Questions related to polyhedra and their properties.

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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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2answers
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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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1answer
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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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1answer
37 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
31 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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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
30 views

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 ...
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1answer
35 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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25 views

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
23 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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43 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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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
70 views

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
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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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1answer
21 views

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
23 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 ...
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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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83 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
32 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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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 ...
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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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1answer
58 views

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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1answer
28 views

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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1answer
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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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1answer
194 views

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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3answers
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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 ...
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0answers
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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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Voronoi sets and polyhedral decomposition.

Let $x_0,\dots,x_K \in \mathbb{R}^n$. Consider the Voronoi region, $V$,around $x_0$ w.r.t. $x_1,\dots,x_k$. $V = \{x\in\mathbb{R}^n | \left\|x-x_0\right\|_2\leq \left\|x-x_i\right\|_2, i=1,\dots,K \}$ ...
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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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Do there exist brief formulas to describe multiple symmetric, platonic , prismatic polyhedra?

Mathematics resources give coordinates of special polyhedra by their coordinates. Is there a mathematical algorithm that can take an input number i.e. 4-5-5-10-12-24- and divide space in to various ...
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Is $S = \{ x\in\mathbb{R}^n | x\geq 0 x^Ty\leq 1 \forall y\text{ with } ||y||_2=1\}$ a polyhedra?

Is the set $S$ polyhedra? where $S = \{ x\in\mathbb{R}^n | x\geq 0 x^Ty\leq 1 \forall y\text{ with } ||y||_2=1\}$ The answer is : S is not a polyhedron. It is the intersection of the unit ball ...
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2answers
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Insect eyes. Tiling spheres with hexagons - pentagons required as facet size decreases?

I’m an amateur naturalist and am fascinated with the underlying geometry of an insect eye primarily made of hexagonal facets. In particular, how the sphere surface is fully packed as the hexagons ...
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1answer
112 views

Faces of Polyhedra

I am starting to study convex geometry. Let $\mathcal{P}$ be a polyhedron in $\mathbb{R}^n$. I want to show that the face $\mathcal{F}'$ of a face $\mathcal{F}$ of $\mathcal{P}$ is still a face of ...
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Looking for polyhedra under two simple but stringent conditionts

I noticed usual polyhedra have some vertices joining exactly 3 edges or some triangular faces (or both). Out of curiosity I started wondering if there is a polyhedron with the following constrains: ...
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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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The smallest 8 cubes to cover a regular tetrahedron

A regular tetrahedron $T$ of edge-length $\sqrt{2}$ fits inside a unit cube:                     (Image from MathWorld.) This means that $8$ ...
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1answer
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About the interior of a polyhedron

Let us consider a polyhedron in $\mathbb{R}^n$ (in this context it must NOT be bounded) $\mathcal{P} = \{ x: A \cdot x \leq c\}$ for some matrix $A$. Let $\mathcal{I} = \{ x: A \cdot x < c\}$ be a ...
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1answer
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The Rhombohedron

I am trying to model a rhombohedron (using Blender) as a first pass to building Dürer's solid so I am trying to calculate the (x,y,z) values for a given side length 'a' and angle 'theta' (starting ...
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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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2answers
67 views

Faster Algorithms for Convex Hulls

I was interested in the following: Given two polyhedra $P_1, P_2$ specified in the form: $$ P_1 = {x : A_1x \le b_1 } $$ $$ P_2 = {x : A_2x \le b_2 } $$ Whereas $ x \in R^n$ and $b_1, b_2$ are ...
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trying to grasp disphenoid tetrahedral honeycomb, what are the dihedral angles?

What are the dihedral angles in a disphenoid with four identical triangles, each having one edge of length $2$ and two edges of length $\sqrt{3}$? Tried to look it up, but couldn't find it...
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2answers
289 views

How do you find redundant constraints for a feasible region?

I've found a few papers that deal with removing redundant inequality constraints for linear programs, but I'm only trying to find the non-redundant constraints that define a feasible region (i.e. I ...
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2answers
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How many face we could make regular convex polyhedron

I want to tile the sphere as many face as possible. And I want every face be the same size and shape. Is it possible to generate more than 100 or 1000 faces of regular convex polyhedron?