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Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

2
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2answers
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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 ...
0
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0answers
13 views

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 ...
2
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1answer
26 views

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 ...
4
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0answers
40 views

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 ...
6
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3answers
146 views

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 ...
0
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1answer
41 views

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 ...
5
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1answer
467 views

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 ...
0
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2answers
46 views

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 ...
0
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1answer
189 views

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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0answers
28 views

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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0answers
58 views

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 ...
5
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3answers
63 views

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 ...
0
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1answer
15 views

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
3
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1answer
67 views

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 ...
1
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1answer
32 views

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?
0
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1answer
77 views

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 $r^1$,......
0
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1answer
38 views

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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0answers
32 views

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 ...
3
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1answer
53 views

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. ...
1
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1answer
64 views

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 $...
8
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2answers
179 views

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. ...
0
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2answers
53 views

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\...
3
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3answers
64 views

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^...
0
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3answers
79 views

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.
4
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2answers
86 views

Combinatorial or polyhedral description for tropicalization of the positive subset of a real linear subspace

I had two questions: one regarding a definition of tropicalized linear subspaces, and the second about how to find similar characterizations for the logarithmic limit set of the positive subset of a ...
1
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1answer
68 views

Is the following function concave? Where does it attain its minimum?Its maximum?

Consider the function $f: \{1 , \ldots, m\} \times \Delta^{m} \rightarrow \mathbb{R}$ where \begin{equation} \Delta^{m}:=\{\lambda \in \mathbb{R}^{m} \ | \ \sum_{i=1}^{m} \lambda_i = 1 , \ \...
4
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1answer
70 views

What kind of polyhedron is this?

What kind of polyhedron is this? I understand 20 faces is the largest regular polyhedron, but this one seems to have 36 faces. Source: It is the logo of the Green Climate Fund
2
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1answer
37 views

Finding all “soccer” polyhedra (Each vertex meets three faces: two $m$-gons and one $n$-gon ($m\neq n$))

I am dealing with the test of the OBM (Brasilian Math Olympiad), University level, 2016, phase 2. As I've said at this topic (question 1), this other (question 2) and this (question 3), I hope ...
0
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1answer
53 views

Local optimality of non-convex quadratic minimization with linear constraints

We are interested in minimizing a quadratic function, which is not convex. The feasible set is a polyhedron. We know that the global minimum is at a vertex when the function is concave. Also, there ...
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0answers
23 views

Prove $C(t)$ varies in an absolutely continuous way

Prove that there exists an absolutely continuous function $v(\cdot): I → R$ such that for any $y\in\mathbb{R}^n$ and $s,t\in [0,T]$, $$|dist (y, C(s)) − dist (y, C(t))| ≤ |v(s) − v(t)|$$ for all $s,t\...
0
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0answers
45 views

Shapes Homeomorphic to a torus

This polyhedron is made up of 8 identical octahedrons. If I were to take away the octahedron at the very top, would the remaining polyhedron still be homeomorphic to a torus? (I'm concerned that ...
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2answers
44 views

$v-e+f=2-2g$ as a topological invariant

Euler's theorem was expanded to encompass polyhedrons homeomorphic to not only spheres but also $g$-holed toruses. I've tried to understand proofs about how $2-2g$ is a topological invariant but have ...
0
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1answer
28 views

Is a structure made up of two polyhedrons connected by a common vertex or edge still called a polyhedron?

This link(https://plus.maths.org/content/eulers-polyhedron-formula) states that two separate polyhedrons joined in this manner cannot be called polyhedrons. But mathematician Hessel once pointed out ...
3
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2answers
47 views

How can you prove that every polyhedron can be dissected into tetrahedrons?

I have always loosely accepted the fact that any polyhedron can be dissected into tetrahedrons just like any polygon can be dissected into triangles. But how can I prove precisely that any polyhedron ...
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0answers
19 views

Does Descartes' theorem work for non-convex polyhedrons?

Does Descartes' theorem work for non-convex polyhedrons? and if it does what are the characteristics of 3 dimensional shapes that don't fall under Descartes' theorem?
9
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1answer
1k views

Rolling icosahedron Hamiltonian path

A cube has 24 orientations. By rolling the cube on its edge within the perimeter of a $2\times4$ rectangle 3 times, all 24 orientations are reached and the next roll returns the cube to both the ...
7
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3answers
620 views

Count the number of shapes in a polyhedron.

So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is: The faces of the following polyhedron are either triangles or squares. Each triangle is ...
3
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2answers
63 views

What is the minimun faces that you need to make a polyhedron with all the faces equilateral triangles, but one(base) square? (not a Pyramid)

I'm not a Mathematical person, but I know some things. I can't imagine a polyhedron with those characteristics. That polyhedron do exists? Is it possible to make something like that? (for that is why ...
0
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0answers
21 views

Given a polyhedron, show that the projection can be made with the extreme rays

I am stuck with the following question: Let $Q = \{ (x,y) \in R^n_+ \times R^p_+ : Ax + Gy \leq b\}$ and $\{v^t\}^T_{t=1}$ be the extreme rays of $V=\{v \in R^m_+ : vA \geq 0\}$. Use Farkas' lemma ...
0
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1answer
44 views

write a point in the plane as a convex combination

Suppose we have a polyhedron with vertices (extreme points) $e_1 = (4,5)$, $e_2 = (0,3)$ , $e_3=(1,2)$, $e_4=(6,0)$ and extreme direction $d_1 = (1,0)$. I want to write the vector $(10,1)$ as convex ...
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0answers
44 views

Volume of polyhedron similar to prism but with different bases

Imagine a polyhedron similar to a prism, with parallel but different bases. Let the bases have the same number of sides, so that every vertex on a base is connected to exactly one vertex on the other ...
0
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1answer
42 views

Convex hull of union of nondisjoint polyhedra

Is there a theorem which proves that the convex hull of the union of nondisjoint polyhedra is also polyhedral?
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0answers
18 views

Set of hyperplanes to polyhedron topology

Problem By intersection of a set of planes (in three-dimensional space) I constructed an edge-vertex topology structure (see image). I was able to reconstruct the polygons in this structure by ...
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1answer
138 views

how to find vertices of polyhedron, given inequalities?

I have a polyhedron which is defined by the following system of inequalities: $$ \left\{ \begin{array}{c} x \leq 2 \\ y \leq 1 \\ x + y + z \leq 1\\ x + y + 2z \leq 1 \end{array} \right. $$ I want ...
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0answers
30 views

Let $S = \{y_1a_1+y_2a_2~|~-1 \leq y_1,y_2 \leq 1\}$ where $a_1 , a_2 \in \mathbb{R}^2$.

Let $S = \{y_1a_1+y_2a_2~|~-1 \leq y_1,y_2 \leq 1\}$ where $a_1 , a_2 \in \mathbb{R}^2$. Show $S$ is a polyhedron. Assume $a_1,a_2$ are linearly independent. Now I believe this question was asked ...
0
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1answer
62 views

Maximal polyhedra of a polyhedral complex

According to this paper by Diane Maclagan (https://arxiv.org/abs/1207.1925, p.10), a polyhedral complex is pure if the dimension of every maximal polyhedron is the same, which is shown in the figure ...
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1answer
86 views

Intersection of Two Polyhedrons Linear Programming

I am stuck on the following linear programming problem: If P and Q are two n-dimensional polyhedra Devise a linear programming such that: If P ∩ Q is nonempty, return a point in P ∩ Q Else: LP is ...
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0answers
18 views

Constructing $n$-Cube, $n$-Cell and $n$-Orthoplex Groups

What are the properties of the rotating cube, tetra- and octahedron that give them the group structures they have, and can those properties be generalised to get at the group structure of their higher-...
0
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1answer
356 views

Find all extreme points of 3 variable polyhedral set

Find all the extreme points of the polyhedral set, $X=\{(x_1,x_2,x_3):x_1-x_2+x_3\leq 1, x_1-2x_2\leq 4, x_1,x_2,x_3\geq 0\}$ I usually start out by drawing the feasible region but I couldn't do it ...
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0answers
45 views

Approximate a convex body by polyhedron

There are many results how many vertices we need to approximate a convex body. It is known, for a given $\tau > 0$, we need $O(\tau^d)$ vertice of set $X = \{x_1, \dots, x_m\} \subset \mathbb{R}^d$ ...