Questions related to polyhedra and their properties.

learn more… | top users | synonyms

0
votes
0answers
15 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 ...
7
votes
4answers
183 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 ...
1
vote
1answer
10 views

General algorithm to cap an n-dimensional convex polyhedra

I am looking for a way to cap an $n$-dimensional ($n$ > 3) polyhedra, that is to say: Given an $n$ dimensional set of vertices and faces (including hyperplane equation), and an $n$ dimensional ...
0
votes
0answers
26 views

permutohedron vs permutahedron

Why are there two spellings for the terms denoting the sets $$\mathrm{Conv}\left(\left\{(\sigma(1),\ldots,\sigma(n))\,\middle|\, \sigma\in S_n\right\}\right)\qquad(n\in\mathbb{N}^+)\,,$$ namely, ...
5
votes
2answers
380 views

Maximum number of vertices in intersection of triangle with box

Suppose we have a triangle and a box in 3D. The intersection of the triangle with the (solid) box will be a polygon with some number of vertices (possibly zero). The vertex count will vary according ...
0
votes
1answer
41 views

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, ...
0
votes
0answers
31 views

Is a snub disphenoid more oblate or prolate?

Figuring out which deltahedra are oblate/prolate (of coarse the platonic solids are spherical) was pretty easy for all of them except for the snub disphenoid... Anyone know what the radia of the ...
3
votes
1answer
383 views

Drawing a Truncated Octahedron

I'm trying to draw a truncated octahedron in MATLAB. This is also known as a permutahedron so my strategy is to link up all the vertices via adjacent transpositions of permutations in $S_4$. What I ...
3
votes
6answers
3k views

How many faces does the resulting polyhedron have?

Take a regular tetrahedron of edge one. Also take a square-based pyramid, whose edges are all one (therefore the side faces are equilateral triangles of same size as the faces of the tetrahedron). ...
1
vote
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\}\...
2
votes
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 ...
2
votes
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 ...
0
votes
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{...
7
votes
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 ...
8
votes
3answers
6k views

Tetrahedron inside a sphere

What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
1
vote
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 ...
1
vote
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: $$ C(V):=\Big\{ \sum_k a_k\,v_k; \; a_k\ge 0 \Big\}\;. $$ ...
1
vote
1answer
35 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 ...
3
votes
1answer
32 views

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 ...
3
votes
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. ...
1
vote
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 ...
2
votes
1answer
43 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 ...
1
vote
1answer
168 views

Question about Euler's polyhedral formula in a proof of minimum distances

I am confused by a step made in a proof of the following result. Let $f_{2}^{\text{min}}(n)$ denote the maximum number of times the minimum distance can occur among n points in the plane. Then $f_{...
6
votes
0answers
92 views

What shape is the Sage logo

Just curious what is this shape used by Sage as its logo?
2
votes
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 ...
0
votes
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 ...
0
votes
1answer
34 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 &...
8
votes
1answer
90 views

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 ...
2
votes
1answer
42 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 $a, b, c, d$ ...
8
votes
4answers
5k views

Angle between lines joining tetrahedron center to vertices

What are the angles formed at the center of a tetrahedron if you draw lines to the vertices? I'm trying to make these: I need to know what angles to bend the metal.
2
votes
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. ...
1
vote
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
votes
1answer
57 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 ...
1
vote
0answers
41 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 ...
0
votes
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 ...
0
votes
0answers
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
votes
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 $...
2
votes
1answer
34 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 $\...
4
votes
2answers
2k views

What are the vertices of a regular tetrahedron embeded in a sphere of radius R

Imagine you had a sphere of radius R centered at the origin. What are the coordinates of the vertices of the regular tetrahedron which is circumscribed by the sphere? One of the vertices of the ...
2
votes
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 ...
0
votes
0answers
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 <...
1
vote
2answers
154 views

Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$? Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral ...
1
vote
1answer
71 views

What is the depth of water above the prism?

I have been practising for a math competition and came across the following question: A fishtank with base $100\,\rm cm$by $200\,\rm cm$ and depth $100\,\rm cm$ contains water to a depth of $...
1
vote
2answers
34 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 ...
5
votes
1answer
207 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
2
votes
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 ...