Questions related to polyhedra and their properties.

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Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there any relationship between dual of an optimalization problem and dual of a polyhedra?
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2answers
93 views

Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?

Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra: http://en.wikipedia.org/wiki/Platonic_solid http://en.wikipedia.org/wiki/Archimedean_solid ...
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4answers
78 views

there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon?

I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use ...
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0answers
32 views

Convex Polyhedron problem

I want to prove it isn't possible to make a football (a convex polyhedron such that at least 3 edges meet at each vertex) out of exactly 9 squares and m octagons where $m>3$.
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1answer
19 views

How do you know at least one face is not simply connected on a polyhedra?

if it has 14 vertices, 21 edges and 9 faces, its boundary is a single surface and there is at least one hole. I dont understand.
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40 views

Proof that a Polytope has vertices

As part of my Discrete Optimization course, I have a homework where I have to prove that a Polytope has vertices. I seems to have all tools in hand (definition of a vertex, polytop, convex hull, etc.) ...
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1answer
39 views

Counting polyhedra

Given $n>6$ points in space are placed in such a way that no three are collinear and no four lie on the same plane. Show number of convex polyhedra with $5$ faces and vertices among the given ...
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1answer
38 views

Could someone explain me this induction.

I'm trying to understand a paper called "Diameter of Polyhedra: Limits of Abstraction" available here : http://sma.epfl.ch/~eisenbra/Publications/designs.pdf My problem is with the first two ...
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17 views

The process of alternation on an n-polytope

I am currently working on a problem involving algebraic geometry and as a part of the research it would be helpful for me to understand the process of alternation, also called partial truncation, ...
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31 views

Generalized permutahedron and random polytopes

The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...
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3answers
130 views

Is the rhombic dodecahedron the only isohedral polyhedron that tiles 3-space (other than the cube)?

Is the rhombic dodecahedron the only face-transitive (or isohedral, i.e. all faces are the same) polyhedron that seamlessly tiles 3-dimensional Euclidean space (other than the cube)? I'm looking ...
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0answers
56 views

Three theorems for Polyhedra, Polytopes, and Cones

Is there anybody have readable proofs of the following theorem? A polytope (bounded polyhedron) is the convex hull of a finite set of points. A polyhedral cone is generated by a finite set of ...
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2answers
404 views

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
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2answers
45 views

Polyhedra having equal quadrilateral faces are cubes?

While discussing with my 11 y.o. daughter about the definition of a cube as regular hexahedron, I observed that actually we can let drop the assumption that the faces are squares, and require only ...
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0answers
40 views

showing that all convex polehedron graphs are 3-connected

I'm trying to figure out how to show that two nonadjacent vertices in the graph of a convex polyhedron can be disconnected from one another by the removal of at least three vertices. I know what a ...
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0answers
18 views

Is $S$ a polyhedral set?

Let $\textbf{x}=(x_1,x_2)^T$, $\textbf{y}=(y_1,y_2)^T$, is $$S=\{\textbf{x}|\textbf{x}^T\textbf{y}\le1 \text{ for all }\textbf{y}\text{ such that }y_1\ge0,y_2\ge0,y_1+y_2=5\}$$ a polyhedral set? How ...
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2answers
37 views

Is this a polyhedron?

Is $S$ a polyhedron? $$S=\{x\in\mathbb{R}^n|\|x-x_0\|\le\|x-x_1\|\}$$ where $x_0, x_1$ are given. $S$ is the set of points that are closer to $x_0$ than to $x_1$. I was thinking the ...
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1answer
47 views

Showing that this family of vectors generates $\mathbb{R}^n$

Suppose that: $$\{x\in\mathbb{R}^n\mid a_i^Tx\le b_i, i=1,2,\dots,m\}=\{x\in\mathbb{R}^n\mid g_j^Tx\le h_j, j=1,2,\dots,k\}$$ How can I show that if the vectors $a_1,\dots,a_m$ span $\mathbb{R}^n$ ...
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42 views

How to name/call this polyhedron?

How to name/call this polyhedron? What's a general method for finding the scientific name of a polyhedron?
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1answer
64 views

Polyhedron's Representations and spanning the Euclidian space

Let's say you have to different representations of the same polyhedron $P\neq \emptyset$: $$P=\{x\in \mathbb{R}^n\;|\;h_i^Tx\leq c_i, i=1,...,k \} =\{x\in \mathbb{R}^n\;|\;g_j^Tx\leq d_i, j=1,...,l ...
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1answer
39 views

Proving faces of polyhedron

let $F(k)$ be the number of faces of a convex polyhedron with k edges how can we prove that $F(k) > 1$ for some $k$? I know Euler's Formula for Polyhedra: $V-E+F=2$, and $\sum k\,F(k) = 2E$. ...
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98 views

Difference between polyhedral, CSG and B-rep

I am working on the 3D object modeling project. I found objects can be represented in the form of Polyhedrol model, CSG (Constructive Solid Geometry) model, and as well as B-Rep (Boundary ...
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1answer
71 views

Does a polyhedron with 7 hexagons and 20 pentagons exist?

A beautiful polyhedron with 20 hexagons and 60 pentagons can be seen here: http://robertlovespi.wordpress.com/2013/11/03/a-polyhedron-with-80-faces/ . Euler formula and the corresponding Diophantine ...
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113 views

Lattice orthogonal polyhedra face-area sequences: Golyhedra?

Let $P$ be a polyhedron, all of whose vertices are at points of $\mathbb{Z}^3$, all of whose edges are parallel to an axis, with every face simply connected, and the surface topologically a sphere. ...
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1answer
80 views

Does a polyhedron with 16 quadrilateral faces exist?

I have just seen here the picture of a polyhedron with 15 quadrilateral faces. In some lists of polyhedra a big variety of quadrilateral sides can be found (12, 13, 15, 18, 20,...) but the number 16 ...
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3answers
959 views

Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that ...
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2answers
57 views

Construct polyhedron from edge lengths

I'm interested in the following problem: I am given the combinatorial structure (vertices, edges, faces) and edge lengths of a polyhedron. From this I'd like to infer the vertex positions. Now, I ...
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0answers
19 views

Polyhedron in $\mathbb{R}^2$.

Let $G\in \mathbb{R}^{4\times 2}$, $x\in \mathbb{R}^{2\times 1}$, $B\in \mathbb{R}^{4\times 2}$, $u$ is an interval matrix in $\mathbb{R}^{2\times 1}$, $H\in \mathbb{R}^{4\times 1}$. Given an ...
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0answers
19 views

traversing faces of a polyhedron Hamiltonian Tour?

I wanted to know if I could start at one point on an icosahedron and traverse to all the others sequentially without visiting any one twice, which I assume I could model as a Hamiltonian path in a ...
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220 views

How to map sphere to faces of an Icosahedron

This is the mathematics behind some graphics I am trying to build in OpenGL. I believe the question belongs here. I want to represent an approximate sphere (let's say the Earth) as an icosahedron and ...
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70 views

Formula for Rhombic Dodecahedron

Here is a diagram for a rhombic dodecahedron: Call the diameter of the solid, the diagonal connecting the two circled vertices (where 4 faces meet). If the length of each edge is $e$, find the ...
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1answer
67 views

Extension of dihedral group to higher dimensions

The dihedral group $D_{2n} = \{x, y \mid x^2=y^n=yxyx=1\}$ is tied with the symmetries of the regular polygon on a plane. What is the natural extension to higher dimension? For instance, in $3$D, does ...
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0answers
21 views

The exact type of my 3d model

I have reconstructed vertical features (hole like objects lie on a vertical face) lie on two connected faces. To understand the situation, I say I have 2 walls with many windows and doors on ...
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0answers
38 views

Equation on the vertices of regular polyhedra

I found in this book, on page 6 that the equation on vertices of icosahedron inscribed in sphere considered as $\mathbb{CP}^1$ by means of stereographic projection is $xy(x^{10}+14x^5y^5-y^{10})=0$. ...
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1answer
55 views

Probability that a $n$-dimensional Gaussian falls into a half-space

For $a \in \mathbb{R}_{\ge 0}^d$ and $b \in \mathbb{R}_{\ge 0}$, we can define a half-space as the set of points $x \in \mathbb{R}^d$ such that $a \cdot x \le b$, namely, $$\mathcal{H}(a,b) = \{x \in ...
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2answers
129 views

Is projection of a convex polyhedron on a plane a convex polygon?

If we have a convex polyhedron with vertices $\mathbf{V}$ and project it on a plane $\mathbf{P}$, is this procedure equivalent to projecting points in $\mathbf{V}$ on the plane $\mathbf{P}$ and then ...
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24 views

Volume of Dodecahedron

A dodecahedron undergoes the transformation $$T(x,y,z) \to (3x,3y,3z)$$ What is the ratio of the new dodecahedron to the volume of the old dodecahedron? A) $3$ B) $3\sqrt3$ C) $9$ D) $27$ My ...
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1answer
208 views

Consistent formula for Regular Polyhedral Volumes

Back in high school I re-discovered the formula for regular polygonal areas like so: $$A = nx^2\frac{\cot(\pi/n)}{4}$$ Where $A$ was the area of the regular polygon, $n$ was the number of sides and ...
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1answer
83 views

Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
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1answer
45 views

What do you call a convex polyhedron whose symmetry group is transitive on the facets?

I'd like to know a name/source for the following concept: Let $P$ be a convex polyhedron in $\mathbb{R}^3$. Let $G$ be its symmetry group, and let $F$ be the collection of (top-dimensional) faces of ...
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0answers
49 views

Geometrically, what is the difference between a “flat face” and a “non-flat” face?

I was curious when I was checking sites like MathisFun, and I came across a pretty unclear system that defines a "flat face" and as a "non-curving" face of a shape; a polyhedron. However, I have to ...
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0answers
58 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...
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1answer
232 views

Coordinates of the Vertices of a Goldberg Polyhedron

I'd like to be able to generate visualizations of the pentagon Goldberg Polyhedra from scratch (i.e. I'm looking for the math, not a software library or package to do this). I can generate truncated ...
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40 views

Farkas lemma corollary and duality

Which is the application of the Farkas lemma corollary on duality? Let $$ Ax \leq b $$ be a system of inequalities, and let $$ P = \{y^TA = 0, y^Tb \leq -1, y \geq 0\} $$ be a polyhedron. May I say ...
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1answer
90 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
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1answer
108 views

Can I specify the edge lengths of a simplicial polyhedron?

Let $X$ be a convex polyhedron in $\mathbb{R}^3$ whose faces are all triangles, and let $\ell$ be a function which assigns a positive real number to each edge of $X$. We say that $\ell$ is realizable ...
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2answers
68 views

Generalization of sum of angles to polyhedra?

The sum of interior angles of a polygon is (n-2)*180. Is there a similar statement for the sum of the solid angles of a polyhedra? Is there any non-trivial relationship, $f(\alpha,\beta,...)=0$, ...
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0answers
60 views

Dodecahedron: How do we get the distance between 2 opposite faces?

I am deciphering a CSS code that Ana Tudor Maria has done. http://codepen.io/thebabydino/pen/qIfbL In her example, she has a formula that calculates the distance between 2 opposite faces. I have no ...
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1answer
58 views

Proving that a polynomial about the volume of a tetrahedron is irreducible

We know that the volume of a tetrahedron $ABCD$ can be represented as ...
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162 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...