Questions tagged [polyhedra]

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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 ...
14
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0answers
468 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one: Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...
11
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0answers
712 views

Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that $S$...
10
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0answers
298 views

Bi-spherical polyhedra

Bicentric polygons have been studied: a polygon all of whose vertices lie on its circumcirle, and whose incircle is tangent to every edge:   I have not been able to find a comparable literature ...
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 ...
9
votes
0answers
474 views

Maximum volume convex body coverable by a unit square

Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected) pieces (where $k=1$ means just the square). Your task is to construct the largest volume convex body ...
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 ...
8
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0answers
164 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 ...
7
votes
0answers
875 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? ...
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 ...
6
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0answers
110 views

Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
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 ...
5
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0answers
166 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 ...
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 ...
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 ...
4
votes
0answers
81 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 ...
4
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0answers
1k views

Regular cross-sections of a dodecahedron; analogous sections of 4-polytopes

One can intersect a dodecahedron with a plane and obtain an equilateral triangle, a square, a regular pentagon, a regular hexagon, and a regular decagon:             &...
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 ...
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 ...
3
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0answers
137 views

Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
3
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0answers
153 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 ...
3
votes
0answers
213 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 ...
3
votes
0answers
120 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 $$f\colon (|M|,|N|) \to (P,Q)$$ ...
3
votes
0answers
94 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 ...
3
votes
0answers
179 views

Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$. The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...
3
votes
0answers
127 views

Convex polyhedra jammed in $k$ disjoint holes

For a given convex polyhedron $P \subset \mathbb{R}^3$, I was imagining finding the optimal "fixing" of $P$ in holes (or jamming them in "mud"), which led to the following question. First, scale $P$ ...
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 ...
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
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 $\mathbb{R}^3$...
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 ...
2
votes
0answers
49 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
0answers
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 ...
2
votes
0answers
74 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 ...
2
votes
0answers
208 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 ...
2
votes
0answers
41 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 ...
2
votes
0answers
134 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, ...
2
votes
0answers
102 views

Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear inequalities....
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\...
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 $$ \...
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 ...
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 ...
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 ...
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 ...
1
vote
0answers
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 ...
0
votes
0answers
57 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 ...
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?
0
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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 ...
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&...
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 ...