Questions related to polyhedra and their properties.

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Is there a 4-dimensional picture making a geometrical proof of Heron's formula

Heron's formula states that if you have a triangle $T \subset \Bbb R^2$of sides $a,b,c$ then the hypervolume of a right-angled hyper-parallelepiped (is there a better word for this) of sides ...
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30 views

Single loop polyhedra

The odd antiprisms are both Eulerian and polyhedral, with the first implying that the edge can be represented with a single closed path. The Cuboctahedron also has that property. With the rule to ...
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Interesting cube subdivisions: what is going on here, and what are these polytopes?

I was messing around recently with a unit cube. If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured ...
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155 views

From Icosahedron to Pentagonal hexecontahedron (Floret Tessellation)

Inspired by this post: Floret Tessellation of a Sphere I tried to transform myself an icosahedron into its simplest Floret tessellation. But I am having trouble when applying the 'method' given in the ...
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153 views

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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128 views

Is Paley-13 graph a unit distance graph in 3D space?

The 13-node Paley graph has vertices 1 to 13 that are connected by an edge when their difference is one of the values $(1,3,4,9,10,12)$ Can this graph be put into 3D space so that all edges have ...
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45 views

Decomposing geodesic tessellations over a sphere into parallelograms

I'm working with some icosahedron-based tessellations of triangles over the surface of a sphere. Class I and Class II tessellations have a nice property where, cutting along the edges of the ...
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58 views

Isomorphism of divisors

Consider the cartier divisor group $CDiv_{T_{N}}(X_{\Sigma})$ defined on the fan $X_{\Sigma}$. I am having trouble proving the following assertion that there is a natural isomorphism ...
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33 views

Computer realisation of automorphism group of polyhedron

I have a polyhedron (I am especially interested in the case of Platonic solids) and the graph corresponding to its skeleton. I also have some data associated with this graph (e.g. different ...
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77 views

Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
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33 views

Asymptotic bounds on the number of faces needed to construct a polyhedron of a certain genus

Let a polyhedron be a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices, where moreover we require that every edge touches exactly two faces, every ...
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Polyhedra with identical faces

The isohedra have identical faces. They have symmetries acting transitively on their faces -- any face can be mapped to any other face to give the same figure. There are also polyhedra where all ...
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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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176 views

Represent numbers on clock by polyhedrons

The "1" is replaced with a four-sided object, then the next one could be a five sided object, then six (the cube), but then after that, it is either a five-sided pyramid, or a eight-sided die. ...
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20 views

How to construct a polyhedron from given planes

This seems to be a basic questions, but I really don't know a good computer algorithm to do this. I have a set of planes (parameterized by normal direction and distance from a given point), and I want ...
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15 views

Tracing the faces of a convex polyhedron from edges and vertices

I have a set of vertices and edges that by construction, form a convex polyhedron. I would like to know how to trace out the faces of such a polyhedron i.e. find a list comprised of set of edges that ...
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47 views

Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub ...
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53 views

Number of deltahedra as a function of the number of faces

How does the number of deltahedra (polyhedra with only equilateral triangles as faces) with no holes grow asymptotically as a function of the number of it's faces? If we have this as $N=g(F)$ for ...
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44 views

Center of mass of vertices without enumeration?

Given a $n$-dimensional convex polytope defined by $A x\leq b$ and $A_{eq} x = b_{eq}$, is there an efficient way to determine the average coordinates of all vertices without enumerating them? (As if ...
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55 views

Stellating the Octahedron

I have a few related questions and I'd be happy to get some help with any one of them. Is the stellation of a polyhedron generally a 'messy' affair that involves trimming away portions of the ...
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27 views

Dinamically generate Goldberg polyhedra G(m,n)

In these pages the autor provided a lot of info about some Goldberg polyhedra (http://en.wikipedia.org/wiki/Goldberg_polyhedron): http://dmccooey.com/polyhedra/DualGeodesicIcosahedra.html ...
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117 views

What honeycomb has the highest volume to edge length ratio?

This question is analagous to the Kelvin Problem where the solution, the Weaire-Phelan Structure, has the highest volume to surface area ratio; however, the cell volume is compared to edge length ...
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76 views

What is the name of convex polyhedra with congruent faces of regular polygons?

The definition of platonic solids is the following (see Wikipedia): In Euclidean geometry, a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same ...
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119 views

Analytic-geometrical properties of dodecahedron

Consider the following projection of a dodecahedron: An equilateral triangle can be projected to make points $A, B, C, D, E, F$ intersect with it's edges. What would be the mathematical proof (if ...
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246 views

Symmetries of Archimedean Solids

There are five platonic solids, and 13/15 (which is correct?) Archimedean Solids. The finite groups of isometries of Euclidean $3$-space are the finite subgroups of $SO(3,\mathbb{R})$ or ...
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Does every polyhedral graph have a path cover with non-empty paths?

I'm looking to prove or disprove the following conjecture: Every polyhedral graph has a path cover with vertex disjoint, non-zero (length $\ge 1$) paths. Any pointers to literature are appreciated. ...
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52 views

On the Name of the Amplituhedron

Shouldn't the 'amplituhedron' really be called an 'amplitutope' since it's really a polytope and not strictly a polyhedron?
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40 views

Create Platonic solids from the coxeter group (vertexes & edges & faces)

How can one define vertexes, edges and faces from the Coxeter group? For example, for all platonic solids? I would like to create a general function that takes the Coxeter diagram as input, and gives ...
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40 views

Discrete Analogue of the Poincaré Conjecture and Simple Connectedness

I apologize if this question is badly worded or obvious, but I have no formal topology background. I have put some effort into trying to find something, but nothing turned up, perhaps due to my lack ...
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313 views

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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76 views

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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21 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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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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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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69 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 ...
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83 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 ...
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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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37 views

The dimension of birkoff polytope

Let $P_m$ be a subset for R^mxm be the polytope given by: $x_i,_j \ge 0$ $x_i,_1 + ... + x_i,_m \le 1$ $x_1,_j + ... + x_m,_j \le 1$ $\sum_{1 \le i,j \le m } \ x_i,_j \ge m-1$ Contruct a ...
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61 views

Convex cone as sum of simplices?

In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices. I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
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192 views

$\{x:Ax\leq 0\}$ contains a subset of type $\{x:A'x=0, ax\leq 0\}$

If $C:=\{x:Ax\leq 0\}\neq\{x:Ax=0\}$, an independent set of rows of $A$ can be chosen, one denoted by $a$ and the others put as rows into a matrix $A'$, such that $\{x:A'x=0,ax\leq 0\}\subseteq C$. ...
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Are there 3D tilings of a 3D projective hyperplane or 3-sphere?

I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
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Geometric Interpretation of $h_1(P)=f_{d-1}(P)-d$ for a polytope

In our lecture "Discrete Geometry 1", we are examining lineare realtions between the components of the f-vector and the h-vector of a polytope, in particular the Euler-Poincaré formula and the ...
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Integration over a polyhedron

We face the problem of computing the integral of a function f over a polyhedron P (defined by a mixed integer linear program) and we were thinking of using Latte (link) for this task. We would like to ...
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198 views

The structure theorem of Tropical geometry

The Structure Theorem of Tropical geometry states that "Let $X$ be an irreducible $d$-dimensional subvariety of $\mathbb T^n$ . Then $\operatorname{trop}(X)$ is the support of a balanced weighted ...
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Does it make sense to compare sets (polygons) with different dimensions?

In the context of integer programming, I am considering 3 different linear models for a given problem. The goal is to determine which formulation is the tightest, that is, the one that gives the least ...
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Regular packing of an infinite number of infinitely long cylinders in 3d space

Is it possible to pack an infinite number of congruent infinitely long cylinders into 3 dimensional space in a regular pattern? Another condition is that an equal number of the cylinders must be ...
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Simplest molecules with multiple local minima

Methane (C + 4 H) goes to a tetrahedral structure. Water (O + 2 H + 2 e-) goes to a slightly skewed tetrahedron. In a computer model, both of these have no local minimum problems. There is a ...
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53 views

Why is the height of a pentagonal antiprism equal to the circumradius of the base?

It is a fact (that one can verify, for example, by plugging in n=5 into the formulae on the Wolfram Mathworld page on antiprisms) that the height of a (regular) pentagonal antiprism (i.e., the ...
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Projection of a convex set in $\mathbb R^n$ onto $\mathbb R^2$

Suppose $A$ is an $n\times n$ matrix and $b$ is an $n\times 1$ column vector. $$X=\{ x\in \mathbb R^n: A x\leq b\}$$ Is it possible to compute the projection of $X$ on $(x_1,x_2) $ ...
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What are the exact critera for a CW-complex being a polytope?

Everybody talks about the fact that polyhedra are special CW complexes, and some of the higher dimensional abstract polytopes are too, but nobody tells the exact criteria for a CW complex being a ...