# Tagged Questions

Questions related to polyhedra and their properties.

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### How many octahedrons in icosahedron

How many different ways can octahedron be inscribed in icosahedron so that all vertices of octahedron are selected vertices of icosahedron? Can it even be done? There are 4 edges in the middle of ...
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### 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 ...
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### Definition of a polyhedral region

I believe the following two conditions on a subset $S$ of $\mathbb{R}^3$ may be equivalent. I would like to know if they are equivalent, and where I can find either a counterexample or a proof of ...
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### Simplest algorithm for edge coloring of a dodecahedron?

I have an origami model of a dodecahedron I am assembling. There are 30 edges with 3 colors of 10 each. I could use a diagram that gives a possible 3 color edge coloring. However, is there some sort ...
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### Proof equivalence of equal dihedral angles and vertices on a sphere for regular polyhedra.

I know that the following theorem is true: Theorem: Provided that all faces of a polyhedron are regular poygons, the statement all the dihedral angles are congruent'' is equivalent to saying ...
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### The relation between face counts and edge counts in a polyhedron, $3f_3 + 4f_4 + 5f_5 +\dots = 2E$

Why does $3f_3 + 4f_4 + 5f_5 + \dots = 2E$ hold for every polyhedron? Notation: $f_k$ is the number of faces with $k$ edges; $E$ is the total number of edges. Is there a specific proof for this or ...
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### Measure of overall misfit between two polyhedra

Imagine I have two arbitrary polyhedrons with the same volume. How could one reasonably measure the misfit between them. E.g. how could one determine the minimum possible volume that they could not ...
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### Polyhedron = polytope + polyhedral cone, how does it look graphically?

We have learned that a polyhedron is the sum of a polytope and a polyhedral cone, but how do you know this graphically? For example if you're a given polyhedron on paper and you have to determine ...
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### Answer check, where did I go wrong with this plane geometry question?

Consider a regular tetrahedron with edge length one (four equilateral triangles joined edge to edge) call it $T$. Set $T$ on the $x,y$-plane with a vertex at the origin and an edge aligned with the ...
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### bounding the hausdorff distance between a convex set and a template polytope.

How can we find an upper bound on the hausdorff distance between a convex set and its enclosing template polytope whose facets directions are given in advance?? Note that the bound should tend to zero ...
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### Why don't Archimedean solids give finite subgroups of $SO(3,\Bbb R)$?

I know that the Platonic solids correspond to finite subgroups of $SO(3,\Bbb R)$. For example, the tetrahedron corresponds to a subgroup isomorphic to $A_4$. The cube and octahedron to one isomorphic ...
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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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### Polyhedron cut along an edge

By cutting along an edge of a net of a polyhedron, you will form 2 pieces. Are there two distinct polyhedra for which this process may result in the same two pairs of pieces? Is there a real example? ...
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### How similar can the nets of distinct polyhedra be?

My school, and most math books do not cover 3-d geometry well, especially the topics of polyhedron nets. However, I see quite a few questions here are being answered about them. I was wondering about ...
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### “The gyroelongated triangular bipyramid can be made with equilateral triangles”

According to Wikipedia article Gyroelongated bipyramid The gyroelongated triangular bipyramid can be made with equilateral triangles I can only imagine that this would result in a cube, could ...
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### What are polyhedrons?

Polyhedrons or three dimensional analogues of polygons were studied by Euler who observed that if one lets $f$ to be the number of faces of a polyhedron, $n$ to be the number of solid angles and $e$ ...
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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 ...
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### Cut up a cube into pieces that form 3 regular tetrahedra?

Everyone knows that a regular tetrahedron fits inside a cube, and that the volume of the tetrahedron is 1/3 that of the cube. (For a picture, see this question or this Google image search.) The ...
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### Chromatic number of staggered icosidodecahedron

Shell 1: points of a unit edge icosidodecahedron centered on the origin. Shell 2: shell 1 / golden ratio Shell 3: shell 2 / golden ratio + origin Connect the 480 pairs of points that are at distance ...
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### Conbinatorial equivalence to cross-polytope

Let $p_1,\ldots,p_n \in \mathbb{R}^n$ be linearly independent and $C=Conv\{p_1,-p_1,p_2,-p_2,\ldots,p_n,-p_n\}$. Is it true that C is combinatorially equivalent to the n-dimensional cross-polytope? (...
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### Vertical visibility graphs — canonical icosahedral graph

Here's a vertical visibility graph for the icosahedral graph. Also called a scheduling graph. And maybe other names. Each open segment corresponds to a vertex. If there is unblocked vertical ...
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### 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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### Linear Algebra Tetrahedron Centroid 3/4 Down Segment Proof

Here is the question: Show that the centroid of the four vertices of a tetrahedron (a solid with four vertices joined by six lines that bound the tetrahedron’s four triangular faces) is the ...
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### uncertain point in textbook's solution of a “distance from point to line” problem

We have a cube with each side equaling $1$ unit of length. We need to determine the distance from $B$ to the line $A_1C_1$ In my calculation, each side of the yellowish-shaded triangle equals $\sqrt2$...
Let $x_0,\dots,x_K \in \mathbb{R}^n$. Consider the Voronoi region, $V$,around $x_0$ w.r.t. $x_1,\dots,x_k$. $V = \{x\in\mathbb{R}^n | \left\|x-x_0\right\|_2\leq \left\|x-x_i\right\|_2, i=1,\dots,K \}$ ...