# Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

685 questions
68 views

### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 (...
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 $\Bbb R^d$ ...
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 ...
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 ...
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
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 ...
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?
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 $r^1$,......
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 ...
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 ...
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. ...
64 views

64 views

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 ...
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 ...
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 ...
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-...
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 ...
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 $X = \{x_1, \dots, x_m\} \subset \mathbb{R}^d$ ...