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Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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Find a polytope of dimension $n-1$ from a $(0,1)$-polytope of $\mathbb R^n$

A $(0,1)$-polytope of $\mathbb R^n$ is the convex hull of a finite set $V \subset \{0,1\}^n$. Problem: Let $P$ be a $(0,1)$-polytope of $\mathbb R^n$. Show that $$P_1 : = \left\{ x \in \mathbb R^n \...
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Is this integral the same over dual polyhedra of equal volume?

The integral $$\int_V \frac{d^3x\,d^3y}{|\mathbf{x}-\mathbf{y}|}$$ over various regions $V$ is of interest in physics, where it is related to the gravitational binding energy of an object with uniform ...
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Hexahedron made with similar polygons

The isohedra are made with congruent polygons. I wondered what could be built with similar polygons that are not congruent. Here is one where edgelengths are powers of $\sqrt{\psi}$, where $\psi$=...
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Average area of the shadow of a convex shape [closed]

What is the average area of the shadow of a convex shape taken over all possible orientations? If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be ...
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Motivation for Projection definition in polytopes

In the following Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization by EGON BALAS polytope projection definitions are given: Given a polyhedron of the form $$Q := \...
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what is the relation between projection of a polytope and this polytope?

suppose we have a polytope $P$ in $R^{4}$ and $-1\leq x_{3}\leq 4$ and $0\leq x_{4}\leq 6$, if I replace the upper and lower bound of $x_{3}$ and $x_{4}$ (it depends on the sign of variables $x_{4}$ ...
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Is the average number of edges per 2-faces of a convex 3-polytope always below six?

Is the average number of edges per 2-faces of a convex 3-polytope always below six? Which theorem can answer this question or how do you prove this?
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Using Euler's formula with an icosidodecahedron

So I have this question: "An icosidodecahedron is a convex polyhedron built out of P regular pentagons and T equilateral triangles, for suitable $P, T \in \mathbb{N}$, with the rule that every ...
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Looking for systems of linear equations to construct polyhedrons

I am looking for linear equation systems for the construction of standard polyhedra in 3 dimensions. I found another one beside the simple cube, the Cuboctahedron: ...
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What is the name of this hexagon/pentagon polyhedron?

What is the name of this convex polyhedron?                     $(V,E,F)=(14,36,24)$. The top and bottom vertices are degree-$6$, spanning ...
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Chebyshev center of a polyhedron: nonnegativity issue

Let us have a polyhedron, defined by the inequalities of the form: $$ \mathcal{P} = \{ x \ | \ a_i^T x \leq b_i, \ i=1,\ldots,m \} $$ Here on page 19, the way to calculate Chebyshev center is given ...
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combinatorics of connected components of a bicolored polyhedron skeleton

Consider the skeleton of a 3-dimensional convex polyhedron with all vertices being either red or black. We have n red and m black vertices. n < m. Take the largest sub-graph that consists of black ...
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Nonperiodic space-filling polyhedra

For periodic space-filling polyhedra, the maximum number of faces seems to be 38, according to On Space Groups and Dirichlet-Voronoi Stereohedra. For non-periodic space-filling polyhedra, the ...
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Why is the permutohedron simple?

I am working with the permutohedron in $\mathbb{R}^n$ which is defined as the convex hull of $n!$ vectors as follows: $$\Pi_n := conv\{(\sigma(1), \ldots, \sigma(n))\ |\ \sigma \text{ permutation of }...
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What solid does the truncation of an icosahedron approach?

i was watching some old maths fun related stuff on youtube, and i stumbled across this video: https://www.youtube.com/watch?v=cwWBpjeyRS0 in which the guy mentions how a football is obtained by ...
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Linear program geometry

I’ve tried to solve a question in my homework, and I don’t really know what to do. In the problem a polyhedron is given and I need to build the set of constraints that defines this polyhedron. The ...
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Construct Polytopes from Arbitrary Edge Sets

I was wondering if given an arbitrary set of edges in 3-space, if there is a way to determine if those edges can be used to construct a single polytope (allowing for the scaling of edge lengths). In $\...
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Area of Convex polyhedron (2D) with unordered vertices

I am aware that an algorithm exist to find area of a convex polyhedron when the vertices are given in order. But, I have a convex polyhedra which does not have vertices in order and I wish to compute ...
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Distribution of points on a sphere

Ok, to start with, please go easy on me - I only have secondary school level maths, and that was 24 years ago. I'm looking to work out how to evenly distribute points over a sphere. Specifically 20 ...
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1answer
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Euler's Formula To Show 2E=3V

I have this question: "Consider a convex polyhedron, all of whose faces are square or regular pentagons. Say there are m squares and n pentagons. Assume that each vertex lies on exactly 3 edges" ...
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What physical features of the tetrahedron prevent its symmetry group from being all of $S_4$?

This is an exercise from the book "Visual Group Theory" by Nathan Carter. Exercise $5.26$. As you know from the chapter, the symmetry group for the tetrahedron is $A_4$. We can think of it, as you ...
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How is a nonnegative orthant a cone?

My textbook says the following: The nonnegative orthant is the set of points with nonnegative components, i.e., $$\mathbb{R}_+^n = \{ x \in \mathbb{R}^n \mid x_i \ge 0, i = 1, \dots, n\} = \{ ...
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Neighboring solids in tetrahedral-octahedral honeycomb

In the tetrahedral-octahedral honeycomb, each vertex seems to be incident to 6 octahedra and 8 tetrahedra: Such simple combinatorial fact is probably well-known, or perhaps even obvious. However, ...
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What are all possible orientations of the Platonic Solids?

Five platonic solids. For each one, label all the faces. It does not matter what order or pattern you assign the labels. All that matters is that this assignment is fixed while you rotate the objects. ...
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Why are there only 13 Archimedean solids and not 14?

I just finished a project on solids for geometry and I couldn’t help but wonder out of curiosity why are there only 13 Archimedean solids and not 14 or more?
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How to understand the Conway recipe C969qD to construct this polyhedron? Canonicalization then quinto?

The answer to my previous question about the shape below is the Conway notation C969qD. Per the linked viewer in that answer: The specification consists of a ...
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What is the name of the shape generated by the vertices of the Soviet pennant on the Luna 2 spacecraft? Is it an polyhedron?

xkcd 2125 titled "Luna 2" refers to the object shown below. All faces seem to be pentagons. (found in this question.) Copy of the Soviet pennant sent on the Luna 2 probe to the moon, at the Kansas ...
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Is there a conformal mapping from the surface of a cube to the surface of a spherical cube that preserves edges?

Is there a conformal mapping (with certain singularities noted below) from the surface of a cube to the surface of a spherical cube that preserves edges? Note that this also implies that vertices and ...
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A convex polyhedron has 20 vertices and 12 faces. Each face of the polyhedron is bounded by the same number of edges. What is this common number?

A convex polyhedron has 20 vertices and 12 faces. Each face of the polyhedron is bounded by the same number of edges. What is this common number? If I am not mistaken , "this common number" is the ...
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Names For Polyhedra

I am trying to enumerate the number of distinct polyhedra that can be formed from a given number of vertices. So far, i have managed to finish the sets for 4, 5 and 6 vertices and is now a third of ...
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Two people take turns coloring a convex polyhedron

Rachel and Beatrice take turns coloring the faces of a convex polyhedron red and blue, respectively. A player wins if she gets her color on three faces that share a common vertex. If Rachel goes first ...
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Proving a set has no extreme points

Suppose we have the set {$x : Ax < b$} and we want to show that it has no extreme points. Graphically, and intuitively this would make sense to me because the points of intersection of all the ...
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Can a polyhedron be an empty set?

A polyhedron is defined as the intersection of finitely many generalized halfspaces. That is, a polyhedron is any set of the form $ \{x \in R : Ax \leq\ b \} $ I would like to understand this ...
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Why is this set not a polyhedron?

(Question from Stephen Boyd and Lieven Vandenberghe - Convex Optimization) $S = \{x \in \mathbb{R}^n |x \ge 0, x^{T}y \le 1$ for all $y \in \mathbb{R}^n$ such that $\lVert y \rVert_2 = 1$}. Is the set ...
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How to apply Euler's formula to polyhedra with pentagons and hexagons

I have just started with polyhedra (know Euler's formula etc..) not sure how to approach this? A shape has faces only consisting of P regular pentagons (all of the same size), and H regular hexagons (...
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Polytopes given by affine relations of their vertices

This is a reference request for a description of polytopes by affine relations of their vertices. Let me give a few examples of what I'm looking for: Taking $n+1$ points $a_0,\dots,a_n$ in $\Bbb R^d$ ...
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Decomposing a polyhedron in $\mathbb{R}^3$ into a lineality space, cone, and polytope.

Consider the set in $\mathbb{R}^3$ given by $$\{(x,y,z) : x+y+z\ge 3, x \ge 0\}$$ I can picture this set; it is simply the intersection of two half-spaces. A theorem states that every polyhedron can ...
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Comparing Regular-Faced Toroidal Polyhedra

Many apologies ahead of time, I have no idea how to phrase this question, and I'm certainly way out of my element. I'll do my best but please go easy on me. I wanted to make a polyhedra that was in ...
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How can I check which face of spherical polyhedron corresponds to a given euler angle?

I am specifically mapping a dodecahedron to a sphere and I am trying to get if a rotator is within the boundaries of a given face. Thank you
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Is there a theoretical maximum number of sides a single die can have? If so what is it?

Is there a theoretical maximum number of sides a single die can have? If so, what is it? The question is fairly self-explanatory. I read the Nerdist article "This d120 is the Largest Mathematically ...
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Is there an analog of the Law of Cosines that applies to polyhedra?

Is there a relationship between the side areas of a polyhedron $A_1, A_2, \ldots, A_n$ similar to the law of cosines for triangles?
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Proving a projection based on Farkas Lemma

In an Integer Programming book, a proof is given for a theorem but i do not understand a certain step: Theorem: Consider a polyhedron $P := \{(x,z) \in R^n \times R^p: Ax+Bz\leq b\}$ and let $r^1$,......
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Total number of ways to paint the faces of a regular icosahedron with $20$ distinct colors

If all the 20 faces of a regular icosahedron are painted with a set of 20 distinct colours then the total number of such icosahera possible. The cube analogue of this is more well known and the answer ...
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On Matrices that are close to Total Unimodularity

Assume we have some matrix $A$ which is totally unimodular(TU). By a well known results we have that the polyhedron $P = \{Ax\le b\}$ has integer vertices for all $b \in \mathbb{Z}^n$. My question is ...
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Does there exist a higher-dimensional 5-sided “tetrahedron + 1”?

The first shape is "0"-sided and is a point. The next shape is a line segment and it's "1"-sided. The next shape is a triangle and it's 3-sided. The next shape is a tetrahedron and it's 4-sided. ...
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Octahedra with four equilateral faces

Let $A_1, A_2, A_3, A'_1, A'_2, A'_3$ be the vertices of a (not-necessarily convex) octahedron; here $X'$ is the vertex not on an edge with $X$. Suppose that the four non-adjacent triangular faces $...
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Volume of an Irregular Octahedron from edge lengths?

Does anyone know how to calculate the volume of an irregular octahedron from the lengths of the edges? The octahedron has triangular faces, but the only information are the edge lengths. ...
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Conditions on boundedness of a polyhedron which makes it polytope. [closed]

Let $P=\{x \in \mathbb{R}^n \mid Ax=b, x\geq 0\}$ be a nonempty convex polyhedron (not bounded). Show that $P$ is bounded (i.e., it is a polytope) if and only if the linear inequality $Ax=0, \,\, x\...
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Connection/Consistency Between Different Definitions of Polyhedron? And How Does This Proof Apply to This Definition?

My textbook gives the following definition of a polyhedron: A polyhedron is defined as the solution set of a finite number of linear equalities and inequalities: $$\mathcal{P} = \{ x \mid a^...
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Prove that in any convex polyhedron, the number of faces that have an odd number of edges is even.

Prove that in any convex polyhedron, the number of faces that have an odd number of edges is even. I attempted to prove this by contradiction but didn't make any progress.