# Tagged Questions

Questions related to polyhedra and their properties.

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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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### 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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### Polyhedra vs Polytope

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?
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### Is a snub disphenoid more oblate or prolate?

Figuring out which deltahedra are oblate/prolate (of coarse the platonic solids are spherical) was pretty easy for all of them except for the snub disphenoid... Anyone know what the radia of the ...
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### The $d$-skeleton of a polytope is strongly connected

A polytope is the convex hull of a finite set of points in $\mathbb R^n$. The $d$-skeleton of a polytope is the set consisting of faces of dimension at most $d$. I would like to show that every $d$-...
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### 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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### How many sewings are there on a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings are ...
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### What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
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### Faces of a Bipyramid over a a Simplicial Polytope

Is there a simple way of expressing the number of faces of a bipyramid built over a polytope that is known to be simplicial, using the number of faces of the original polytope? This seems an easy ...
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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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I have a convex polytope of arbitrary dimension. Let $\mathcal{F} (A)$ denote the set of facets that vertex $A$ belongs to. If two vertices share an edge, is it true that the disunion of $\mathcal{F} (... 1answer 25 views ### What is the answer to the Kelvin problem if restriced our selves to isohedral polyhedra? A few days ago I asked this question. It turned out to be a known unsolved problem in mathematics: The Kelvin problem. Now I'd like to slightly change the question: What is the optimal way to ... 0answers 29 views ### relationship between polyhedron and its polar set Polyhedron is defined by intersection of finite half space in Euclidean space. Let$P$be a polyhedron, denote $$P^*=\{u\in\mathbb R^n|<u,x>+1\geq0,x\in P\}$$ as polar set. Theorem$0\in P$... 2answers 96 views ### Non-convex polyhedron with 18 edges, 12 faces and 8 vertices [closed] Which non-convex polyhedron has 8 vertices, 12 faces and 18 edges? 1answer 36 views ### What is the analogon of the hexagonal grid in 3-dimensional space? Rhombic dodecahedral honeycomb? Conjecture: The optimal way to divide 3-space into pieces of equal volume with the least total surface area is the rhombic dodecahedral honeycomb. Reasoning: "(The rhombic dodecahedral honeycomb) is ... 0answers 53 views ### The number of facets of an affine image I have a full dimensional polyhedron$P_1 \subseteq \mathbb{R}^d.$Now i define another polyhedron as follows: $$P_2 = AP_1 \oplus B$$ with$A \in \mathbb{R}^{(d-1) \times d}, \,\, B \in \mathbb{R}^{...
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 $e$ ...