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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1answer
53 views

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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3answers
871 views

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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3answers
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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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15 views

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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8 views

Polyhedral Sets and $min$-function

I'm asked to verify if the following set is polyhedral, $$ X = \{[x_1;x_2]: min(x_1,x_2) \leq 0\}$$ Definition of a polyhedral set, A set $Y$ is polyhedral if $Y = \{y: Ay \leq b\}$, for finite $...
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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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Intersecting rational polyhedral cones

Call A the cone generated by the rays (1,0,0) and (0,1,0) and B the cone generated by the rays (1,1,0),(1,0,1), and (0,1,1). I want to compute the intersection of these polyhedral cones, but I am ...
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1answer
35 views

Regular Triangulations of Cube

I want to figure out which triangulations of the cube (i.e., partitions into tetrahedra using only the $8$ given vertices) are regular, but I'm not sure how to easily tell whether a given ...
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Sphere inside cylinder vs polyhedra?

Comparing a cylinder with a polyhedra that has a symmetric coxeter $\ge 3$. Both have their centers hollowed out by $k\%$, in the shape of their outer, i.e.: relative to top face Which can better ...
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1answer
55 views

Is closed convex set with finite number of extreme points convex polyhedron

I have this simple question related to convex set and convex polyhedron. As the content in the title, it's basically my question: Is closed convex set with finite number of extreme points convex ...
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0answers
27 views

$OABCD$ tetrahedron with $OA ⊥ OB ⊥ OC ⊥ OA$

I've got stuck at this problem: Let $OABCD$ be a tetrahedron with $OA ⊥ OB ⊥ OC ⊥ OA$. If $OH$ is the orthocentre of triangle $ABC$, show that $OH$ is perpendicular on plan $(ABC)$. Then ...
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1answer
45 views

The limit of infinite truncations?

When a regular polyhedron is made to undergo repeated truncations, is there a solid that acts as a kind of limit for this iterated process? That is, say a cube is truncated N times. As N gets larger ...
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4answers
1k views

Volume of 1/2 using hull of finite point set with diameter 1

It's easy to bound a volume of a half. For example, the points $(0,0,0),(0,0,1),(0,1,0),(3,0,0)$ can do it. The problem is harder if no two points can be further than 1 apart. Bound a volume of 1/2 ...
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1answer
142 views

Possible all-Pentagon Polyhedra

If a polyhedron is made only of pentagons and hexagons, how many pentagons can it contain? With the assumption of three polygons per vertex, one can prove there are 12 pentagons. Let's not make that ...
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1answer
75 views

Topological surface covered by hexagons and heptagons

I've found an interesting exercice that I don't know how to approach. It goes like this. We have a topological space which is Hausdorff, compact, connected and locally homeomorphic to $\mathbb{R}^{2}...
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1answer
43 views

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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1answer
665 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 ...
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2answers
270 views

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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1answer
58 views

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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37 views

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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2answers
95 views

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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1answer
44 views

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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1answer
38 views

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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1answer
34 views

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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26 views

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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1answer
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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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All pointed polyhedral cones are finitely generated.

A set $C \subset \mathbb{R}^n$ is called a cone if $x + y\in C$ for all $x\in C$ and $y\in C$ and $\lambda x \in C$ for all $x\in C$ and all real numbers $\lambda \geq 0$. A set $C \subset \mathbb{R}^...
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1answer
41 views

On a convex polytope

Let $e_i$-s denote the standard unit vectors of $\mathbb{R}^n.$ Denote $$\mathcal{C}_k = \left\{ \sum_{i \in S} \pm e_i \colon S \subseteq \{ 1,2,..., n\} \mbox{ and } |S| \leq k \right\}$$ the set ...
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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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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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how to find a tetrahedron in $R^n$ to bound an ellipsoid (again in $R^n$)

Assume you are given the following ellipsoid in $R^n$: $E: (c+\sum_{i=1}^n \alpha_ix_i)^2$, where $x_i$ 's are the coordinate variables. c and $\alpha_i$'s are constant. now the question is how to ...
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Kelvin problem with restriction to only one type of polyhedron

If we only allow one type of polyhedron, what would be the answer to the Kelvin problem? Would the Kelvin conjecture be true?
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1answer
24 views

Is $2n$ the smallest number of halfspaces to determine a segment in $\mathbb{R}^n$?

I proved that a segment in $\mathbb{R}^n$ is a polyhedron, and it is determined by $2n$ halfspaces. My question follows: Is $2n$ the smallest number of halfspaces to determine a segment in $\...
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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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1answer
48 views

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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1answer
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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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2answers
60 views

“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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1answer
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Facets shared by two points on a convex polytope

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} (...
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1answer
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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 ...
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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 $0\in P$ ...
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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?
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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 ...
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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}^{...
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2answers
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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 ...