Questions related to polyhedra and their properties.

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Integrating Over a Product of (Non-Separable) Piecewise Functions (Hyper-Solid Angle of a Convex Polyhedral Cone)

My problem is as follows: given a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ where $n$ is some integer of order 10 and $f$ is defined by a product of (non-separable) linear piecewise functions, ...
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3answers
145 views

Maximum area of triangle inside a convex polygon

Prove that within any convex polygon of area $A$, there exists a triangle with area at least $cA$, where $c=\tfrac{3}{8}$. Are there any better constants $c$? I'm not sure how to approach this ...
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0answers
13 views

efficiency of different whole-number-mass-to-a-power in balancing a regular triangle/tetrahedron

I saw this qustion: http://puzzling.stackexchange.com/questions/186/whats-the-fewest-weights-you-need-to-balance-any-weight-from-1-to-40-pounds Suppose you want to create a set of weights so ...
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0answers
11 views

Proof of the existence of a rational finitely generated cone

Let $P$ be a rational polyhedron and $F$ be the inclusion-wise minimal face. Then we define: $C_F= \left\{c\in \mathbb{R}^n : F \subseteq \left\{x \in P:c^Tx=\max\left\{ c^Ty:y \in P\right\}\right\}\...
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1answer
67 views

Truncated objects coloring

I am looking for ways to color a truncated tetrahedron allowing rotations and reflections. I know the ways to color a tetrahedron in a similar way but stumped on this. From wikipedia, both tetrahedron ...
2
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1answer
20 views

Dual polyhedron & dual cone

From Wiki: Def. of dual of polyhedral (polytope): polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. EX: 2. Def. of dual ...
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0answers
7 views

To show total total dual integrality

Let P be the unit cube with, $P=conv\{\left(\begin{array}{c} 1/2 \\ 1/2\\1/2 \end{array}\right),\left(\begin{array}{c} -1/2 \\ 1/2\\1/2 \end{array}\right),\left(\begin{array}{c} 1/2 \\ -1/2\\1/2 \end{...
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2answers
189 views

Coloring the pentagonal hexecontahedron

So, I'd like to color the pentagonal hexecontahedron in a way that is satisfying aesthetically and mathematically. For me this equates to, in order of priority - 1. No same-colored faces can share an ...
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0answers
16 views

Convex 4-polytopes requiring 6 or more colors

Projected into 3-D space, a convex 4-polytope looks like a collection of convex polyhedra. If any two convex cells sharing a face have different colors, how many colors are required? In the paper ...
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0answers
27 views

Width of a cone

Let $V=\{v_k\}$ be a collection of vectors of $\Bbb{R}^n$, and define their cone to be the set of all their non-negative linear combinations: $$ C(V):=\Big\{ \sum_k a_k\,v_k; \; a_k\ge 0 \Big\}\;. $$ ...
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1answer
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Relationship between circumscribed sphere radius and edge length of a dodecahedron? [duplicate]

Hello and I'm a secondary student doing a math exploration, but I'm currently stuck with this problem... Can anyone kind enough to show me the derivation of the relationship between the circumscribed ...
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1answer
34 views

Polyhedron, understanding face vs facet.

I've the two following definitions, for which I was trying to understand the difference. For a given polyhedron $P$ a face $F$ is both $P$ itself or the intersection of $F$ with $P$. A facet is ...
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1answer
35 views

Finding a convex decomposition of a point in a polytope

Suppose I'm given the set of vertices, $\{v_i \} $, of a convex polytope. Suppose that I'm also given a point $p$ in terms of its coordinates, and I'm promised that $p \in \mbox{conv} \{v_i \}$. How ...
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1answer
42 views

Understanding definition of “dimension” of a subset of $\mathbb{R}^n$

In a book of combinatorial optimization the following definition is stated: A polyhedron in $\mathbb{R}^n$ is a set of type $P = \left\{x \in \mathbb{R}^n \;:\; Ax \leq b \right\}$ for some matrix ...
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0answers
92 views

What shape is the Sage logo

Just curious what is this shape used by Sage as its logo?
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2answers
61 views

Counting faces of each type in a rhombicosidodecahedron

If I know that, in a rhombicosidodecahedron, at every vertex one triangle, one pentagon, and two squares meet, then how can I compute the number of faces and edges that are needed to build it? There ...
3
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0answers
127 views

Maximal unit lengths in 3D with $n$ points.

Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. ...
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0answers
36 views

Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
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1answer
32 views

Which vertex-transitive planar graphs represent non-self-intersecting polyhedra?

Consider an infinite planar graph with the following properties. Its vertices all have valence $3$. The faces all have $5$ edges. Now put it in cartesian space and require that the faces are all ...
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0answers
8 views

Finding a polytope in the Cartan Subalgebra

The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can. Some examples of ...
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2answers
36 views

Polyhedron with 12 pentagons and 1 hexagon

In this answer http://mathoverflow.net/a/19823/5239, it is indicated that it is impossible to make a polyhedron (with 3 faces meeting at each vertex) out of 12 pentagons and 1 hexagon. There is ...
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0answers
5 views

Complementary Facets of Pointed Cone

I am looking at a particular full-dimensional pointed cone $C \subset R^{11}$ with $14$ generators. In matrix form, with each column being a generator, I have the matrix \begin{pmatrix} 1 & 1 &...
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0answers
17 views

Which rhombuses can make a rhombic dodecahedron

There are two dodecahedra I know of whose faces are identical rhombuses. One has rhombuses whose diagonals have a ratio of $\sqrt{2}$ -- this one is often simply called the "rhombic dodecahedron". The ...
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1answer
41 views

How to prove a set of inequalties in not satisfiable?

For the set of inequalities $$\begin{cases} 10 a - b - c \ge d\\ 5 b - a - c \ge d\\ 2 c - a - b \ge d\\ d \ge a + b + c\end{cases}$$ how can I show these cannot all be satisfied for $a, b, c, d$ ...
2
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1answer
29 views

Existence of direction for polyhedral set

I refer to Lemma 4.42 of this lecture notes on linear programming, about the relationship between the boundedness of a polyhedral set and its directions. Let $P$ be a non-empty polyhedral set. ...
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0answers
63 views

Twisty Puzzle Solving Program

I'm writing a program to help me solve a twisty puzzle. In this case it's the face-turning octahedron. I'm representing the puzzle as a group with face twists as generators. The facelets are in a list ...
4
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1answer
53 views

Computing volume of concave polyhedron

I have a circular grid with points uniformly distributed throughout it. See this: Each point has some nonnegative height assigned to it (i.e. height can be 0 on up). I'm trying to accurately ...
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0answers
40 views

What are the exact angles of the triangular faces of each of the rhombic pyramids of the icosahedron stellation, the compound of five octahedra?

I'm talking about this compound of five regular octahedra. Based on looking at the icosahedron stellation diagram and making an educated guess about where the golden ratio comes in, I calculated the ...
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0answers
10 views

constrain general solution of ill-posed linear system to $\Re_+$?

I have a solution space of an under-determined linear system Ax = b with n x m matrix A: $$x= x0 + V2 * c (1)$$ with [U, S, V] = svd(A); V2 = V(:,r+1:end); $x0 = A^+ b; $ r = rank(A); I ...
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1answer
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can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ...
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14 views

maximal volume/diameter of a set of simplexes

I am trying to develop a simplicial integral in $R^n$ and I am faced with the problem of controlling the "compacity" of a set of simplexes: Let $S$ be a finite set of n-d simplexes in $R^n$. Define ...
2
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1answer
42 views

maximal volume/diameter of polyhedron

I am trying to develop an integral in $R^n$ and I am faced with the following problem: Given a polyhedron $P$ in $R^n$ of diameter d, define the "compactness" of $P$ as the quotient of the volume of $...
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1answer
32 views

Dual of the Minkowski Sum

Suppose $X$ and $Y$ are convex sets in $\mathbb{R}^d$ such that the origin is in each of their interiors. Then the dual of $X$, $X'$ is defined as the set of linear functionals $\alpha$ such that $\...
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0answers
18 views

Does it make sense to compare sets (polygons) with different dimensions?

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one that gives the least ...
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20 views

$\{x\in R^n | Ax \leq b\} \cap \{x \in R^n | Dx \leq d\}= \emptyset$ iff there is a vector $c \in R^n$ such that $c^Tx < c^T y$

Consider two non-empty polyhedra $P := \{x\in R^n | Ax \leq b\}$ and $Q := \{x \in R^n | Dx \leq d\}$. Show that $P \cap Q = \emptyset$ if and only if there is a vector $c \in R^n$ such that $c^Tx <...
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2answers
33 views

Is a convex cone a convex polyhedron?

Say that I have a convex cone $C=\{t|Ax = t, x\geq 0\}$. where $x\in R^n$, and $t\in R^m$, $A\in R^m\times R^n$. Can I say that this is a convex polyhedron? and why? EDIT: Just in order to avoid ...
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0answers
15 views

Regular packing of an infinite number of infinitely long cylinders in 3d space

Is it possible to pack an infinite number of congruent infinitely long cylinders into 3 dimensional space in a regular pattern? Another condition is that an equal number of the cylinders must be ...
2
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2answers
44 views

Polygon with curved sides, and higher-dimensional generalizations

I am trying to find references about generalizations of polygons with non-straight sides. I am interested in both the convex and non-convex cases, and particularly in polynomial boundaries, and ...
0
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1answer
41 views

Find the edge angle of a dodecahedron using spherical trigonometry?

How can I find the edge angle (the angle at the center of a polyhedron subtended by an edge of the polyhedron) of a dodecahedron (a polyhedron with 3 pentagonal faces meeting at each vertex)? I know ...
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1answer
77 views

Which Archimedean solid takes up the most volume in its circumscribed sphere? [closed]

I have a question that has really kept me wondering: Which Archimedean solid takes up the most volume in its circumscribed sphere?, meaning Which solid takes up the greatest percentage in its ...
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0answers
31 views

Simplest molecules with multiple local minima

Methane (C + 4 H) goes to a tetrahedral structure. Water (O + 2 H + 2 e-) goes to a slightly skewed tetrahedron. In a computer model, both of these have no local minimum problems. There is a ...
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0answers
33 views

Asymptotic bounds on the number of faces needed to construct a polyhedron of a certain genus

Let a polyhedron be a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices, where moreover we require that every edge touches exactly two faces, every ...
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0answers
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How to construct a polyhedron from given planes

This seems to be a basic questions, but I really don't know a good computer algorithm to do this. I have a set of planes (parameterized by normal direction and distance from a given point), and I want ...
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0answers
31 views

Single loop polyhedra

The odd antiprisms are both Eulerian and polyhedral, with the first implying that the edge can be represented with a single closed path. The Cuboctahedron also has that property. With the rule to ...
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0answers
10 views

Estimate size of smallest solution to linear program

I have a linear program: a system of linear inequalities of the form $$Ax \le b, \qquad x \ge 0.$$ where $x \in \mathbb{R}^n$, $b \in \mathbb{R}^m$, and $A$ is a $m\times n$ matrix. I am looking ...
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0answers
22 views

Formal definition of a face of a polyhedron

Given an $n$-dimensional convex polyhedron, an $(n-1)$-dimensional face of it can be defined as an intersection of the polyhedron with a supporting hyperplane. What is the formal definition in the ...
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0answers
23 views

Number of Edges in Unimodular Triangulation of Simplex

Let $d\Delta$ be the simplex that's the convex hull of $(0, 0, 0, 0), (d, 0, 0, 0), (0, d, 0, 0), (0, 0, d, 0), (0, 0, 0, d)$. A unimodular triangulation of $d\Delta$ is a subdivision of it into ...
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0answers
76 views

Difference between invariant and contractive sets

I came across this particular notion of contractive sets. I know what an invariant set is, but can anyone explain what a contractive set is and the difference between invariant and contractive sets?
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1answer
93 views

Lower bound on the number of faces of a polyhedron of genus g

Is there a lower bound on the number of faces of a polyhedron of topological genus g? For example: it seems very reasonable that $g$ < $F$ i.e. the genus of a polehydron is less than the number ...
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0answers
18 views

Tracing the faces of a convex polyhedron from edges and vertices

I have a set of vertices and edges that by construction, form a convex polyhedron. I would like to know how to trace out the faces of such a polyhedron i.e. find a list comprised of set of edges that ...