Questions related to polyhedra and their properties.

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1answer
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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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1answer
66 views

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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2answers
476 views

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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1answer
144 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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0answers
45 views

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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1answer
529 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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1answer
174 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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1answer
144 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
87 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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1answer
133 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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1answer
54 views

what is a vector of polyhedron?

What does v mean in the following, does it a point inside the polyhedron?
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1answer
251 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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1answer
74 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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1answer
182 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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1answer
144 views

Curvature of a vertex in a polyhedral surface

Can anybody define what we mean by curvature of a vertex?
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2answers
188 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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0answers
58 views

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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1answer
205 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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3answers
1k views

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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2answers
95 views

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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1answer
71 views

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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1answer
311 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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1answer
478 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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1answer
454 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 ...
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3answers
302 views

Three-dimensional art galleries

The well-known art gallery problem starts with an "art gallery" (a simple polygon in the plane, not necessarily convex) and asks for the minimum number of "guards" (points on the polygon) required to ...
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1answer
138 views

Question about Euler's polyhedral formula in a proof of minimum distances

I am confused by a step made in a proof of the following result. Let $f_{2}^{\text{min}}(n)$ denote the maximum number of times the minimum distance can occur among n points in the plane. Then ...
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1answer
632 views

Icosahedron coordinates

Wikipedia says (link)that cartesian coordinates of icosahedron are: (0, ±1, ± φ) (±1, ± φ, 0) (± φ, 0, ±1) Where φ = (1 + √5) / 2 is golden ratio ≈ 1.618. I ...
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1answer
385 views

How to calculate vertices of dodecahedron using W|A?

Can I calculate for example with wolfram alpha coordinates of vertices of dodecahedron? I know coordinates of center point of dodecahedron (center of gravity) and the height of dodecahedron. Thanks ...
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2answers
1k views

What are the vertices of a regular tetrahedron embeded in a sphere of radius R

Imagine you had a sphere of radius R centered at the origin. What are the coordinates of the vertices of the regular tetrahedron which is circumscribed by the sphere? One of the vertices of the ...
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0answers
116 views

Grothendieck's schematic point of view of regular polyhedra

I am asking this question out of curiousity. Wikipedia lists Grothendieck's main mathematical achievements the last of which is "schematic point of view, or "arithmetics" for regular polyhedra and ...
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0answers
95 views

Represent numbers on clock by polyhedrons

The "1" is replaced with a four-sided object, then the next one could be a five sided object, then six (the cube), but then after that, it is either a five-sided pyramid, or a eight-sided die. ...
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0answers
168 views

The structure theorem of Tropical geometry

The Structure Theorem of Tropical geometry states that "Let $X$ be an irreducible $d$-dimensional subvariety of $\mathbb T^n$ . Then $\operatorname{trop}(X)$ is the support of a balanced weighted ...
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2answers
373 views

On gluing regular tetrahedra together to form a ring

The above are part of the articles and some background of it. And there is one claim in the articles, saying that " in order to show that no trivial ring can be formed, it is sufficient to show ...
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3answers
395 views

Platonic Solids

It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses ...
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1answer
200 views

The difference between polyhedral complex and support of a polyhedral complex?

A polyhedral complex is a collection of polyhedra such that intersection of any two polyhedron is a face of of both the polyhedron or empty. Support of a polyhedral complex is the set of all points ...
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4answers
1k views

How to cut a cube into an icosahedron?

Edit: Originally I asked this about a using a cube, but it is not a requirement to start with a cube, just how to end up with an icosahedron as on of the answers showed how to make dodecahedron a ...
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2answers
292 views

How to prove there are exactly eight convex deltahedra?

A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by ...
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1answer
514 views

Calculating the area of a cross-section of a tetrahedron

Ok, I completely revised my question. For those interested about my purpose with this question, see the older versions. So, I would like to calculate the area of a cross-section of a tetrahedron. The ...
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1answer
64 views

Determine if two polyhedrals are the same shape and if so, map their vertices

I have a polyhedron and want to determine whether it is combinatorially equivalent to another polyhedron. I know how many faces comprise each polyhedron and for each face, I know all of its vertices, ...
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1answer
160 views

convex polyhedron edge property

I have a convex polyhedron (with integral nodes). I only calculate in euclidian spaces. Let N be the set of nodes, c the center (arithmetic mean) of the polyhedron. I now want to determine if a line ...
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1answer
141 views

The set of distances between three points chosen with uniform probability on a finite interval

I pick three numbers $(x_1, x_2, x_3)$, where the value each $x_i$ is a real number selected with uniform probability on the interval $[J, K]$. I then plot a point in three-dimensions where {$x, y, ...
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0answers
132 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
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1answer
89 views

Polyhedra from Cayley Graphs

I was playing around with the Cayley graphs for some simple groups today and stumbled across something interesting, but can't quite figure out if there's something deeper going on. Here's what I did: ...
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1answer
196 views

What's the correct name for a geometric solid that's a beanbag?

enter image description hereMy daughter is in the first grade, and I'm having a good deal of fun trying to determine the shapes of irregular geometric solids. I'm stuck on the good, old beanbag. ...
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2answers
319 views

Find points of a regular tetrahedron

I'm given one of the vertices of a regular tetrahedron and the radius of the circumsphere. I also know the center point of the circumsphere. How can I find the remaining three vertices? (It was ...
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1answer
77 views

Icosahedral group. Need the angles of all diagonals.

I want to implement the icosahedral pointgroup. For that I need all angles of the lines between two opposite vertices, between the midpoints of two opposite faces and between the midpoints of two ...
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0answers
88 views

volume of polyhedra which correspond to icosahedral fullerenes

There is, I believe, a sequence of polyhedra whose shape approaches that of the icosahedron (they all have twelve pentagonal faces and the rest hexagons), and starts: regular dodecahedron (C$_{12}$, ...
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3answers
97 views

Finding the number of edges that connect to a single vertex in a dodecahedron

Please note my geometry background is very weak (high school geometry is all I have), so I would appreciate it if someone could explain it in very layman terms how to do this. I am trying to solve ...
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1answer
98 views

Determining if two tetrahedra in $\mathbb R^3$ are identical or have reflection symmetry

I have two tetrahedra in $\mathbb R^3$, $T_1$ and $T_2$, and access to the coordinates of their vertices. $T_1$ and $T_2$ are tetrahedra in the sense that they each have four vertices, each vertex is ...
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1answer
132 views

Convex polyhedron with five, six, or seven vertices at distinct corners of a cube

What are the names of the convex polyhedron with five, six, or seven vertices, where all vertices lie at distinct corners of a cube? I'm particularly interested in the five vertex case.