Questions related to polyhedra and their properties.

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1answer
114 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
69 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
170 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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1answer
158 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 $S$...
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2answers
155 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
59 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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8answers
948 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
98 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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1answer
27 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 each. ...
2
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0answers
54 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
73 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
380 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
644 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
98 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
53 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
319 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
95 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
86 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 +...
3
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1answer
763 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
218 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
165 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
266 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
148 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
100 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
61 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 $$144V^2=(a^2b^2d^2+b^2c^2e^2+c^2a^2f^2+b^2a^2e^2+c^2b^2f^2+a^2c^2d^2+c^2e^2f^2+a^2f^2d^2+b^2d^2e^2+c^2d^2f^2+a^2e^2d^2+b^2f^2e^2)-...
2
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0answers
199 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 (i.e....
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0answers
74 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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2answers
403 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
475 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 idea....
2
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1answer
291 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
439 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
135 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
33 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
218 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
38 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 (...
2
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0answers
74 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 ...
2
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0answers
89 views

Convex polyhedron is union of simplices

Given a convex polyhedron $P$, how can we prove that every point $x \in P$ is in some simplex whose vertices are vertices of $P$? One proof is to inductively build a triangulation of $P$. If $P$ is ...
2
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0answers
70 views

About the relation between a tetrahedra and spheres moving in a tetrahedra

I found the following question in a book: There exists a regular triangle $OAB$ which has edge-length $2$. Let $H, I, J$ be a foot of the perpendicular line drawn from a point $P$ in $OAB$ to the ...
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4answers
284 views

Inequality for each $a, b, c, d$ being each area of four faces of a tetrahedron

We know 'triangle inequality'. I'm interested in the generalization of this inequality. Here is my question. Question: How can we represent a necessary and sufficient condition for each positive ...
3
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2answers
127 views

Forming a polyhedron from concave polygonal faces.

A polyhedron is a convex, three dimensional region bounded by a finite number of polygonal faces. So is it possible that some of those polygonal faces be concave ? Can concave polygons be used in the ...
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1answer
489 views

How do I determine the Tait-Bryan angles (yaw, pitch, and roll) of polyhedron faces to its center?

I'm modeling a pentagonal hexecontrahedron by placing faces and then rotating them. I've determined the center of each face by using the Cartesian coordinates of the vertices of its dual polyhedron (...
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1answer
142 views

Integral Polyhedra: Integer on each face

The general topic is unimodular matrices and integral polyhedra. I am really new to this field and I am studying for an exam in an advanced operations research course. In this case we are always ...
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1answer
104 views

Bounded polyhedra closed under rotation, intersection and complement

Are there any known types of bounded polyhedra, which exist in all Euclidean dimensions, and are are closed under intersection, rotation and relative complement? In other words, I am looking for a set ...
3
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0answers
55 views

Regular apeirohedra?

Have been toying with structures that I think are best describe as unbounded regular polyhedra. More specifically I arrived at non-convex polyhedra that are unbounded in one direction: Alternate ...
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2answers
2k views

Angles between two vertices on a dodecahedron

Say $20$ points are placed across a spherical planet, and they are all spaced evenly, forming the vertices of a dodecahedron. I would like to calculate the distances between the points, but that ...
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0answers
49 views

$X$ is a point in a bounded polyhedron $\ \in R^n $ with $n-1$ active constraints

Lets take a vector $d$ which is orthogonal to the active constraint. Since the polyhedron is bounded: We'll move to a point $x+\alpha*d$ where we will activate another constraint let's name it j. ...
2
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1answer
1k views

coordinates of icosahedron vertices with variable radius

I was looking on the wikipedia page about icosahedrons and it says that for edge length $a$ the radius of the circumscribed sphere around the icosahedron is given by $r = a \times sin(\frac{2\pi}{5})$....
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2answers
193 views

Computing bounding box of polytope (system of linear inequalities)

Given a N real valued variables and a set of linear inequality constraints, I would like to find a minimal bounding box which encapsulates the convex polytope defined by these constraints. I think (...
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1answer
55 views

Mathematical word for geometrical object?

Is there a mathematical word to designate the concept of a geometrical object like: square cube tesseract N-dimensional cube circle sphere hypersphere regular and non-regular polygons regular and ...