Questions tagged [polyhedra]

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"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 ...
Dasha Poliakova's user avatar
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 ...
ARG's user avatar
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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}\} ...
The Number Theorist's user avatar
3 votes
0 answers
192 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 ...
user avatar
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142 views

Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?
Denis T's user avatar
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220 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 ...
Joseph O'Rourke's user avatar
3 votes
0 answers
294 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 ...
gradstudent's user avatar
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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 ...
Ben Knudsen's user avatar
3 votes
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127 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 $$f\colon (|M|,|N|) \to (P,Q)$$ ...
Joao Faria Martins's user avatar
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102 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 ...
Oliver Nash's user avatar
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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 $P$ ...
Joseph O'Rourke's user avatar
2 votes
2 answers
1k views

Mathematical tools appropriate to analyse convex polyhedra

What mathematical tools (means: set of areas of mathematical knowledge) are appropriate to begin with to analyse (to enumerate face vectors associated with polyhedron, to calculate the combinatorial ...
Ali Dino Jumani's user avatar
2 votes
2 answers
778 views

Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$. A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$. The four ...
Trimok's user avatar
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2 answers
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A Jordan Separation Theorem for Polyhedral Surfaces

Let me begin by defining what a polyhedral surface is. A path-connected subset $ P $ of $ \mathbb{R}^{3} $ is called a polyhedral surface iff it is the union of a finite collection $ \mathcal{C} $ of ...
Transcendental's user avatar
2 votes
1 answer
894 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\...
Keshav Srinivasan's user avatar
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| \, \...
Joaquín Moraga's user avatar
2 votes
1 answer
213 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 ...
Mike's user avatar
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1 answer
273 views

Digital topology, animal problem, 2-sphere and torus

I have the following question relating digital topology, surfaces, particularly $S^2$ and torus. Can a body $B$ constructed with cubes (without cavities or tunnels) and with frontier homeomorphic to ...
Martin Garcia Fernandez's user avatar
2 votes
1 answer
168 views

coarser than triangulations "almost partitions" into simplices

The total space $T$ of an embedded into $\mathbb{R}^n$ pure $n$-dimensional simplicial complex (in other words, the union of finitely many $n$-dimensional compact convex polytopes) sometimes admits an ...
Dima Pasechnik's user avatar
2 votes
1 answer
140 views

Tangent cone on polyhedral spaces

Let $X$ be an n-dimensional polyhedral space with, say, $n\geq 3.$ Let also $p\in X$ be a vertex on a triangulation $\tau$ of $X,$ so a vertex on the polyhedral space. The tangent cone (as a metric ...
Lucas L.'s user avatar
2 votes
1 answer
66 views

Generic infinitesimal rigidity of polyhedra

Let $M$ be a 1-skeleton of a triangulation of a sphere with $V$ vertices and $E$ edges. Definition 1 A polyhedron is a map $M\to \mathbb R^3$ that is affine on edges (and non-degenerate on faces). The ...
Dmitrii Korshunov's user avatar
2 votes
1 answer
122 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 ...
Ilya Palachev's user avatar
2 votes
1 answer
132 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)=\...
ElliptCg's user avatar
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1 answer
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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 ...
MHenry's user avatar
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2 votes
1 answer
151 views

Polyhedra containing hexagones only [closed]

It is well-known that Euler's theorem gives raison d'être to polyhedra containing exactly 12 pentagons if they are connected by 3 in a vertex. The number of hexagons may be arbitrary (in fact >1). In ...
Vladimir Sotirov's user avatar
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 ...
Joseph O'Rourke's user avatar
2 votes
1 answer
117 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 $$ \{ x \in \mathbb{R}^n \mid Ax \le b \}.$$ ...
Pietro Paparella's user avatar
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 ...
vangel86's user avatar
2 votes
1 answer
207 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 ...
user2367663's user avatar
2 votes
1 answer
223 views

How can pushing a vertex in a polytope lead to merging facets?

I'm trying to understand the Lemma 2.2 and Corollary 2.3 of Francisco Santos paper "A counterexample to the Hirsch conjecture": http://arxiv.org/abs/1006.2814 Corollary 2.3 is a proof of a result of ...
nomatter's user avatar
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 ...
DNQZ's user avatar
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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 ...
John's user avatar
  • 185
2 votes
1 answer
131 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 \...
Student's user avatar
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0 answers
240 views

Why is it impossible to create a numerically balanced die with more than 120 sides?

I allow myself to contact you as a mathematics enthusiast. I have recently been intrigued by the concept of balance in dice and the assertion that it would be impossible to create a numerically ...
Matthieu Nauly's user avatar
2 votes
1 answer
82 views

Is there a way to parametrize the configuration space of all convex polyhedra of a given combinatorial type as a convex set?

I'm sure this is easy/known, but I'm not hitting an appropriate search term for finding the answer and the coffee hasn't kicked in enough to come up with it myself: Let $T$ be a simplicial 2-complex ...
John's user avatar
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0 answers
82 views

Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance: https://math.stackexchange.com/a/14469/454016 Poincaré duality is explained through a duality of ...
Alexander Praehauser's user avatar
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0 answers
48 views

Endpoints of intrinsic diameter of a convex polyhedron

Let $P$ be a convex polyhedron in $\mathbb{R}^3$, and $d(P)$ its intrinsic diameter, i.e., the longest shortest surface path between two points. Say that $P$ is of class $D_0$ if neither endpoint of $...
Joseph O'Rourke's user avatar
2 votes
0 answers
228 views

Generalization of the Napoleon equilateral triangle to higher dimention

When I researched the Fermat-Dao-Nhi equilateral triangle in preamble before points X(33602) of the Kimberling triangle center. I discovered the general result for polygon as follows: Let $A_1$, $A_2$...
Đào Thanh Oai's user avatar
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?
Daniel Sebald's user avatar
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\|_\...
Turbo's user avatar
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2 votes
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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 ...
David G. Stork's user avatar
2 votes
0 answers
84 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 $\mathbb{R}^3$...
Joseph O'Rourke's user avatar
2 votes
0 answers
29 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 ...
VS.'s user avatar
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2 votes
0 answers
74 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 $(n-1)$ ...
Christoph's user avatar
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0 answers
40 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 ...
Turbo's user avatar
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2 votes
0 answers
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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 ...
A.2's user avatar
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2 votes
0 answers
51 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 ...
Sidharth Ghoshal's user avatar
2 votes
0 answers
185 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, ...
Seonhwa  Kim's user avatar
2 votes
0 answers
105 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....
John Doe's user avatar
  • 170
1 vote
1 answer
285 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 ...
Amr's user avatar
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