Questions related to polyhedra and their properties.

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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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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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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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$OABCD$ 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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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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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
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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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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 ...
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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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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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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
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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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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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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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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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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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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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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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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 ...
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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. $x_1,\dots,x_k$. $V = \{x\in\mathbb{R}^n | \left\|x-x_0\right\|_2\leq \left\|x-x_i\right\|_2, i=1,\dots,K \}$ ...