# Questions tagged [polyhedra]

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### 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 ...
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### 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 ...
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### 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,...
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### 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 ...
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### 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}\} ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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'. ...
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### 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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### 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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### 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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### 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 $n$ ...
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### 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 $d$ ...
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### 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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### 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 $\mathbb{R}^3$...
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### 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 $P$ ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...