Questions related to polyhedra and their properties.

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1answer
41 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 ...
2
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0answers
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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0answers
41 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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1answer
66 views

Formula for greatest cross section of regular dodecahedron

Is there a formula for the area of greatest cross section of a regular dodecahedron? For example, this can be viewed as finding a hole big enough for it to fit into.
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4answers
248 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
66 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 ...
26
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2answers
668 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 ...
2
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2answers
61 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
46 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
22 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
56 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
49 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
50 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
93 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
48 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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3answers
104 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 ...
2
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1answer
116 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
1k 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
81 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
26 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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5answers
358 views

Cleverest construction of a dodecahedron / icosahedron?

One can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedra, cube, and octohedra all admit very intuitive ...
2
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1answer
74 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 ...
2
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0answers
23 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 ...
2
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0answers
41 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
64 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
171 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 ...
2
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1answer
343 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
91 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
48 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 ...
2
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1answer
91 views

Symmetries of a dodecahedron

Suppose we want to measure order of the group symmetries of a dodecahedron, and we know that If $G$ is a group and $S$ is a set on which $G$ acts and $s\in S$, then Order of G=(Order of stabiliser of ...
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0answers
67 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
74 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
398 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 ...
1
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1answer
141 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 ...
6
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1answer
121 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 ...
2
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2answers
90 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
74 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 ...
2
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1answer
81 views

How to find that Dehn invariant of a dodecahedron?

What is the Dehn invariant of a regular dodecahedron with center (0,0,0), and radius 1?
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1answer
59 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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0answers
169 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 ...
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0answers
60 views

Net for both cube and regular tetrahedron

At how to fold it by Joseph O'Rourke, there is a net given that can be folded into a cube or irregular tetrahedron. Is there a net that can be folded into either a cube or regular tetrahedron?
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1answer
193 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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1answer
340 views

Finding the vertices of a rhombic dodecahedron

I'm trying to figure out a straightforward way to find the vertex (x,y,z) coords for a rhombic dodecahedron. Besides starting with a rhombus and rotating it around at the proper angles, I have no ...
2
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1answer
182 views

Dimension of polyhedron defined by inequalities and rank of implied equalities

I'm looking at "Optimization Over Integers" by Bertsimas and Weismantel and I have a question about one of the examples in the book. I'm getting a conflicting answer and I'm not sure what I'm ...
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2answers
328 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
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0answers
104 views

Indexing Goldberg (0,n) polyhedron faces

I would to know how to uniquely identify a face of a Goldberg (0,n) polyhedron: http://en.wikipedia.org/wiki/Goldberg_polyhedron#Icosahedral_G.280.2Cn.29_polyhedra It's possible to uniquely assign ...
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1answer
32 views

Give examples of polytopes $\Delta$ in $\mathbb{AR}^3$ such that

With Sym $\Delta$ of the set $\Delta$ consisting of all isometries of $\mathbb{AR}^n$ that map $\Delta$ onto $\Delta$, Sym $\Delta$ acts transitively on the set of vertices of $\Delta$ but is ...
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2answers
118 views

Instrinsic definition of concave and convex polyhedron

Is it possible to distinguish a concave polyhedron from a convex one by mesurements made only on its surface, without a reference to the 3d space around it?
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0answers
32 views

Standard for intrinsic polyhedron definition using angular deficit?

Is there a standard definition of a given polyhedron using only intrinsic properties (those which can be measured by a 2d being living on its surface) and particularly angular deficit at a vertex ...
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0answers
45 views

How to fit a cuboid into a polyhedra?

I have multiple points which create a solid (polyhedra). And now I want to place a cuboid inside this solid in a way that it uses the maximum amout of space inside. Are there any solutions for this ...