Questions related to polyhedra and their properties.

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1answer
25 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 ...
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2answers
21 views

LP: An algorithm to decide whether a polyhedron is a subst of another polyhedron

I've encountered the following question which I am unable to solve: $$ P = \{\vec x | A\vec x \geq \vec a\} \\ Q = \{\vec x | B\vec x \geq \vec b\}\\ P, Q \subseteq R^n $$ Find an algorithm to ...
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1answer
34 views

a general definition of the volume of a high dimensional polytope

I would like to find a general definition of the volume for a full dimensional polytope in $R^n$. Could anyone give me a hint please! Thank a lot
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1answer
38 views

Why is the 24-cell (also called Icositetrachoron or Hyperdiamond) the unique regular convex polychoron which has no direct three-dimensional analog?

The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog. http://mathworld.wolfram.com/24-Cell.html I don't understand why that is ...
2
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1answer
27 views

Time to roll through subterranean chords

Imagine a spherical airless body. It is small enough that central pressure allows a tunnel to be built from north pole to the south pole. I jump in the tunnel at the north pole and fall to the south ...
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1answer
36 views

If we inscribed all the 6 regular convex four-dimensional polytopes in a sphere, which one would have the highest volume?

When a dodecahedron is inscribed in a sphere, it occupies more of the sphere's volume (66.49%) than an icosahedron inscribed in the same sphere (60.54%). But what about for the 6 regular convex ...
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0answers
27 views

What are the formulas for the number of vertices, edges, faces, cells, 4-faces, …, $n$-faces, of convex regular polytopes in $n \geq 5$ dimensions?

I know that in dimension $n \geq 5$ there are only 3 kind of convex regular polytopes in each dimension: the $n$-simplex, the $n$-cube and the $n$-orthoplex. I would like to know if there are ...
2
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1answer
81 views

Which polyhedron has 17 vertices, 34 edges and 19 faces?

on exam I had task to check that there is polyhedron with 8 triangle faces, 11 quadrangle, each vertices have degree 4. after calculate I obtain that it have 34 edges, 17 vertices and 19 faces but i ...
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0answers
13 views

Volume of overlap between two convex polyhedra

I have two convex polyhedra represented by triangle meshes. I can easily determine if they are in contact or not, but when they are in contact then I would like to determine the volume of their ...
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1answer
40 views

Minimum distance of extreme points of polyhedra

Let $P = \{x \in \mathbb{R}_{\geq0}^n \colon Ax \leq b\}$ with $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R^m}$. Let $E \subseteq P$ be the extreme points of $P$. Can anything be said about ...
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1answer
15 views

what is and how to generate a Net representation for a given polyhedron?

The so called Net representation for a Tetrahedron is depicted in the following image ( link to wolfram ) : What is this for ? How to reason about this and how to generate this very same ...
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0answers
24 views

Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there relationship between dual of az optimalization problem [see: http://en.wikipedia.org/wiki/Duality_(optimization) ] and dual of a ...
4
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2answers
81 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
70 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
19 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
17 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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0answers
38 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
38 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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0answers
14 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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0answers
22 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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2answers
66 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
36 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
258 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
36 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
37 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
16 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
30 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 ...
2
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1answer
45 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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0answers
39 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
33 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 ...
1
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1answer
34 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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0answers
49 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
65 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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0answers
98 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
77 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 ...
13
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3answers
917 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 ...
2
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2answers
48 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
18 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
15 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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0answers
156 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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0answers
59 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
63 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
36 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
48 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
101 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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0answers
20 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
168 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 ...
4
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1answer
80 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 ...