Questions tagged [polyhedra]

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2 votes
3 answers
2k views

non-convex Polytope definition

I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$, we can define a convex polytope in the following way: $$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, \...
0 votes
1 answer
343 views

The dimension of the normal cone of a face in a polytope

Let $P$ is a polytope in $\mathbb{R}^n$ if $F$ is one of its faces of dimension $d$ then the dimension of its normal cone $\mathcal{N}(F)$ is $n-d$.\ This seems to be intuitively obvious but I can't ...
5 votes
2 answers
306 views

Dimension of configuration space of triangulated convex polyhedron

The configuration space of all tetrahedra is $5$-dimensional, perhaps a non-obvious fact. There are $12$ face angles, but the sum of each of the four faces angles is $\pi$, reducing $12$ to $8$ ...
2 votes
0 answers
90 views

Dodecahedron deformation II

(Follow-up to this question) Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
6 votes
1 answer
252 views

Can a dodecahedron be deformed into a great stellated dodecahedron?

Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
2 votes
0 answers
76 views

Polyhedron coordinate bound

Given a polyhedron $$Ax\leq b$$ where we assume $A\in\mathbb Q^{m\times n}$ and $b\in\mathbb Q^{m}$ and it takes $L$ bits to represent the inequalities what is a good bound on the quantity $\|y\|_\...
3 votes
1 answer
342 views

Request for an article by Jim Lawrence

Jim Lawrence has a very important paper on the topic of valuations on polyhedra called "Rational-function-valued valuations on polyhedra", published in the DIMACS volume Discrete and ...
28 votes
5 answers
2k views

Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
24 votes
1 answer
1k views

Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?

A six year old question, Which unfoldings of the hypercube tile $3$-space?, has just been answered by Moritz Firsching: All $261$ unfoldings tile space! So now we know: For $d=2$, the unfolding of ...
5 votes
1 answer
237 views

Convex polyhedra with non-congruent faces

Question: Are there convex polyhedra wherein all faces are convex polygons with same area and perimeter and no two faces are mutually congruent? Remarks: If the answer to above is "no", then,...
27 votes
3 answers
12k views

Which unfoldings of the hypercube tile 3-space: How to check for isometric space-fillers?

Recently Mark McClure constructed and displayed the 261 unfoldings of the hypercube (tesseract) in response to the question, "3D models of the unfoldings of the hypercube?": The first 9 unfoldings ...
1 vote
0 answers
118 views

Regularity of Laplace equation on non-convex polyhedral domain

This might be a known problem, but I could not find a precise answer. I have the following Laplace equation \begin{equation} \begin{cases} -\Delta u = f & x \in \Omega;\\ \quad\: u = g & x \in ...
3 votes
0 answers
110 views

"Slim" directed polytopes: any established name for them?

This is a "looking for context" question. Let's say that a polytope is directed if its 1-skeleton is an oriented graph with no cycles, one source, one sink. (Edit: let us additionally assume ...
0 votes
1 answer
108 views

Detecting non-negativity of a single constraint by polyhedral constraints - $I$

We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...
1 vote
0 answers
63 views

Counting $\bmod 2$ number of vertices of sparsely represented polyhedra

Given a polyhedron $$Ax\geq b$$ is there an $NC^1$ or an $NC^2$ algorithm to count the number of vertices $\bmod2$? Assume $A\in\{0,1\}^{m\times n}$ and $b\in\mathbb Z^{m}$ ($m=O(n)$) and assume rows ...
1 vote
0 answers
43 views

Detecting non-negativity of a single constraint by polyhedral constraints - $II$

Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
0 votes
1 answer
135 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?
20 votes
1 answer
876 views

Tetrahedra passing through a hole

Assume a plane $P\subset\mathbb R^3$ has a hole $H$, and that the hole is topologically a compact disc. Being so, $P\setminus H$ does not separate the space. A regular tetrahedron $\sigma^3$ (of edge-...
1 vote
0 answers
118 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
0 answers
92 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
1 answer
163 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 ...
2 votes
0 answers
128 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 ...
7 votes
2 answers
292 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 $...
5 votes
1 answer
299 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 ...
4 votes
1 answer
156 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 ...
26 votes
4 answers
894 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 =\...
1 vote
0 answers
66 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
0 answers
271 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
1 answer
91 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(\...
48 votes
4 answers
3k 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 ...
3 votes
1 answer
277 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&...
4 votes
2 answers
227 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 ...
10 votes
2 answers
359 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 ...
6 votes
0 answers
222 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 ...
8 votes
1 answer
1k 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 ...
14 votes
0 answers
379 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 ...
2 votes
2 answers
112 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 ...
13 votes
0 answers
464 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 ...
6 votes
1 answer
210 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
0 answers
125 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 ...
2 votes
2 answers
198 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 ...
4 votes
0 answers
79 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
1 answer
75 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
1 answer
77 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
103 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
72 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
112 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
0 answers
83 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
194 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 ...
7 votes
1 answer
294 views

Embedding of flat surfaces

Let $S$ be a orientable compact surface with a flat euclidean structure with conical singularities (cf. [T] for instance). Let also $\mathcal P$ be a polyhedral euclidean decomposition of $S$ (with ...