Questions related to polyhedra and their properties.

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Vertices of Polyhedral

Suppose there are matrix $A\in\mathbb{R}^{n \times m}$ and vector $b\in\mathbb{R}^n$. Consider a non-empty polyhedron $P = \{Ax \leq b\} $. Then, there exists a vector $\bar{x}\in P $ such that ...
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How many arrangements (correct and incorrect) possible with cube puzzle pieces in a particular unfolded form?

I am trying to do a computer program for arranging cube puzzle pieces in an unfolded form. A cube can be unfolded into 11 forms or nets. I have chosen a single net for now for simplicity purposes and ...
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Explanation regarding Kaleido index for polyhedra

I can't seem to find any information about the Kaleido index number used in geometry (see 'K# at http://en.wikipedia.org/wiki/List_of_Wenninger_polyhedron_model'). I found an abstract called "Uniform ...
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Linear programming - Textbook recommendations

Next term, I will attend a course on linear programming. Due to the assignments, we will have to write many thorough proofs. I anticipate that we will be supposed to cope with in-depth background ...
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Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
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55 views

Visualizing Platonic Solid group symmetries

How do you visualize the rotation symmetries, to classify a icosahedron for example as Ih, H3, [5,3], (*532)
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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 ...
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13 views

For an app teaching about polyhedra, what are some core characteristics to include?

For fun: I'm building a 3d app that teaches about polyhedra. What should I include? The obvious didactic elements for each polyhedron would be: Fundamental polygon's Vertices 
Edges
 Faces
 (and ...
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1answer
116 views

Showing that group of orientation preserving isometries of Icosahedron is a simple group

Let $G$ denote the group of orientation preserving isometries of Icosahedron. To prove the claim, I have shown that $\nexists \ N \ \triangleleft \ G$ such that $|N|=5.$ $\nexists \ N \ ...
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31 views

How to locate sphere resting in a corner of a polyhedron?

I am working on an interface to a computational solid geometry program. I would like this program to be able to fillet corners (although fillet sometimes seems to refer to an internally smoothed ...
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1answer
51 views

Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
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33 views

Is there a nice characterization of circumscribed hexahedra?

A convex quadrilateral that contains a circle tangent to its sides is called a tangential or circumscribed quadrilateral. There is a very nice characterization of tangential quadrilaterals known as ...
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Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
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2answers
41 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
26 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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41 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
67 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 ...
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1answer
28 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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40 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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30 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 ...
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93 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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25 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
48 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
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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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31 views

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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87 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
76 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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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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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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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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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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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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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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26 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
100 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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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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341 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
43 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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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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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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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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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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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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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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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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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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69 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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107 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
79 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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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 ...