Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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1answer
39 views

Trying to create a 3D model of Kelvin’s Tetradecahedron / Tetrakaidecahedron polyhedra

How can I go about creating a 3D model / 3D image of a Kelvin’s Tetrakaidecahedron Cell / Tetrakaidecahedron. I planned on using Octave to 3D model an image it mathematically then convert that into a ...
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15 views

The point of intersection of the four space diagonals of a general parallelepiped, the centroid

I know that the centroid of a parallelogram is the intersection of the diagonals, but is it true that the centroid of a parallelepiped is the intersection of the space diagonals? I'm talking about the ...
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20 views

Embedding a polyhedron given a relative interior point

How can I embed a polyhedron given in an H-representation (hyperplane inequalities) which is not full-dimensional, given a relative interior point?
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1answer
72 views

Platonic solids - topological and geometrical conditions

With V, E, F as the numbers of vertices, edges and faces of a given polyhedron and based on Euler‘s polyhedron formula $$ V - E + F = 2 $$ it is quite simple to derive a necessary topological ...
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28 views

Algorithm for variant of Alexandrov's uniqueness theorem

Alexandrov's uniqueness theorem states for each development, there is a unique convex polyhedra with the metric induced by the development. For our purposes, we can view developments as finite planar (...
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1answer
66 views

Every map of a sphere can be homotoped to a map whose fiber is finite: using simplicial approximation

Let $f:S^n\to S^n$ be an arbitrary continuous map. It is a part of Exercise 15 in Hatcher's AT, chapter 4.1, to show that $f$ is homotopic to a map $g$ such that there exists a point $q\in S^n$ with $...
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1answer
50 views

Adjacent vertices of a polyhedron.

Given a polyhedron $P$ we have two vertices $x'$, $y'$ to be adjacent, if their convex hull is a one-dimensional face of the polyhedron. I want to show that this is equivalent to the fact, that there ...
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1answer
90 views

Smallest non-4-space-filling polytesseract?

As a follow-up to Smallest non-space-filling polycube?, what's the smallest polychoron produced by fusing tesseracts 3-face to 3-face which does not fill 4-space?
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31 views

Piecewise smooth set does not contain interior points

In the middle of page 4 of this paper the author states : It is obvious that any set of dimension less than n which is piecewise-smooth in V does not contain interior points. which I do not ...
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1answer
71 views

How small can the average dihedral angle be in a large polyhedron?

Given a non-degenerate non-self-intersecting polyhedron $P$, consider the average of the dihedral angles at each edge in $P$. For small polyhedra, this average can be fairly small; for instance, a ...
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2answers
153 views

Smallest non-space-filling polycube?

The title nearly says it all: what is the fewest number of cubes that can be fused face-to-face into a polyhedron that does not fill space? The smallest that seemed like a sure non-tiler to me was 9: ...
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31 views

Is $\big\{x \in \mathbb{R}^3 :\ \forall t \in [1,\infty)\bullet x_1 + e^{-t}x_2 + e^{-2t}x_3 \leq 1.1\big\}$ a convex polyhedron?

A convex polyhedron is a set $S$ that satisfies $S = \{x \in \mathbb{R}^n :\ Ax\preceq b,\ Cx = d\}$ for some positive integers $m, n, p \in \{1,2,\dots\}$, some matrices $A \in \mathbb{R}^{m\times n}$...
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41 views

Understanding Peter Engel's space-filling tricontaoctahedron: What is the notation for these bounding planes?

In Peter Engel's 1991 paper Über Wirkungsbereichsteilungen von kubischer Symmetrie, a 38-sided convex polyhedron is described which apparently can tile space. I'm interested in better understanding ...
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2answers
269 views

Name of this polyhedron with 17 faces

I recently saw a jungle gym in a playground constructed with ropes and small vertices. It formed a really strange polyhedron that I had never seen before. I've searched all over for it, but cannot ...
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59 views

Octahedron to tetrahedron ratio in generalized polyominoes in the tetrahedral-octahedral honeycomb

Drake Thomas and I have proposed a sequence A343909 to the On-Line Encyclopedia of Integer Sequences (OEIS), which counts "generalized polyforms": generalizations of free polyominoes (Tetris ...
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1answer
40 views

Representation of a Face of a Polytope

In Ziegler's book "Lectures on Polytopes" it is proven that each face $F$ of a polytope $$P=\mathrm{conv}(V)=\{x\in\mathbb{R}^n:~Ax\leq\textbf{1}\}$$ (assuming $0\in\text{int}(P)$) can be ...
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2answers
55 views

Is $\mathbb{R}^2$ a polyhedron? [duplicate]

I'll appreciate if anyone can explain to me if $\mathbb{R}^n$ is a polyhedron? I saw this link here which says $\mathbb{R}^n$ is not a polyhedron, but could one think of $\mathbb{R}^n$ as an ...
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1answer
53 views

A geometric realization of a 4-polytope with 7 vertices

I am looking for a geometric realization of a 4-polytope with 7 vertices. A list can be found in "An Enumeration of Simplicial 4-Polytopes with 8 vertices" by Grünbaum and V. P. Sreedharan. ...
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1answer
36 views

Do a polytope $P$ and its polar dual $P^\circ$ have the same centroid?

One natural choice for a center point of a convex polytope $P\subset\Bbb R^d$ is the average of all its vertices $$c(P):=\frac{v_1+\cdots +v_n}{n}.$$ Call $P$ centered if $c(P)=0$ Question: If $P$ is ...
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1answer
23 views

Number of Deformation Parameters of Simple Polyhedron Equals Number of Edges

Suppose $P$ is a simple convex polyhedron with $n$ faces. Euler's formula and the handshaking lemma tell us that the number of edges $E=3n-6$. By a deformation of $P$, I mean a polyhedron, ...
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24 views

Finding the minimum volume of all possible regions made by a concave polyhedron and its convex hull

I'm working on a project creating 3d-printed joint models and I would like to minimize the amount of plastic needed in adding protrusions that would form the perfect makebed. A visual example of the ...
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1answer
30 views

Prove that this linear relaxation has half-integral extreme points

Given a graph $G=(V,E)$, here is a Linear Relaxation of the edge cover polytope: (1) For each $v \in V, \sum_{e \in \delta(v)} x_e \geq 1.$ (2) For each $e \in E$, $0 \leq x_e \leq 1.$ Here $\delta(S)$...
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2answers
79 views

How do I find or derive circumspherical radii, surface area, and volume for these 5 non-traditional concave polyhedra?

How do I find or derive circumspherical radii, surface area, and volume for these 5 non-traditional concave polyhedra? As much of a mathematics enthusiast I am, I'm stuck on a huge roadblock in ...
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1answer
19 views

Finding the Canonical Polyhedron associated with a 3-connected simple graphs.

I am not a professional mathematician but I am a reasonably competent programmer and I am also no stranger mathematics, though I must say that my usual domain is closer to calculus and functions ...
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29 views

Is there any new developments on the Barnette's conjecture?

When I searching for interesting math problems. I find there is a graph theory conjecture called the Barnette's conjecture. The statement is: Is every bipartite simple polyhedron Hamiltonian? A early ...
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1answer
20 views

Prove the solution set of a Linear programming problem is a polyhedron

Problem: Prove the solution set of a Linear programming problem is a polyhedron. I have proved the feasible set of an LPP is a polyhedron (as the constraints are inequations). Now I want to show the ...
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19 views

Every convex polyhedron has a stable face

Recently stumbled across the idea of monostatic polytopes, and I was reminded of an old book I'd read that gave a "proof" that every polyhedron has at least one stable face. Since such a ...
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1answer
85 views

How to understand the reduced cost in simplex method?

I’m reading a note on the simplex method. The author mentions a quantity called “reduced cost”, yet no interpretation of it is provided. Here’s the setup: $c \in \mathbb{R}^n$ is the cost vector, $A \...
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272 views

What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?

Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with unit perimeter? A reasonable first guess would be the regular tetrahedron of side ...
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1answer
150 views

Do there exist uniform triangular prisms with all vertices in $\mathbb Z^3$?

It's quite easy to find a regular square prism (cube) or a regular triangular antiprism (octahedron) with vertices in $\mathbb Z^3$. Take for instance, take the convex hulls $$ \begin{align*} &\...
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9 views

Hyperplane definitions from set of half-spaces

A $d$-dimensional polyhedron in $\mathbb{Q}^d$ can be described by a set of (closed) half-spaces $AX \ge B$, where $A \in \mathbb{Q}^{n \times d}$, $X = (x_1, \dots, x_d)$, $B \in \mathbb{Q}^n$. If ...
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1answer
42 views

Existence of an inner point in a nonempty polyhedron

I was reading some notes on polyhedral analysis and encountered a proof that confused me. The proof is in the book named "Integer and Combinatorial Optimization" by Wolsey and Nemhauser. Let ...
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18 views

Criterion for polytope being full-dimensional using bounding hyperplanes

Suppose $P$ is a polytope of $\mathbb{R}^n$ defined by a (bounded) finite family of half-spaces $t_i \cdot x \leq q_i$ with every $t_i \in \mathbb{R}^n$ non-zero and $q \in \mathbb{R}$. Let $V \...
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155 views

Are there “close” solutions to Hilbert's third problem?

Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra $P,Q$ of equal volume are equidecomposable by cutting $P$ into finitely many polyhedral pieces and rearranging them ...
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2answers
50 views

Having trouble understanding difference between polyhedron and polytope

Hi i´m reading a pdf about linear programming and i´m having trouble understanding the difference between a polyhedron and polytope between those two definitions A polyhedron P ⊆ $R^{n}$ is the set ...
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1answer
112 views

Finding the best (yet suboptimal) extreme point in an unbounded LP / open polyhedron?

Given the following open example polyhedron: \begin{equation} \begin{aligned} x_1 & -x_2 & -s & \le 0 \\ x_1 & +x_2 & & \le 3 \\ x_1 & & & \ge 1 \\ ...
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39 views

An example of polytope with exponential number of vertices?

I am looking for an example of polytope with exponential number of vertices? (Like $2^n$ vertices) I guess that dual of cyclic polytopes has exponential number of vertices. Are there simpler & ...
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1answer
44 views

Does this tetrahedron have a name?

Consider the tetrahedron with vertices at: $$(0,0,0)$$ $$(2,-\sqrt2,0)$$ $$(2,\sqrt2,0)$$ $$(2,0,2)$$ This tetrahedron is not regular but does it have any notable properties? It appears to have some ...
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5 views

Computation of maximal angle of polyhedral cone with respect to a subspace

Let $\mathbb R^n_{\geq 0} = \{x\in\mathbb R^n:x\geq0\text{ entrywise}\}$ as usual be the non-negative orthant, and suppose that $U$ is a $k$-dimensional subspace of $\mathbb R^n$ that intersects $\...
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1answer
40 views

Proof of dimension of Polyhedron

Before proving the theorem the book introduces a new definition called implicit equality. What does exactly the below equality mean? $$ A^{=}x=b^{=} $$ What is inside this set? $$ \{x \in \mathbb{R}^...
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1answer
26 views

Is the following set polyhedral at all of its points?

I'm trying to understand the following definition: Let $x \in A \subset \mathbb R^n$. We say that $A$ is polyhedral at $x$ iff there is a neighborhood $U$ of $x$ and a polyhedron $B$ such that $A \...
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34 views

Voronoi Cell of the Dual Lattice?

A lattice is a discrete (we may assume full-rank) subgroup of $\mathbb{R}^n$, often written as the image of $\mathbb{Z}^n$ under a particular matrix $\mathbf{B}\in\mathbb{R}^{n\times n}$ (a basis of ...
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33 views

Determine whether $S_1=\{{x\in{R^2}}\mid |x_1|+|x_2|\le 1\},S_2=\{{x\in{R^2}}\mid |x_1|+|x_2|\ge 1\}$ are polyhedral.

Determine whether $$S_1=\{{x\in{R^2}} \mid |x_1|+|x_2|\le 1\} $$ $$S_2=\{{x\in{R^2}} \mid |x_1|+|x_2|\ge 1\} $$ are polyhedral. The definition of polyhedron is that $S =\{{x\in{R^n}} \mid Ax\ge b\} $,...
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2answers
172 views

How to construct a chamfered dodecahedron with equilateral or coplanar faces?

According to wikipedia, "The chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (geometry) (edge-...
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2answers
61 views

Can each edge of a Wythoffian polytope be flipped by a reflection?

A Wythoffian polytope $P\subset\Bbb R^d$ is an orbit polytope of a finite reflection group, that is, $$P:=\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in\Gamma\},$$ where $\Gamma$ is a finite ...
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1answer
77 views

Proving that a $3$-dimensional compact polytope such that every two vertices are adjacent is a tetrahedron

So I have to prove that given a 3-dimensional compact polytope such that every two vertices are adjacent, then it is a tetrahedron. Somehow I'm able to visualise from the fact that every two vertices ...
2
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1answer
142 views

Given dihedral angles, find a set of edges

In the paper Space Vectors Forming Rational Angles a special set of tetrahedra is mentioned. "The remaining three are in the R-orbit of the tetrahedron with dihedral angles (π/7, 3π/7, π/3, π/3, ...
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0answers
37 views

Extreme points of a “boundary” of a polyhedral set are the extreme points of the polyhedral set

Consider a matrix $A_{m\times n}$ and $b_{n\times 1}$. Let the space be $\mathbb R^n$. Let $S:=\{ x\in\mathbb R^n:Ax\le b, x\ge 0\}$ and $T:=\{x\in\mathbb R^n:Ax=b, x\ge 0\}$. I want to show that if $\...
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1answer
238 views

Given isosceles triangles $\triangle ABC$ and $\triangle DBF$ (all spherical chords), identify the chord $\overline {DF}$ so that $|AD| = |DF| = |FC|$

tl;dr: As shown in the image below, find the chord $\overline {DF}$ so that $|\overline {AD}| = |\overline {DF}|$, and have the answer be in the form of the ratio between $|\overline {AC}|$ and $|\...
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25 views

farkas-minkowski theorem specific exmaple

I was reading about farkas-minkowski theorem, which basically said a convex cone is polyhedral iff it's finitely generated. The theorem make sense, but when I played with some example I met the ...

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