Questions related to polyhedra and their properties.

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0answers
79 views

What shape is the Sage logo

Just curious what is this shape used by Sage as its logo?
2
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2answers
56 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 ...
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0answers
70 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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29 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
30 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
33 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 ...
2
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1answer
40 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
votes
1answer
28 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
votes
1answer
43 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
39 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 ...
8
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1answer
88 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 ...
0
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0answers
11 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
41 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
25 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
16 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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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 <...
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2answers
34 views

Construct a homemorphism $\phi : T^2/A \rightarrow X/B $

Construct a homemorphism $\phi : T^2/A \rightarrow X/B $ $T^2=S^1 \times S^1$ and $A \subset T^2$ is given by $A=S^1 \times\{1\}$. $X=S^1 \times [-1, 1]$ and $B = S^1 \times\{-1, 1\}$. $T^2/A$ ...
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2answers
31 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
14 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
42 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
38 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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votes
1answer
75 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 ...
3
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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
26 views

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 ...
5
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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
17 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
71 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 ...
2
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0answers
17 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 ...
3
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0answers
108 views

Polyhedra with identical faces

The isohedra have identical faces. They have symmetries acting transitively on their faces -- any face can be mapped to any other face to give the same figure. There are also polyhedra where all ...
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0answers
47 views

Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub ...
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1answer
46 views

Polyhedra vs Polytope

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?
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0answers
15 views

The $d$-skeleton of a polytope is strongly connected

A polytope is the convex hull of a finite set of points in $\mathbb R^n$. The $d$-skeleton of a polytope is the set consisting of faces of dimension at most $d$. I would like to show that every $d$-...
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0answers
56 views

Number of deltahedra as a function of the number of faces

How does the number of deltahedra (polyhedra with only equilateral triangles as faces) with no holes grow asymptotically as a function of the number of it's faces? If we have this as $N=g(F)$ for ...
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3answers
857 views

How many sewings are there on a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings are ...
5
votes
3answers
115 views

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
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0answers
14 views

Faces of a Bipyramid over a a Simplicial Polytope

Is there a simple way of expressing the number of faces of a bipyramid built over a polytope that is known to be simplicial, using the number of faces of the original polytope? This seems an easy ...
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0answers
7 views

Polyhedral Sets and $min$-function

I'm asked to verify if the following set is polyhedral, $$ X = \{[x_1;x_2]: min(x_1,x_2) \leq 0\}$$ Definition of a polyhedral set, A set $Y$ is polyhedral if $Y = \{y: Ay \leq b\}$, for finite $...
5
votes
0answers
69 views

Interesting cube subdivisions: what is going on here, and what are these polytopes?

I was messing around recently with a unit cube. If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured ...
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0answers
79 views

Intersecting rational polyhedral cones

Call A the cone generated by the rays (1,0,0) and (0,1,0) and B the cone generated by the rays (1,1,0),(1,0,1), and (0,1,1). I want to compute the intersection of these polyhedral cones, but I am ...
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1answer
35 views

Regular Triangulations of Cube

I want to figure out which triangulations of the cube (i.e., partitions into tetrahedra using only the $8$ given vertices) are regular, but I'm not sure how to easily tell whether a given ...
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0answers
12 views

Sphere inside cylinder vs polyhedra?

Comparing a cylinder with a polyhedra that has a symmetric coxeter $\ge 3$. Both have their centers hollowed out by $k\%$, in the shape of their outer, i.e.: relative to top face Which can better ...