Questions related to polyhedra and their properties.

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11
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551 views

Any other Caltrops?

This question has been edited. The regular tetrahedron is a caltrop. When it lands on a face, one vertex points straight up, ready to jab the foot of anyone stepping on it. Define a caltrop as a ...
6
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217 views

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 ...
5
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36 views

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 ...
5
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139 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 ...
5
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125 views

Symbols to represent each distinct symmetry of polyhedra

Is there a pictorial or symbolic way to represent each distinct symmetry of a polyhedron?
5
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123 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 ...
4
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33 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 ...
4
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57 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 ...
4
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32 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 ...
4
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0answers
76 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?
4
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178 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
3
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54 views

3D models of the unfoldings of the hypercube?

There are (apparently) 261 distinct unfoldings of the 4D hypercube, a.k.a., the tesseract, into 3D.1 These unfoldings (or "nets") are analogous to the 11 unfoldings of the 3D cube into the plane.2 ...
3
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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 ...
3
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171 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. ...
2
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43 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 ...
2
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48 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 ...
2
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0answers
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 ...
2
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103 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 ...
2
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60 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 ...
2
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110 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 ...
2
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224 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 ...
2
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24 views

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. ...
2
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49 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?
2
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36 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 ...
2
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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 ...
2
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278 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 ...
2
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0answers
71 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 ...
2
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0answers
19 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, ...
2
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0answers
53 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 $xy(x^{10}+14x^5y^5-y^{10})=0$. ...
2
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0answers
191 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 ...
2
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0answers
66 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
79 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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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 ...
2
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0answers
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 ...
2
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0answers
59 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 ...
2
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0answers
190 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 $\{x:A'x=0,ax\leq 0\}\subseteq C$. ...
2
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0answers
83 views

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 ...
2
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0answers
52 views

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 ...
2
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0answers
86 views

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 ...
2
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0answers
195 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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0answers
48 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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0answers
38 views

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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0answers
37 views

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 ...
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0answers
25 views

What does “modulo polyhedra with lines” mean?

I have done some reading about integer points in polyhedra, and in one of the books I have come across the definition: "Let $f$,$g$ be polyhedra. $f$ $\equiv$ $g$ modulo polyhedra with lines provided ...
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19 views

The Walking Saddle Curve and other weirdly moving shapes

The link at the end of this sentence seems to be a video of a walking saddle curve. Where can we obtain these? What are more exact parameters of this curve? What is the relation between wind speed ...
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18 views

For polytopes, does union and linear transformation commute?

Given two (convex) polytopes $P_1$ and $P_2$ and a linear transformation $T$, is it true that: $$T(P_1 \cup P_2) = TP_1 \cup TP_2$$ What if $P_1$ and $P_2$ are not convex?
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25 views

Polyhedral surface with infinitely many triangulations with same combinatorics

Is there an example of a polyhedral surface that has infinitely many triangulations with the same combinatorics?
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64 views

How to determine the outward normal vector of a face of a hexahedron if the orientation (CW or CCW) of vertices of the face is unknown

Consider a single quadrilaterally-faced hexahedron. If given the co-ordinates of the vertices, $\mathbf{v}_i$, of a face in counter-clockwise orientation, I can compute the corresponding unit outward ...
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0answers
97 views

The skeleton of Eulerian polyhedra

There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
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81 views

How to find out the circumscribed radius of a snub dodecahedron?

A snub dodecahedron is produced by partially rotating all 12 regular pentagonal faces of a small rhombocosidodecahedron. A snub dodecahedron has 80 congruent equilateral triangular faces, 12 congruent ...