Questions tagged [polyhedra]

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4 votes
3 answers
799 views

Is there a pyramid with all four faces being right triangles? [closed]

If such a pyramid exists, could someone provide the coordinates of its vertices?
Humberto José Bortolossi's user avatar
1 vote
0 answers
79 views

A face and all its neighbors: terminology?

Suppose $F$ is a face of a 2-complex, and $F_1,\dotsc,F_n$ are the faces that are adjacent to (i.e., share an edge with) $F$. Is there a standard term for a collection of faces of the form $\{F,F_1,\...
James Propp's user avatar
4 votes
1 answer
159 views

Regular polyhedral spaces

By symmetrically gluing together opposite faces of a dodecahedron together, one of three spaces can be obtained, depending on the angle the faces are rotated by before twisting. In fact, this can be ...
Daniel Sebald's user avatar
2 votes
0 answers
214 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
4 votes
1 answer
146 views

Orienting the dual of the associahedron

Let $A_n$ be the dual simplicial complex to the associahedron on $n$ letters. The complex $A_n$ is thus a simplicial triangulation of an $(n-3)$-dimensional sphere. The vertices of $A_n$ correspond ...
Dylan's user avatar
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1 vote
1 answer
53 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
  • 175
0 votes
1 answer
104 views

Which simplicial complexes are completely determined by the 1-skeleton of their dual polyhedral complexes?

Consider the following line of reasoning that shows certain simplicial complexes (of arbitrary dimension) are completely determined by corresponding graphs: The facet complex of any simplicial ...
hasManyStupidQuestions's user avatar
2 votes
0 answers
177 views

Does this sequence stop?

Let $\{ X_i\}$ ($i=1,2,\ldots $) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
M.Ramana's user avatar
  • 1,160
1 vote
0 answers
28 views

Inside-out dissections of solids

We add to Inside-out dissections of polygons - a generalization. The inside-out (fully inside-out) dissections are defined on pages linked there. How does one inside-out dissect a tetrahedron into ...
Nandakumar R's user avatar
  • 4,945
1 vote
0 answers
37 views

About the number of faces of the conification of a polytope

Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
ElliptCg's user avatar
5 votes
1 answer
164 views

The bounded complex of a polyhedral decomposition

Let $\mathscr{P}$ be a polyhedral decomposition of a real vector space $V$. By that I mean that $\mathscr{P}$ is a finite set of polyhedra in $V$ satisfying the following three properties: The union ...
Nicholas Proudfoot's user avatar
0 votes
1 answer
111 views

How can I find the hyperplane passing through a 600-cell

I have a 600-cell, whose coordinates are given by $$\begin{array}{ccc} \text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\ \text{16 vertices} & \frac{1}{2}\left(\pm1,\...
Dac0's user avatar
  • 275
1 vote
0 answers
96 views

What is the difference between a simple polyhedron and a triangulated graph?

On a famous website I've seen the following: The skeletons of the simple polyhedra correspond to the triangulated graphs, the smallest of which are illustrated above. That "illustration above&...
PatL's user avatar
  • 11
1 vote
0 answers
25 views

Does Hoffman constant keep the same after a very tiny perturbation on the polyhedron such that the bases are even unchanegd?

Suppose that $P$ is a polyhedron represented by $$P:=\{x \in \mathbb{R}^n: A x \le b \} \text{ for }A \in \mathbb{R}^{m\times n},\ b \in \mathbb{R}^m,$$ and $P$ contains interior points. Moreover, the ...
ZZZZZZ's user avatar
  • 11
2 votes
0 answers
73 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
1 vote
1 answer
82 views

Intersection of conical neighbourhoods on a polyhedral space

Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...
Lucas L.'s user avatar
2 votes
1 answer
102 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
11 votes
0 answers
261 views

How many ways to flatten a Tesseract onto a table?

A cube can be cut and flattened out onto a table in a way that the faces stay connected and none of them overlap. There are $384$ ways to make the cuts and $11$ distinct meshes emerge (see here). And ...
ryu576's user avatar
  • 171
2 votes
0 answers
47 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
0 votes
0 answers
73 views

Polynomial-time algorithm for exact projection to polyhedral cone

Given $c \in \mathbb{R}^d$ and $A \in \mathbb{R}^{n \times d}$, project $c$ to the polyhedral cone $\{x \in \mathbb{R}^d \mid A x \leq 0\}$. Is there an algorithm that outputs an exact solution to ...
user76284's user avatar
  • 1,721
2 votes
0 answers
224 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
4 votes
1 answer
124 views

On polyhedrons with specified numbers of congruent faces

Basic question: Given 3 integers n, n1 and n2 such that n1+n2 = n, to form an n-face polyhedron such that n1 of its faces are mutually congruent and the remaining n2 faces are different but congruent ...
Nandakumar R's user avatar
  • 4,945
0 votes
1 answer
181 views

Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?

Let ‎$‎‎X_1$ ‎‎be ‎the suspension of ‎$‎‎‎\mathbb{R}P^2‎$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$. Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...
M.Ramana's user avatar
  • 1,160
-4 votes
1 answer
129 views

Hilbert’s third problem and what a polyhedron is [closed]

What is the definition of a polyhedron used by Hilbert’s third problem?
Daniel Sebald's user avatar
5 votes
1 answer
266 views

Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?

Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...
R. W. Prado's user avatar
3 votes
2 answers
345 views

Secondary polytope

Given a polytope $P$, what do the points of the secondary polytope correspond to? I know that the vertices of the secondary polytope correspond to regular triangulations of $P$. But what do the ...
André Henriques's user avatar
10 votes
2 answers
2k views

Great polyhedra: What does "great" signify?

Great Cubicuboctahedron Great Icosacronic Hexecontahedron Great Rhombic Triacontahedron Great Snub Icosidodecahedron Great Stellated Dodecahedron Great Triakis Octahedron ... There are many polyhedra ...
Joseph O'Rourke's user avatar
7 votes
0 answers
196 views

Tiling space with supertile of hypercube unfoldings

Two students in my class asked and answered what might be a novel question. It is well known that the cube has exactly $11$ edge-unfoldings (or "nets"), as shown below:         (Image from ...
Joseph O'Rourke's user avatar
0 votes
0 answers
78 views

Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form $$ M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\} $$ where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$. Goal I am looking for an (explicit) representation ...
Felix B.'s user avatar
  • 305
2 votes
1 answer
62 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
3 votes
1 answer
188 views

Well-behaved trajectories

Call trajectory any continuous function $f: \mathbb{R}_{\geq 0} \to \mathbb{R}^n$ (here, $\mathbb{R}_{\geq 0}$ is interpreted as time). A polyhedral partition of $\mathbb{R}^n$ is a finite set of ...
Marco's user avatar
  • 141
10 votes
2 answers
481 views

Are there Monohedra with odd number of faces?

A monohedron is a convex polyhedron with all faces mutually congruent but with no other symmetry necessarily needed. So obviously, this is a wide class of polyhedrons that includes the Platonic solids ...
Nandakumar R's user avatar
  • 4,945
5 votes
2 answers
303 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 $8$ ...
Joseph O'Rourke's user avatar
0 votes
1 answer
261 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 ...
Mathlover's user avatar
2 votes
0 answers
87 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
6 votes
1 answer
247 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?
Daniel Sebald's user avatar
3 votes
1 answer
337 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 ...
efs's user avatar
  • 2,959
5 votes
1 answer
219 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,...
Nandakumar R's user avatar
  • 4,945
16 votes
3 answers
1k views

If I have zeros at the vertices of an icosahedron, where should the poles go?

I've been tinkering with Newton's method applied to polynomials. E.g., Newton's method for $z^5 - 1 = 0$ gives: There aren't a lot of symmetric patterns of finite sets of points in the plane, so I ...
Geoffrey Irving's user avatar
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 ...
Joseph O'Rourke's user avatar
1 vote
0 answers
103 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 ...
MathMax's user avatar
  • 203
3 votes
0 answers
109 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 ...
Dasha Poliakova's user avatar
2 votes
0 answers
75 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
  • 13.5k
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 ...
Turbo's user avatar
  • 13.5k
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_{\...
Turbo's user avatar
  • 13.5k
0 votes
1 answer
107 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 $\...
Turbo's user avatar
  • 13.5k
1 vote
0 answers
117 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\...
Turbo's user avatar
  • 13.5k
0 votes
0 answers
89 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 ...
Turbo's user avatar
  • 13.5k
1 vote
1 answer
151 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 ...
Turbo's user avatar
  • 13.5k
0 votes
1 answer
129 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?
Turbo's user avatar
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