argument for elliptic regularity
M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...
0
votes
0answers
42 views
What are the expected values of the volumes of two classes of ellipsoids contained within the unit 3-ball, and/or what is their ratio?
Consider the class of all ellipsoids contained in the unit 3-ball, and also the subclass of those ellipsoids also contained within tetrahedra also contained in the unit 3-ball. What are the expected ...
8
votes
1answer
180 views
Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?
A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
5
votes
0answers
101 views
What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?
Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
4
votes
2answers
127 views
Clustering of vertices in an $n$-dimensional cube
Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices.
If we ...
12
votes
1answer
702 views
Two questions on the permutohedron
The $n$-dimensional permutohedron $P_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes.
I ...
9
votes
2answers
312 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 ...
14
votes
0answers
289 views
Minimum number of distinct triangles for tesselating the sphere
Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
8
votes
0answers
108 views
What are the known convex polyhedra with congruent faces?
Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
9
votes
0answers
159 views
How to correctly state Cauchy's rigidity theorem?
Cauchy's rigidity theorem is usually cites briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...
0
votes
0answers
29 views
Distance to the “boundary” of a polyhedral complex
Suppose I have a polyhedral complex $\{P_1, \ldots, P_k\}$ and let $S := \cup_{i = 1}^k P_i$. I am interested in a function which measures the distance from a point $x \in S$ to the "boundary&...
6
votes
1answer
124 views
Is Sydler's theorem concerning Dehn invariants constructive?
Sydler proved something of a converse to Dehn's negative resolution
of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that
"every two Euclidean polyhedra with the same volumes and Dehn ...
0
votes
0answers
51 views
Find tetrahedron vertex given 3 vertices of a face and the 3 opposite angles
I have the following tetrahedron:
which I know the coordinates of $P$, $Q$ and $R$ and the value of angles $\theta_0$, $\theta_1$ and $\theta_2$.
I need to find the coordinates of vertex $E$. Is that ...
4
votes
0answers
55 views
Classification of space-filling (by identical copies) convex polyhedra in R^3
Is the classification of space-filling (by identical copies) convex polyhedra in R^3 is known ?
There are only 5 "parallelohedra" - filling by translation.
But if relax that property to ...
2
votes
2answers
69 views
How to define and compute the degree of congruence of two rigid polyhedra in same type with knowing vertex coordinates?
If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertices of a polyhedron. The two polyhedra have same type, so we don't need to consider the topological ...
2
votes
1answer
44 views
Do continuous motions of the vertices of convex polyhedra that maintain local convexity imply global convexity? (Reference request)
A convex polyhedron has all of its internal dihedral angles in $(0, \pi)$. However, if I start with an abstract polyhedron $P$, let's say a triangulated one, so I don't have to worry about planarity ...
3
votes
1answer
49 views
Tilings of lattice polytopes by transformations of lattice polytopes
A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
1
vote
0answers
66 views
Untruncate permutohedron of order 5
I would like to understand commutation classes of reduced expressions of the longest element in $S_5$ a little better. For this, it makes sense to look at the permutohedron of order 5. Since I am only ...
0
votes
0answers
44 views
Visualization of higher Bruhat order B(5,2)
I made the following images of the higher Bruhat order B(5,2) (in the sense of Manin/Schechtman) with vZome:
image 1
image 2
image 3
Unfortunately, in vZome its not possible do have regular octagons,...
3
votes
0answers
98 views
Pyramids whose volume can be computed by simple cutting and glueing
Since this question remained without answers even after a bounty, I thought it might be time to ask it here.
For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn ...
2
votes
0answers
32 views
An exponential integral over a closed convex polytope
For any $T\geq 2$, let us define the polyhedron $S$ given by
\begin{align*}
S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\}
...
2
votes
1answer
135 views
Self-intersecting path of stacked regular tetrahedra
(This question occurred to me after reading
@IanAgol's reminisces
of Conway's spiral tetrahedron billiard path.)
Let $T_i$ be a regular tetrahedron,
and $P$ a collection of regular tetrahedra
glued ...
1
vote
0answers
40 views
fast V representation update of polytope
Say that I have both the V and the H representation of a (possibly unbounded) polytope $P$. I want to append a some rows to the H representation, how can I quickly update the V representation to ...
2
votes
1answer
78 views
Distance between two polyhedra that takes incidence structure into account
Suppose that we have two polyhedra $P_1$ and $P_2$ in $\mathbb{R}^3$. I would like to define such a metric $\rho(P_1, P_2)$ that depends on several factors, but currently I don't know how to do it ...
3
votes
2answers
138 views
Polyhedra that can pack 3-space only in a non-vertex-to-vertex fashion
Question: Are there polyhedral units (convex or otherwise) that can pack 3D space without gaps only such that the arrangement is not vertex-to-vertex?
Same question can be asked with 'edge to edge'.
...
7
votes
1answer
140 views
Alexandrov's rigidity in higher dimensions
If $\Phi_1,\Phi_2$ are convex polyhedra in $\mathbb{R}^3$ such that the sets of outer normals to facets coincide, but $\Phi_1$ is not a translate of $\Phi_2$, then there exist two corresponding ...
1
vote
0answers
46 views
Essential rays in fan structure
Let $|\Sigma|$ be the underlying set of some fan $\Sigma$ in $\mathbb{R}^n$. It is well known that in general there is no coarsest fan structure on $|\Sigma|$. However, there may be some special rays ...
2
votes
1answer
49 views
Polytopes and polyhedral cones in complex Euclidean space
Given $A \in \mathsf{M}_{m \times n}(\mathbb{R})$ and $b \in \mathbb{R}^m$, the polyhedron with respect to $A$ and $b$, denoted by $P(A,b)$, is defined by
$$ \{ x \in \mathbb{R}^n \mid Ax \le b \}.$$
...
2
votes
0answers
58 views
Iterated polyhedron face twisting
Let $Q$ be a polygon in the plane. Modify $Q$ by rotating each edge about its
midpoint by $180^\circ$. The result is $Q$ again: No change.
This suggests exploring a similar operation in $\mathbb{R}^3$...
88
votes
4answers
4k views
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
6
votes
0answers
202 views
Complexity of scissors congruence?
Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
2
votes
2answers
95 views
Products of polytopes and the normals of their facets
I need to compute the normals of the facets of certain polytopes that can be represented as products of polytopes in smaller dimensions.
Searching the bibliography I found that the facets of the ...
5
votes
3answers
527 views
Alexandrov's generalization of Cauchy's rigidity theorem
Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...
3
votes
0answers
181 views
Motivic strong bellows conjecture
There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...
2
votes
0answers
28 views
When do projection maps of polyhedra factor?
Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...
1
vote
0answers
75 views
Mixed integer formulation of union of polytopes?
Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...
4
votes
1answer
331 views
What is the average area of the shadow of a convex shape taken over all possible orientations?
If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape?
1
vote
0answers
123 views
Is there a method to cut a hypercube into disjoint cubes [closed]
Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$
so we can cut a hypercube into at least $n+1$ disjoint parts.
Is there a method how can one do that?
3
votes
0answers
137 views
Can bellows make loops?
Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
4
votes
0answers
108 views
Name for facet of a cone containing all but one edge
Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In ...
