# Questions tagged [polyhedra]

The polyhedra tag has no usage guidance.

176
questions

**3**

votes

**1**answer

39 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

**0**answers

52 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

**0**answers

35 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

**0**answers

91 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

**0**answers

26 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

**1**answer

110 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

**0**answers

33 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

**1**answer

67 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

**2**answers

123 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'.
...

**6**

votes

**1**answer

122 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 ...

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votes

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24 views

### Variants of Melzak and Aberth problems

Melzak's problem: Among all polyhedrons with edge length sum = 1, which one has the max volume?
Aberth's problem: Among all polyhedrons with edge length sum = 1, which one has max total surface area?...

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vote

**0**answers

40 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 ...

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**0**answers

28 views

### Face computation and support function minimisation of polytope projection

Let $P \subset {\mathbb R}^m$ be a convex polytope containing 0, given by system of inequalities $Ay \leq b$. Let ${\mathbb R}^n$, $0<n<m$, be a subspace of ${\mathbb R}^m$ given by first $n$ ...

**0**

votes

**0**answers

21 views

### Relatively prime polytope extension complexity

What is the extension complexity of the $0/1$ vertexed polytope in $2d$ dimensions with property that integer represented by first $d$ coordinates is coprime to integer represented by second $d$ ...

**2**

votes

**1**answer

39 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
$$ \{ x \in \mathbb{R}^n \mid Ax \le b \}.$$
...

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votes

**0**answers

52 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

**4**answers

3k 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

**0**answers

194 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 ...

**1**

vote

**2**answers

75 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

**3**answers

484 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

**0**answers

168 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

**0**answers

27 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 ...

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69 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 ...

**3**

votes

**1**answer

207 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

**0**answers

118 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

**0**answers

135 views

### Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?

**4**

votes

**0**answers

101 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 ...

**2**

votes

**0**answers

44 views

### An implementation of Minkowski reconstruction in 3 dimensions

By a theorem of Minkowski from 1903, an $n$-dimensional polytope $P\subset \mathbb R^n$ is determined up to translation by its unit face normal $u_1,\dots,u_k\in S^{n-1}$ and the corresponding $(n-1)$ ...

**2**

votes

**1**answer

323 views

### Does the Lebesgue Differentiation Theorem hold for regular polytopes?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...

**1**

vote

**1**answer

130 views

### computing the boundary of a union of polytopes

Let $P_1,\dots ,P_m\subset \mathbb{R}^n$ be $m<\infty$ convex polytopes in $\mathbb{R}^n$, and $U:=\bigcup_{j} P_j$ their set-theoretic union. What algorithms are known for computing the boundary $\...

**2**

votes

**1**answer

88 views

### Separation of two pointed polyhedral cones using hyperplanes generated by facets

Let $C_1$ and $C_2$ two pointed (that is, with vertex in $0$) polyhedral cones in $\mathbb{R}^n$ with $\dim(C_1)=\dim(C_2)=n$. If
$$\mbox{relative interior}(C_1)\cap \mbox{relative interior}(C_2)=\...

**1**

vote

**1**answer

187 views

### A possible characterization of the cube?

Let $P$ be the $1$-skeleton of a convex polyhedron fixed in $\mathbb{R}^3$,
and $|P|$ the sum of the Euclidean lengths of the edges of $P$.
Let $P_1, P_2, P_3$ be the perpendicular projections of $P$
...

**3**

votes

**0**answers

109 views

### Reconstructing plane graphs from degree- and face-sequences

Let $G$ be a plane $3$-connected graph; so it partitions the plane
into regions bounded by faces.
Let $\mathrm{deg}_v$ be the sequence of vertex degrees of $G$,
and $\mathrm{deg}_f$ be the sequence of ...

**5**

votes

**3**answers

236 views

### Average caliper diameter (mean width) of a polyhedron

Define the caliper diameter of a polyhedron as follows:
Let $P_1$ and $P_2$ be two planes both of which are parallel to the x axis such that the perpendicular distance between $P_1$ and $P_2$ is the ...

**3**

votes

**1**answer

63 views

### Faces of polyhedral cones and open immersions of affine toric schemes

Let $V$ be an $\mathbb{R}$-vector space of finite dimension, let $N$ be a $\mathbb{Z}$-structure on $V$, and let $M$ be its dual $\mathbb{Z}$-structure on the dual space $V^*$.
Let $\sigma\subseteq V$...

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votes

**0**answers

38 views

### Efficient $H$ representation of matrices with distinct cyclic shift permuted entries

Given points $v_1,\dots,v_n\in\mathbb Z^n$ in codimesion $1$ hyperplane $x_1+\dots+x_n=t$ with $0\leq x_{i}$ and a cyclic shift permutation $\sigma$ where
$v_1,\dots,v_n$ when written as columns of ...

**4**

votes

**2**answers

118 views

### How many dihedral angles need to be specified to uniquely specify a triangulated polyhedron?

Suppose you are given a simplicial complex $K$ homeomorphic to the sphere and for each each edge of the complex a label specifying a length of that edge (this gives us a polyhedral metric on $K$). In ...

**1**

vote

**1**answer

239 views

### What is the status of the smooth version of bellows conjecture

Bellows conjecture for polyhedra was setteled in 1997. How about the smooth version of it, ie bending of closed 2D submanifolds in $\mathbb{R}^3$ while preserving the Riemannian structure/intrinsic ...

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votes

**0**answers

179 views

### Factorization of tropical polynomials

I am referring to the definition of a tropical polynomial given on page 8 (top of section 3) here in this review, https://arxiv.org/pdf/math/0306366.pdf. I understand that here a tropical polynomial ...

**11**

votes

**2**answers

357 views

### Dodecahedral rolling distance

Let a dodecahedron sit on the plane,
with one face's vertices on an origin-centered unit circle.
Fix the orientation so that the edge whose indices are $(1,2)$ is horizontal.
For any $p \in \mathbb{R}...

**39**

votes

**3**answers

2k views

### How many ways can you inscribe five 24-cells in a 600-cell, hitting all its vertices?

You can inscribe five tetrahedra in a dodecahedron so that each vertex of the dodecahedron is the vertex of just one tetrahedron, as drawn here by Greg Egan:
Warmup question: How many ways can you do ...

**20**

votes

**4**answers

558 views

### Why do some uniform polyhedra have a “conjugate” partner?

While browsing through a list of uniform polynohedra, I noticed that the square of the circumradius $R_m$ of the small snub icosicosidodecahedron ($U_{32}$) with unit edge lengths is,
$$R_{32}^2 =\...

**6**

votes

**2**answers

175 views

### Homotopy domination of a wedge of two polyhedra

The topological space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$.
Question: Suppose that $...

**9**

votes

**1**answer

206 views

### Thinnest covering of the plane by regular pentagons

Q. Is it known what is the thinnest covering of the infinite plane by regular pentagons?
By covering I mean every point of the plane is covered.
By thinnest I mean the proportion of the plane covered ...

**2**

votes

**1**answer

116 views

### On the realization of a quotient group

Let $P$ be a finite polyhedron and $N$ be a normal subgroup of $G=\pi_1 (P)$. It is known that there exists a covering space $(\tilde{P},p)$ so that $p_* \pi_1 (\tilde{P})=N$. It follows that for the ...

**27**

votes

**1**answer

1k views

### Are Minkowski sums of upward closed “convex” sets in $\mathbb{N}^k$ still “convex”? (WAS: Comparing mana costs in Magic: The Gathering)

This was originally a question about comparing mana costs in Magic: The Gathering, but it's turned into a question about Minkowski sums of upward-closed convex sets in $\mathbb{N}^k$. The original ...

**14**

votes

**1**answer

248 views

### The space of triangles that fit inside a given triangle, parametrized by edge lengths

Given a triangle T with sides a, b, and c, describe its "fitting set," the set of all points (x,y,z) in 3-dimensions for which a triangle with sides x, y, z exists that fits in T.
Such a set lies in ...

**11**

votes

**11**answers

795 views

### Database of integer edge lengths that can form tetrahedrons

Is there a collection of lists of six integer edge lengths that form a tetrahedron? Is there a computer program for generating such lists? I need to find approximately thirty such tetrahedral ...

**3**

votes

**1**answer

88 views

### Any visualization software for the intrinsic metric of a convex polyhedron?

I'd like to find a visual simulation of what it would be like to 'live' in a polyhedron with the intrinsic, piecewise-Euclidean length metric. Of course, to make it easier to visualize, I'd prefer to ...

**2**

votes

**1**answer

131 views

### Constructing a Polyhedron given areas of its faces

I want to visualize a set of data as a polyhedron in 3d space. Imagine set A includes areas of such polyhedron's faces. I assume the first step is to check if there exist a polyehdron by making sure ...