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Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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6 views

how to find extreme rays of a given polyhedral cone? [on hold]

I want to know the process of finding the extreme ray of a polyhedral equation.
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27 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 ...
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1answer
48 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. ...
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Facet inducing inequality for polyhedron with nonnegative recession cone

I have the following statement: Let $P \subseteq \mathbb{R}^n$ be any polyhedron that has the non-negative orthant as its recession cone. If a valid inequality $a^\top x\geq b$ is facet defining for ...
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1answer
51 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 $...
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2answers
151 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. ...
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2answers
46 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\...
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3answers
59 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^...
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3answers
49 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.
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2answers
79 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 ...
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1answer
56 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 , \ \...
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1answer
67 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
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1answer
32 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 ...
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1answer
35 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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22 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\...
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0answers
34 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
41 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 ...
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1answer
22 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 ...
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2answers
41 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
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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?
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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 ...
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3answers
610 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
57 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 ...
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10 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 ...
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1answer
40 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
38 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 ...
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1answer
33 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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15 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
76 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
25 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 ...
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1answer
58 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
53 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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17 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-...
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1answer
127 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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42 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 $X = \{x_1, \dots, x_m\} \subset \mathbb{R}^d$ ...
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1answer
33 views

For a polytope, why is the intersection of faces a face?

First I fix definitions: A polytope in $V$ is a subset that is bounded and is the intersection of half-spaces of $V$. A half-space is defined with respect to an affine form, and is that part of $V$ ...
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1answer
80 views

Why no higher-genus regular polyhedra?

It seems to be a fact that there are only five bounded connected non-selfintersecting polyhedra with identical regular-polygon faces and congruent vertices (i.e., you can pick a neighborhood of every ...
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1answer
152 views

Are the corner hypercubera polytopes self-dual?

Motivation: The polyhedron whose vertices are seven of the vertices of a cube (four on the bottom and three on top) - called a cubera - is self-dual. Does an analogous construction produce a self-dual ...
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2answers
44 views

Polyhedrons exclusively made out of even sided polygons

I know that the cube is the only 3 d shape which falls in polyhedrons but still is composed of squares, exclusively although its a even sided shape. I have noticed that after square there is no single ...
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1answer
73 views

Calculate the surface area of each face of a hexahedron and it's outward normal,given coordinates

Basically, I have a hexahedral finite element mesh. I know the coordinates of elements, I used the coordinate transformation into an isoparametric structure and shape(basis) functions to calculate the ...
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1answer
61 views

Concept on Euler's formula

Is there a much better way to proof and derive Euler's formula in geometrical figures? In that,F+V-2=E. For example an enclosed cube with 8 vertices, 6 faces and 12 edges. It is true that the edges, ...
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22 views

Determining a polyhedron from a series of inequalities

Given the following equations: $x+y+2z \leq 8$ $y+6z \leq 12$ $z \leq 4$ $y \leq 6$ $x,y,z \geq 0$ we have to determine all the vertices of the polyhedron in $R^3$, and indicate which vertices ...
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Isohedron splitting into isohedra

An order 3 dipyramid can be split into two regular tetrahedra. An order $n$ dipyramid can be split into $n$ isosceles tetrahedra. A rhombic dodecahedron can be split into four rhombic hexahedra. A ...
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1answer
36 views

How to determine the faces of an array of points?

Is there a way of determining the faces of any polyhedron? I have an array of $3$-dimensional points and no information what polyhedron should come out.
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0answers
62 views

Obtuse triangle octahedron packing

Any triangle can make an octahedron. Here are two of them for a 7-12-17 triangle. The green edges correspond to the middle length of 12. Is there any triangle $a-b-c$ where the concave octahedron'...
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0answers
62 views

What planar concave polyforms bound 3D space?

A square can bound a cube or a heptacube if a cube is glued to each face. An equilateral triangle can bound a tetrahedron and other shapes. The diamond (2 equilateral triangles) can make a ...
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0answers
45 views

Polyhedron and Gravity

I am currently looking for a method for pulling a polyhedron to the xy-plane. This means, that an arbitrary polyhedron (i.e. with holes and concavities) is not simply projected to the xy-plane, but "...
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0answers
48 views

Symmetric 3-pire map

Awhile ago, Martin Gardner introduced Scott Kim's symmetric 2-pire map. There are 12 empires, each with two regions. All twelve empires share a border. This is a 2-pire solution for the Empire ...
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2answers
408 views

Minimal surfaces for planar octagons and nonagons

4, 6, 8 triangles can make a tetrahedron and up. 6, 8, 9, 10 quadrilaterals can make a cube and up. 12, 16, 18, 20 pentagons can make a tetartoid or dodecahedron and up. 7, 8, 9, 10 hexagons can make ...
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4answers
524 views

How does this proof of the regular dodecahedron's existence fail?

On Tim Gowers' webpage he has an example "proof" of the regular dodecahedron's existence which he claims contains a flaw. He writes Of course, I have not written the above proof in a totally ...