The polyhedra tag has no usage guidance.

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159 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

48 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 $\sigma\subseteq V$...

**2**

votes

**0**answers

36 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 ...

**0**

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**0**answers

59 views

### Faster Mixed Integer Linear Programming Searchless Feasibility

We know Lenstra's Mixed Integer LP with Kannan's modificiation solves feasibility Mixed Integer LP in $n$ integer variables, $r$ real variables and $m$ constraints by solving the search version in $n^{...

**4**

votes

**2**answers

73 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

203 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 ...

**3**

votes

**0**answers

93 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 ...

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votes

**2**answers

310 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 ...

**17**

votes

**4**answers

443 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 =\...

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votes

**2**answers

162 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 $...

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votes

**1**answer

184 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 ...

**0**

votes

**0**answers

97 views

### Is there a transformation that scales or shifts number of lattice points?

Unitary transformation preserves volume of a polytope while $SL(n,\Bbb Z)$ preserves lattice points.
Let polytope $\mathcal K$ in $\Bbb R^n$ be presented with polynomial in $n$ linear inequalities.
...

**2**

votes

**1**answer

115 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 ...

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votes

**1**answer

227 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

734 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

70 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

81 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 ...

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votes

**1**answer

181 views

### Hypersurfaces whose equation is not known

I would like to find some well-known/interesting hypersurfaces which arise as parametrizations where implicitization is computationally too difficult.
I have software which computes the Newton ...

**3**

votes

**0**answers

44 views

### Perimeters of nested convex spherical polygons

I seek a reference—not a proof—that if $P_1$ and $P_2$
are two convex polygons on a sphere composed of geodesic segments,
contained in a hemisphere, and
$P_1 \subseteq P_2$, then the ...

**7**

votes

**1**answer

209 views

### The $32$-deg polynomial for the tetrahedron inscribed in the icosahedron?

This MO answer discusses this table involving the maximal side lengths of the five Platonic solids $T,C,O,D,I$ inscribed in the other solids,
This table is also found in Moritz Firsching's paper. I ...

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votes

**0**answers

70 views

### Making a polyhedron integral by selecting value for a specific co-ordinate of constraint vector

I am currently trying to solve a binary integer programming(maximization) problem, where the first row of the constraint matrix corresponds to the constraints on the total number of 1's in the vector ...

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vote

**1**answer

78 views

### Are there two tetrahedrons with the same volume that share their opposite edge lengths and arent the same or a different chirality of the same? [closed]

I have been coming up with an efficient way to decide if two tetrahedrons are similar. I believe that it is enough for a computer to check for the ordered by length list of pairs of opposite edges on ...

**2**

votes

**1**answer

83 views

### Polyhedral structure of functions writable as a finite signed sum of max of linear functions

For any two positive integers $k,n$ consider the space of functions writable as,
$\sum_i \sigma_i \max \{ L_{i1},L_{i2},..,L_{ik} \}$ (a finite sum) where each $L_{*} : \mathbb{R}^n \rightarrow \...

**2**

votes

**1**answer

159 views

### Polyhedra names question [closed]

So I've been playing around with polyhedra for my own amusement, but I ended up with some that I couldn't find names for. I have been trying to find them on my own by Googling for polyhedra with these ...

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**0**answers

257 views

### Are all Dehn invariants achievable?

The Dehn invariant of a polyhedron is a vector in $\mathbb{R}\otimes_{\mathbb{Z}}\mathbb{R}/2\pi\mathbb{Z}$ defined as the sum over the edges of the polyhedron of the terms $\sum\ell_i\otimes\theta_i$ ...

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votes

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120 views

### Is there a well-established terminology for polyhedra/polytopes?

I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At ...

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102 views

### Boundary regularity of the solution of a Poisson equation in a polyhedron

Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be bounded and open
$f\in L^2(\Lambda,\mathbb R^d)$
$u\in H_0^1(\Lambda,\mathbb R^d)$ with $$-\langle\nabla\phi,\nabla u\rangle_{L^2(\Lambda,\:\...

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vote

**0**answers

56 views

### Sampling in a polyhedral complex

Assume one is given a polyhedral complex $P$ in $\mathbb{R}^n$. Now consider picking uniformly at random a $D \subseteq \{0,1\}^n$. Is there way to upper bound the probability that $D$ (a subset of ...

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vote

**0**answers

147 views

### Cubical approximation theorem for cubical complexes

A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain.
I have found a claim ...

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**0**answers

71 views

### Stellar moves on pairs of polyhedra

Let $P$ and $Q$ be polyhedra of $\mathbb{R}^n$ with $Q \subset P$. Let $(M,N)$ and $(M',N)$ be pairs of abstract simplicial complexes.
Consider two triangulations
$$f\colon (|M|,|N|) \to (P,Q)$$
...

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votes

**0**answers

76 views

### Is there any connection between Lagrange points and the icosahedron?

Given the Newtonian two-body problem, one can ask if there are any orbits that allow a test particle to maintain a fixed configuration relative to the two bodies. In other words, in a frame that ...

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votes

**2**answers

333 views

### What is Kept Fixed for Flexible Spheres

For background to this question much recent exciting related things, see this videotaped lecture by Alexander Gaifullin.
Consider a triangulation $K$ of a two-dimensional sphere and consider maps ...

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**0**answers

495 views

### Maximum volume cross-section of a hypercube

This is surely well known, but:
Q1. What is the $(d{-}1)$-dimensional polytope
that realizes the maximum volume cross-section of a unit hypercube
by a $(d{-}1)$-dimensional hyperplane?
...

**1**

vote

**1**answer

66 views

### Derive a vertex representation of a permutohedron from its linear-inequalities form

Let us define the $n$-permutohedron $P_n$ as the set of all $x\in\mathbb{Q}^n$ such that
$$\sum_{i=1}^n x_i = \binom{n{+}1}{2}\ \ \ \land\ \ \ \forall\,\text{nonempty}\ S\subsetneq\mathbb{N}_n\colon\ ...

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votes

**0**answers

39 views

### Facet counting argument for polytopes

Consider a pair of piecewise-linear cobordant $n$ dimensional polyhedra $P_1, P_2$ sitting in $\mathbb{R}^{n+2}$ (with some fixed orientation).
Let $O$ be an $n+1$ dimensional piecewise-linear ...

**0**

votes

**0**answers

103 views

### Number of polyhedra with N faces?

A. Up to isomorphism, how many polyhedra with N faces are there? Assume each face can be a triangle, square, pentagon, hexagon, etc... Furthermore each edge can be resized to any nonzero positive ...

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votes

**2**answers

120 views

### Unit-Distance Polyhedra

What polyhedra are known to have two vertices adjacent if and only if they are of distance $d$ apart, for fixed $d$? For example, regular Platonic solids satisfy this condition, so I am looking for ...

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vote

**1**answer

74 views

### Testing whether two vertices are neighbours

I face the following problem: I am given a high-dimensional, convex, bounded polyhedron in both vertex description: $X = \mathrm{conv} \, \{ v_1, \ldots, v_K \}$ and halfspace description: $X = \{ x \...

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**0**answers

56 views

### Distinguishing (possibly lower dimensional) $1$-skeleton of a regular graph inscribed in a sphere

Consider you have two (possibly same) convex $1$-skeleton of a regular graph $A$ and $B$ in $m$-dimensions inscribed in a sphere with possibly exponential number of vertices in $n$-dimension with ...

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**0**answers

88 views

### Harborth conjecture and polyhedra

Harborth conjecture state that every planar graph can be drawn on a plane only using staight line segments of rational or integral edge length.
( There is a good mathoverflow page for this conjecture, ...

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votes

**1**answer

195 views

### Convex polyhedra, combinatorial types and Symmetry

Steinitz theorem says that combinatorial types of convex polyhedra is identified with 3-connected planar graph, called by a polyhedral graph.
A symmetry of polyhedral graph means that a vertex ...

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votes

**3**answers

326 views

### Labeling edges of an icosahedron with sum constraints

The question is inspired by this previous MO question. There it was shown that it's possible to label the edges of a cube by the numbers $\{1,2,\ldots,6,8,9, \ldots, 13\}$ in such a way that:
Three ...

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votes

**1**answer

133 views

### Self-avoiding/reflecting geodesics on a convex surface

Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...

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votes

**1**answer

120 views

### symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let $c:=c(...

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votes

**0**answers

157 views

### Which -icial sets produce the “standard” representations of symmetric groups?

Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...

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**0**answers

43 views

### Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?

Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...

**8**

votes

**4**answers

189 views

### Uniform Sampling Subject to Linear Equalities and Non-Negativity Constraint

I'm trying to sample uniformly on the intersections of faces of several simplicies, with all coordinates being non-negative. That is, given constraints
$$A\vec{w}=\vec{b} \ \ and \ \ \vec{w} \geq \vec{...

**1**

vote

**1**answer

272 views

### Convex polyhedron and its Gauß-curvature [closed]

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here.
What I am trying to prove:
A convex polyhedron has positive Gauß-Curvature at every vertex.
...