Questions related to polyhedra and their properties.

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How to define polyhedra?

Wikipedia does not provide a concise definition of "polyhedron" in $\mathbb R^n$. What is the "best" - in whatever sense - definition of this class of objects? I am interested in a definition where ...
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480 views

Euler's formula for triangle mesh

Can anyone explain to me these two facts which I don't get from Euler's formula for triangle meshes? First, Euler's formula reads $V - E + F = 2(1-g)$ where $V$ is vertices number, $E$ edges number, ...
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191 views

Has anyone discovered a convex space-filling 15-faced polyhedron?

I've been looking for extensive surveys regarding space-filling polyhedra, but have only come across Michael Goldbergs "Convex polyhedral space-fillers of more than twelve faces" from 1979, stating ...
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97 views

Calculate polyhedra vertices based on faces

I have some origami polyhedra which I know the type of faces it has and how they are connected (such as this torus) and I want to calculate the co-ordinates of the vertices to use as an input to ...
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210 views

Number of edge colorings in a tetrahedron with three colors. Is my solution correct?

I've tried to count rotationally distinct edge colorings (both proper and improper) in a regular tetrahedron with three colors. Could you take a look if it's correct? First the improper colorings. ...
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91 views

Is this graph coloring problem solved correctly?

On this Wikipedia page about Burnside's lemma, it is calculated that there are 57 rotationally distinct colorings of the faces of a cube with three colors. I'm confused by the way it is done. They ...
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1answer
263 views

Goldberg polyhedra coordinates

I would 3D-print some Goldberg Polyhedra importing in Sketchup, the coordinates provided on these links: 72 faces (2,1) - (coordinates) 132 faces (3,1) - (coordinates) 192 faces (3,2) - ...
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189 views

How many $n$-colorings up to rotation using exactly 2 of each color are there on a $2n$-polyhedron?

I'm a high-school student and I stumbled across a YouTube video explaining how Rubik's cubes work. A Rubik's cube has 6 colors, one for each side, but I started thinking about ways to $n$-color the ...
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Nested Tetrahedrons

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely ...
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Convex cone as sum of simplices?

In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices. I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
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184 views

$\{x:Ax\leq 0\}$ contains a subset of type $\{x:A'x=0, ax\leq 0\}$

If $C:=\{x:Ax\leq 0\}\neq\{x:Ax=0\}$, an independent set of rows of $A$ can be chosen, one denoted by $a$ and the others put as rows into a matrix $A'$, such that $\{x:A'x=0,ax\leq 0\}\subseteq C$. ...
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Estimate the number of integral solutions inside a convex polyhedron

How can I compute an estimate of the number of integral solutions (points) inside a bounded convex polyhedron with dimension $d$? I'm interested more in an efficient way to estimate the number of ...
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1answer
160 views

How tell if a polyhedron contains a lattice point

So given a polyhedron $Ax \le b$ Is there an Algorithm or formula to determine whether said polyhedron contains a lattice point (integer point) I was thinking a couple things: brute force ...
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1answer
122 views

What is the minimal isoperimetric ratio of a polyhedron with $5$ vertices?

I'm asking and answering this question to provide a partial answer to this question and a comment on this answer at MO. The isoperimetric ratio $\mu$ of a solid is the ratio $A^3/V^2$, where $A$ is ...
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181 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
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1answer
99 views

Which polyhedra have an even number of faces touching each vertex?

A two-coloring of the faces of a polyhedron is always possible when an even number of faces meet at each vertex. http://www.georgehart.com/virtual-polyhedra/colorings.html Is there a name for ...
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120 views

Is There a Formalization of Cauchy's $F - E+V = 2$ proof?

Can anyone provide, or direct me to a formalized version of Cauchy's proof that for any convex polyhedron with $F$ faces, $E$ edges and $V$ vertices that $F - E + V = 2$. I am willing to accept the ...
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Mappings preserving convex polyhedra

It is known that linear mappings between euclidean spaces map convex polyhedra to convex polyhedra. Can you give a characterization of the class of mappings that preserve convex polyhedra?
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480 views

Calculate Spherical Distance between points

I have googled this and not come up with an answer yet, but basically, I'm trying to find out the distance between each point or vertice on a sphere (all points are evenly spaced). I already know this ...
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150 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
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51 views

Maximum Number of Divisions in Octahedron into congruent parts?

I am trying to divide octahedron into congruent parts. I found octahedron inside tetrahedron sided by four smaller tetrahedrons. I found some division here to 12 congruent parts. I can divide ...
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Smallest amount of planes to enclose a closed space in extended projective geometry $\mathbb R^3_{\pm\infty}$

The smallest amount of planes to enclose a polyhedron is 4 in the euclidean $\mathbb R^3$ where it encloses a tetrahedron. What is the smallest amount of planes to enclose a closed space in extended ...
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153 views

Is there a forumla to find out if n faces can be made into a 'regular polyhedron'?

I'm not too sure about the exact terminology since Wikipedia is throwing me all over the place. I'm looking for a formula to find out if for n faces a 'regular polyhedron' can exist. In case that's ...
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Are the face–centroid pyramids of a convex congruent-faced polyhedron congruent?

Let a convex polyhedron $P$ be given, all of whose faces are congruent. Consider any pyramid formed by a face of $P$ as its base and the centroid of $P$ as its vertex. Allowing congruence to admit ...
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Geodesics on a polyhedron

Which sequences of adjacent edges of a polyhedron could be considered to be a geodesic? The edges of a face most surely will not, but the "equator" of the octahedron eventually will. But for what ...
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Part from “Regular polytopes” which I don't understand

This is a paragraph from "Regular polytopes" by Coxeter that I don't understand. Although it is not always possible to include all the vertices of a polyhedron in a single chain of edges, it ...
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The structure of realization spaces of polyhedral graphs

Given a polyhedral graph with $v$ vertices, $e$ edges and $f$ faces, each possible realization of the graph as a geometric (convex) polyhedron corresponds to a point in $\mathbb{R}^{3v}$, ...
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insphere/circumsphere ratio of a polyhedron the same as its dual polyhedron?

Is the $r/R$ ratio for any polyhedron always the same as the $r/R$ ratio of the dual of that polyhedron? Given any polyhedron, we can find the biggest sphere that fits inside it (its insphere) and ...
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179 views

Does Voronoi tessellation in 3D always produce convex polyhedrons?

I'm more or less certain that the Voronoi tessellations (using Euclidean distance measure) produce convex polygons/polyhedrons. Is there a way to prove this mathematically? Or am I wrong? I am ...
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Geometric Interpretation of $h_1(P)=f_{d-1}(P)-d$ for a polytope

In our lecture "Discrete Geometry 1", we are examining lineare realtions between the components of the f-vector and the h-vector of a polytope, in particular the Euler-Poincaré formula and the ...
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655 views

Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right ...
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209 views

Altitude of tetrahedron?

I'm really curious to know any relationships between the altitude of a tetrahedron and how the foot of this altitude splits the base triangle. For example if you have a tetrahedron PABC with apex P, ...
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167 views

Unfolding Polyhedra

I'm interested in learning more on unfolding polyhedra. Are there any known algorithms that unfold polyhedra into nets? I'm interested in writing code on this in either MATLAB, Python, or C#. On ...
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1answer
101 views

Classification of, Information Regarding Particular Family of Polyhedra:

Let $n\ge4$. Let $n$ vertices be distributed on a spherical boundary. Let the vertices lie on this boundary as would electrons on a spherical boundary. That is, they are distributed "equally" by ...
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141 views

Relationship between angles in tetrahedron

Let's say I have a tetrahedron like this in image: Do angles $CAD$ and $CBD$ equals in general tetrahedron?
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what is a vector of polyhedron?

What does v mean in the following, does it a point inside the polyhedron?
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263 views

What hexahedra have faces with areas of exactly 1, 2, 3, 4, 5, and 6 units?

I tried for a while, not very hard, to construct a polyhedron with exactly six faces, whose areas were respectively 1, 2, 3, 4, 5, and 6 units. I did not meet with any success. Still, it seems that ...
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84 views

Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb H3$, has a boundary $\mathbb CP1$. A ideal tetrahedron in $\mathbb H3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb CP1$. The four vertices ...
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203 views

Graph Isomorphisms, Delaunay Triangulation on a sphere, and Kulikowski's Theorem

Suppose I have a collection of $n$ non-collinear points on a sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n $. And I construct a mapping from this collection of points to the Delaunay Triangulation ...
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161 views

Curvature of a vertex in a polyhedral surface

Can anybody define what we mean by curvature of a vertex?
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242 views

Combinatorially equivalent polyhedra?

What does it mean for two polyhedra to be combinatorially equivalent? I've looked on the internet but in vain. If it's not a standard definition, then it might help to say that I found this term in a ...
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Integration over a polyhedron

We face the problem of computing the integral of a function f over a polyhedron P (defined by a mixed integer linear program) and we were thinking of using Latte (link) for this task. We would like to ...
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273 views

How to compute the image of a polyhedron under a linear transformation

Suppose $P$ is a polyhedron whose representation as a system of linear inequalities is given to us: $$ P = \{ x ~|~ Ax \leq b\}$$ Define $P'$ be the image of $P$ under the linear transformation which ...
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Given a tetrahedron, how to find the outward surface normals for each side?

Say I have a triangle in $3$D space. I can get the surface normal by calculating the vector cross product of two of the edges. But, lets say I make this a tetrahedron. How can I work out the outward ...
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Regular Polyhedrons

In $\mathbb{R}^3$, there are five regular polyhedrons (up to similarity), and can be parametrized by number of vertices, edges and faces. What is the number of regular polyhedrons in $\mathbb{R}^n$, ...
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Is $\mathbb R^7$ minimally sufficient for embedding 3 tetrahedra - ABCD, ABEF, and CEGH - of equal edge length?

I've got 8 points - A, B, C, D, E, F, G, H - and I need three specific sets of four (ABCD, ABEF, and CEGH) to describe tetrahedra of equal edge length in some multidimensional space. I can embed ABCD ...
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356 views

Regular polygons that touching to a sphere surface

What is the possible number of n sided polygons(every face is the same regular polygon) that touching their corners to sphere surface and also touching each other ? I would like to know the relation ...
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588 views

Conventional ordering of faces of regular polyhedron?

e.g. For an icosahedron defined as follows: Diagram: A regular icosahedron (courtesy of Microsoft Visio): We define position and orientation w.r.t. this body's frame of reference as follows: ...
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561 views

How would you make a (physical) dodecahedron with edges instead of faces?

Call this a "math problem disguised as a woodworking problem" or vice versa. Background: You can construct a dodecahedron by cutting 12 identical, regular pentagon faces, beveling all the edges of ...