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Questions tagged [polyhedra]

For questions related to polyhedra and their properties.

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Proof verification: A cube and a tetrahedron are not scissors congruent?

I am familiar with a proof that the cube and tetrahedron are not scissors congruent along the following lines: Given a polyhedron or collection of polyhedra $\mathcal{P}$ whose edges form a set $E$, ...
Numeral's user avatar
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1 vote
1 answer
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Chvatal-Gomory integer rounding method to find facets of $\operatorname{conv}(S)$

The question: "given a set $S = \{x \in \mathbb{Z}^2 : 4x_1 + x_2 ≤ 28, x_1 + 4x_2 ≤ 27, x_1 − x_2 ≤ 1, x ≥ 0 \}$. we are tasked with deriving each facet of $\operatorname{conv}(S)$ as a Chvatal-...
alex's user avatar
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6 votes
3 answers
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Euler's polyhedron formula modulo 2

Recall that Euler's polyhedron formula states that, for every (orientable) polyhedron with $V$ vertices, $E$ edges, $F$ faces, and $g$ holes (or more formally, with genus $g$), $$V-E+F=2-2g.$$ As an ...
Kenta S's user avatar
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6 votes
2 answers
548 views

Mismatching Euler characteristic of the Torus

Why is it that when I try to compute the Euler characteristic for the Torus using a drawing like the following , then the number that I get is not the number that the Torus should have? Which is $0$? ...
Tutusaus's user avatar
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Can Cauchy's polyhedron rigidity theorem be generalized to affine transformations?

Conjecture: Suppose $f$ and $g$ are two convex realizations of an abstract polyhedron $P$. (In other words, $f(P)$ and $g(P)$ are two convex polyhedra whose face lattices are isomorphic.) Also suppose,...
mr_e_man's user avatar
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If for a LP P = {x | Ax = b, x>=0} a sol can be degenerate, x is not a BFS

I have the following problem: Consider the polyhedron in standard form P = { x | Ax=b, x>=0} Suppose that the matrix A of dimensions m x n has linearly independent rows and that every BFS is non-...
Techno's user avatar
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Extension complexity - Equivalent definitions

I'm confused by two, apparently equivalent, definitions of extension complexity. See the attached screenshot: From: Sparse sums of squares on finite abelian groups and improved semidefinite lifts ...
Matteo's user avatar
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Has anyone tried “building” any convex uniform polyhedra from combinatorial tree graphs?

CONTEXT Convex uniform polyhedra, like tree graphs, can be described with vertices and unitary edges. QUESTION 1 Has anyone tried “building” any convex uniform polyhedra from combinatorial tree graphs ...
olivierlambert's user avatar
3 votes
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70 views

Which faces does "sphere" lattice polyhedron $\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq4^a(8b+7)$ have?

$\operatorname{hull}(p\in\mathbb{Z}^3, \operatorname{norml}2(p)==n)$ for $n\neq 4^a(8b+7)$ is a lattice polyhedron. By Legendre's three-square theorem, such $n$ have representation(s) as the sum of $3$...
HermannSW's user avatar
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Can a polyhedron be uniquely indexed by its number of facets, edges and vertices?

Let $F,E,V$ be the number of facets, edges and vertices of a polyhedron in 3D space. Euler's characteristic is given by: $$ V-E+F =2 $$ For 3D polyhedra. I was also told that two other inequalities, ...
urquiza's user avatar
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Euler's formula for polyhedron [closed]

I'm curious about Euler's formula and how it applies to polyhedra. Specifically, I'm wondering if there are simple formulas or methods to figure out the number of vertices, faces, and edges for ...
Lei Silvestre's user avatar
39 votes
0 answers
1k views

Dividing a polyhedron into two similar copies of itself

The paper Dividing a polygon into two similar polygons provides that there are only three families of polygons that are irrep-2-tiles (can be subdivided into similar copies of the original). Right ...
Numeral's user avatar
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What class of subgraphs of the $n$-hypercube graph characterizes the region graphs of arrangements of $n$ hyperplanes in $\mathbb{R}^d$?

I am looking for a reference that answers or at least discusses the question in the title. I browsed Sergei Ovchinnikov's book "Graphs and Cubes" and several lecture notes on hyperplane ...
axelniemeyer's user avatar
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The cone generated by a finite set of points is a polyhedron

Prerequisites A subset $K$ of $\mathbb{R}^n$ is a cone if for every $x \in K$ and every $\lambda > 0$, $\lambda x \in K$. If $x_1,\dots, x_p$ are points in $\mathbb{R}^n$, a conic combination of $...
Logos's user avatar
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Describe all cones in $\mathbb R^n$ that are simplicial and cosimplicial.

For any finite set of vectors $(v_1, \ldots, v_k \in \mathbb{R}^n)$, let's define: $Cone(v_1,…,v_k)= \{λ_1v_1+…+λ_kv_k,∣,λ_1,…,λ_k \ge 0\}$ and $Poly(v_1,…,v_k)= \{ x \in \mathbb{R}^n ∣(v_i, x) \ge 0, ...
PikaPika's user avatar
1 vote
0 answers
29 views

How Much To "Twist" A Polygon To Match A Face On A Regular Polyhedron

Suppose I want to create an icosahedron by building a set of twenty triangular pyramids (aka tetrahedrons, but see below) of an appropriate size and then rotating each into position. I need to do ...
BoCoKeith's user avatar
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2 votes
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Does {7,4|3} have a realization on the Klein quartic?

It was pointed out to me recently that the polyhedron {7,4|3} has the same automorphism group as the Klein quartic. Specifically {7,4|3} is: $\langle \rho_0,\rho_1,\rho_2 \mid \rho_0^2, \rho_1^2, \...
Sriotchilism O'Zaic's user avatar
8 votes
2 answers
177 views

Probability of each type of inscribed octahedron

Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. ...
Jeppe Stig Nielsen's user avatar
2 votes
1 answer
52 views

Can a oblique antiprism be constructed?

Can a oblique antiprism be constructed? Intuitively, it would seem oblique antiprism exist: Take any right antiprism and translate one of the parallel faces within it's plane. I'm baffled, though, ...
SRobertJames's user avatar
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Basis of the intersection of a cone and its dual.

Let $a_1,\dots, a_m$ be vectors in $\mathbb R^n$, let $A\in\mathbb R^{n\times m}$ whose columns are $a_i$s. The cone generated by $A$ (denoted $\operatorname{cone}A$) is $C=\{ Aw: 0\leq w \}$, its ...
P. Quinton's user avatar
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8 votes
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146 views

Is it possible for the dihedral angles of a polyhedron to all grow simultaneously?

Suppose $P$ and $Q$ are combinatorially equivalent non-self-intersecting polyhedra in $\mathbb{R}^3$, with $f$ a map from edges of $P$ to edges of $Q$ under said combinatorial equivalence. Let $\theta(...
RavenclawPrefect's user avatar
8 votes
3 answers
388 views

Volume of a great icosahedron

This is the image of a Great Icosahedron that I obtained starting from the coordinates of the vertices, as $A$ ,$B$ $C$, etc.. Now I want to calculate the volume of the solid. In internet (as in this ...
Emilio Novati's user avatar
0 votes
1 answer
34 views

Expected Length of Walk on Truncated Icosahedron

Consider a truncated icosahedron with 12 pentagons and 20 hexagons. Starting from a hexagonal face, we go to any neighboring polygon randomly with equal probability. What is the expected number of ...
Sny's user avatar
  • 151
1 vote
1 answer
142 views

What are the formulas for the circumradius, surface area and volume of each Kepler-Poinsot polyhderon based on the length of the entire edge?

Every formula I've found online is based on only a part of the total edge. If anyone knows the formulas based on each red edge below I would greatly appreciate it. A derivation of those formulas ...
John's user avatar
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Enumeration of uniform polyhedra

It is known that there are two infinite classes of polyhedra (prisms and antiprisms) together with 75 uniform polyhedra that do not belong to these classes. For regular convex polyhedra (i.e., ...
Martin's user avatar
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Simple proof that an extreme point belongs to the boundary of a halfspace containing all other points

Definitions Let $\mathbf{A}\in \mathbb{R}^{m\times n}, \mathbf{b}\in \mathbb{R}^{m}$ and $\mathbf{a} \in \mathbb{R}^n\setminus\{ \mathbf{0} \}, b \in \mathbb{R}$. The set $\{ \mathbf{x} \in \mathbb{R}...
Yuumita's user avatar
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0 answers
52 views

Optimality criterion for convex optimization problem when the feasible set is the polyhedral set

Consider this specific optimization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} & & f_0(x) \\ & \text{subject to} & & f_i(x) \leq 0, \; i = 1, \...
Đặng Hải Đăng's user avatar
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0 answers
52 views

Tilling of polygons on a sphere

Which convex polygons can be used to tile a sphere? For example, if I had a 20 equilateral triangles, I could form a sphere in the shape of a icosahedron. If I had 2 hexagons for every pentagon, I ...
Anirudh Yamunan Govindarajan's user avatar
6 votes
1 answer
214 views

Does there exist a polynomial $f(x, y, z)$ which is positive iff |x|, |y|, |z| are sides of a triangle?

The following problem appeared in the Kömal (Hungarian mathematical journal) in year 1997. Official Source: Problem N 145 at the end of the page here. Problem. Does there exist a polynomial $f(x, y, z)...
Prism's user avatar
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4 votes
4 answers
502 views

Dodecahedral number visualization

The dodecahedral numbers, 0, 1, 20, 84, 220, 455, 816, 1330, 2024, 2925, 4060, ... numbers of the form ${3 n \choose 3}$ (A006566). Does anyone have a good visualization of these? In particular, I'd ...
Ed Pegg's user avatar
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0 votes
0 answers
32 views

Looking for good books on convex polyhedra theory

My knowledge on convex polyhedra and systems of linear inequalities (facets, edges, Farkas Lemma, projections, duality, etc.) is very scattered, and I'l like to go through a book to solidify it. I'm ...
user56202's user avatar
  • 920
3 votes
1 answer
132 views

Combinatorial Identity from $\textrm{sinc}(x)$ integrals

Setup I've been reading/looking into variations on the $si(n) := \int_0^{\infty}\textrm{sinc}^n(x)$ for some power $n$ an integer, or the more general $si(a,b):= \int_0^{\infty}\frac{\sin^a(x)}{x^b}$ (...
Moya's user avatar
  • 5,298
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0 answers
60 views

German online explanation of Schwarz lantern

I'd like to share this thing I just came upon with a friend who doesn't read English. Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of ...
ariola's user avatar
  • 101
2 votes
1 answer
108 views

Why is There a Repeating Structure of the Distance Matrices of the Graphs of Regular Polytopes?

In some work I've done recently. I have been analyzing the distance matrix of the regular polyhedra in multiple dimensions and have found some curious results. Starting in 2 dimensions, I have found ...
Aidan R.S.'s user avatar
1 vote
0 answers
49 views

Minkowski sums of polyhedra given by inequalities

Suppose we are given two $n$-dimensional polyhedra \begin{align} P &= \{ x \in \mathbb{R}^n | \alpha_i(x) \leq a_i, i = 1, \dots, k \} \\ Q &= \{ y \in \mathbb{R}^n | \beta_i(y) \leq ...
Eike Schulte's user avatar
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0 votes
0 answers
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Extreme rays of a polyhedral

I was reading the following https://faculty.coe.drexel.edu/jwalsh/eces811/linProg.pdf, and was confused at one of the lines. They defined a polyhedral cone to be $$\{y\in \mathbb{R}^N | Hy≤0\}.$$ Also,...
David's user avatar
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0 votes
0 answers
28 views

Simplicial representative of the Thom class

Assume $p \colon Q \to P$ is a piecewise linear $n$-dimensional disc bundle. Let's consider triangulations $L$ and $K$ of $Q$ and $P$ in which $p$ is simplicial, and the subpolyhedron of spheres in $Q$...
Alex's user avatar
  • 141
2 votes
0 answers
68 views

Derivation of volume of regular rhombicosidodecahedron?

I understand how to find the surface area of a regular rhombicosidodecahedron. The formula given on Wikipedia is simple to use. Surface area is simply the sum of the areas of each face, which I can ...
Patrick O'Brien's user avatar
0 votes
0 answers
18 views

Is rδηδ an identity on ARPs of type {p,p}?

Background In their book Regular Complex Polytopes, Coxeter remarks that the three regular complex polytopes are "remarkably similar" to the tetrahedron, cube and octahedron. (p. 127) ...
Sriotchilism O'Zaic's user avatar
2 votes
2 answers
110 views

Generalising Thales theorem for points on a sphere to form a 3-orthoscheme (tetrahedron.)

I am trying to find the condition that four points $p_1,p_2,p_3,p_4$ on the unit sphere $\mathbb{S}^1$ need to statisy in order to form a 3-orthoscheme (Tetrahedron with all faces as right angled ...
Vishesh's user avatar
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1 vote
0 answers
9 views

Are there any applications for editing 2d Schlegel diagrams and then projecting them back onto a 3d polyhedron? [closed]

I'm trying to create some (hopefully) interesting polyhedra with Robert Webb's Great Stella / Stella 4D, but even its power users seem to only be able to do this kind of thing indirectly (or via ...
nickpelling's user avatar
1 vote
1 answer
157 views

Uniqueness of vertex optimal solution in linear programming

When we consider a linear programming of the following form: $$\begin{equation} \begin{aligned} \min_{x} \quad & { c^Tx} \\ \textrm{subject to} \quad & x\in P \\ \...
南洋小學生's user avatar
5 votes
1 answer
122 views

Reinhardt's Polyhedron as the first couterexample to the second part of Hilbert's 18th problem

In 1928 Karl Reinhardt published a first solution to the second part of Hilbert's 18th problem "Über die Zerlegung der euklidischen Räume in kongruente Polytope" in "Sitz. Ber. Preuß. ...
Thomas Preu's user avatar
  • 2,002
0 votes
0 answers
53 views

Prove the existence of a halfspace containing the polyhedron.

I have been working on the following problem for a whole day: Let $F$ be a face of polyhedron $P$ in $R^d$. Let $H\subset R^d$ be a halfspace containing F. Show there exists a halfspace $H'$ ...
Yunfei's user avatar
  • 1
4 votes
1 answer
170 views

The intersection of the facets on the polyhedron

Given a point set $E=\{\alpha_j\}_{j=1}^m\subset \mathbb{N}^{n}$ ($1\leq m< \infty$). Define the polyhedron $\mathcal{N}(E)$ to be the convex hull of the set \begin{equation*} \bigcup_{j=1}^m \...
cbi's user avatar
  • 59
0 votes
3 answers
319 views

What regular polyhedra can be tightly packed?

By "tightly packed", I mean that 3-dimensional space can be occupied solely by a collection of these same-sized regular polyhedra with no air gaps in between. I can think of three: ...
robert bristow-johnson's user avatar
2 votes
2 answers
56 views

C3 symmetry, chirality and polytopes [closed]

Is there some analogy for the phenomenon of chirality related to enantiomorph polyhedrons, but where the family of enantiomorphs is more than two polyhedra, in three or more dimensions, and related ...
Esteban R Q's user avatar
2 votes
0 answers
47 views

Is Kelvin structure the optimal solution for a three-dimensional foam when restricted to only one shape?

Weaire–Phelan structure is known as a more optimal solution for Kelvin problem than Kelvin structure, which is Bitruncated cubic honeycomb. However, it uses two different shapes. When restricted to ...
rus9384's user avatar
  • 411
1 vote
0 answers
23 views

Is there a theory of isometrically embedded polyhedra on manifolds?

There is a book that is called embeddings in manifolds that studies topological embeddings and how they relate to each other (by homeomorphisms). I was wondering if there is a study of isometrically ...
MakKa's user avatar
  • 31
6 votes
0 answers
89 views

What tools can show that (possibly irregular) dodecahedra do not fill space?

Here is a fairly natural question: Can three-dimensional space be filled with convex polyhedra of the same incidence structure (if not the same geometry) as the regular dodecahedron, such that four ...
RavenclawPrefect's user avatar

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