Questions related to polyhedra and their properties.

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4
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2answers
792 views

Coloring dodecahedron

I found some months ago that there are the Polya's enumeration theorem to compute number of colorings of dodecahedron. I got interested to find how to show by using only Burnside's lemma that there ...
8
votes
4answers
5k views

Angle between lines joining tetrahedron center to vertices

What are the angles formed at the center of a tetrahedron if you draw lines to the vertices? I'm trying to make these: I need to know what angles to bend the metal.
9
votes
5answers
32k views

Height of a tetrahedron

How do I calculate the height of a regular tetrahedron having side length $1$ ? Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point ...
29
votes
2answers
1k views

What is the Coxeter diagram for?

I understand that Coxeter diagrams are supposed to communicate something about the structure of symmetry groups of polyhedra, but I am baffled about what that something is, or why the Coxeter ...
15
votes
2answers
429 views

Decomposable Families of Shapes

There are two types of golden triangles in the world, as shown in the following picture: Here $\varphi = \dfrac{1+\sqrt{5}}{2}$ denotes the golden ratio. Each of these golden triangles can be ...
8
votes
4answers
16k views

Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real ...
28
votes
4answers
19k views

Why are there 12 pentagons and 20 hexagons on a soccer ball?

Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated ...
12
votes
8answers
875 views

Cleverest construction of a dodecahedron / icosahedron?

One can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedra, cube, and octohedra all admit very intuitive ...
6
votes
1answer
2k 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, ...
11
votes
1answer
283 views

Is the Euler characteristic $\chi =2$ for the prism with a hole?

I keep getting $\chi=2$ for the solid in the picture. It's a prism with a hole joining two opposite sides. I remember reading that $\chi=0$ for such solids. Help me find my error. I'd appreciate if ...
6
votes
1answer
557 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 ...
5
votes
1answer
103 views

Finding generators of toric ideals

Consider the affine toric variety $V \subset k^{5}$ parametrized by $$\Phi(s,t,u) = (s^{4},t^{4},u^{4},s^{8}u,t^{12}u^{3}) \in k^{5}$$ where k is an algebraically closed field of characteristic 2. ...
9
votes
1answer
345 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)? ...
4
votes
2answers
2k 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 ...
3
votes
4answers
276 views

Inequality for each $a, b, c, d$ being each area of four faces of a tetrahedron

We know 'triangle inequality'. I'm interested in the generalization of this inequality. Here is my question. Question: How can we represent a necessary and sufficient condition for each positive ...
1
vote
1answer
215 views

Linear isoparametrics with dual finite elements

The subject presented here is some content of the Wikipedia page about Platonic solids combined with my own experience on Finite Elements.To start with the latter, there is a standard piece of Finite ...
0
votes
2answers
648 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 ...
16
votes
2answers
463 views

What property of certain regular polygons allows them to be faces of the Platonic Solids?

It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular ...
13
votes
1answer
2k 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 ...
8
votes
1answer
176 views

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 $8$ ...
7
votes
1answer
178 views

Showing that group of orientation preserving isometries of Icosahedron is a simple group

Let $G$ denote the group of orientation preserving isometries of Icosahedron. To prove the claim, I have shown that $\nexists \ N \ \triangleleft \ G$ such that $|N|=5.$ $\nexists \ N \ ...
14
votes
3answers
1k views

Making a convex polyhedron with two sheets of paper

Suppose that we have two sheets of paper $S,T$ and that each of $S,T$ is in the shape of a convex quadrilateral. Also, suppose that the length of the perimeter of $S$ equals that of $T$. (Note that ...
13
votes
1answer
296 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 ...
11
votes
2answers
448 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 ...
6
votes
1answer
65 views
+50

can a convex polygon have only one boundary point at locally maximum distance from its centroid?

It's easy to see that given any convex polygon P and any point c in its interior, there is at least one point m on the boundary of P at locally maximum distance from c: simply choose m to be a vertex ...
4
votes
1answer
410 views

Five Cubes in Dodecahedron

I will demonstrate why the group of rotational symmetries of a Dodecahedron is $A_5$. For that, we have to find five nice objects, on which the group of symmetries acts. One object is "Cubes" ...
4
votes
0answers
78 views

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?
13
votes
1answer
662 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 ...
12
votes
1answer
431 views

Largest $n$-vertex polyhedron that fits into a unit sphere

In two dimensions, it is not hard to see that the $n$-vertex polygon of maximum area that fits into a unit circle is the regular $n$-gon whose vertices lie on the circle: For any other vertex ...
4
votes
1answer
93 views

Lower bound on the number of faces of a polyhedron of genus g

Is there a lower bound on the number of faces of a polyhedron of topological genus g? For example: it seems very reasonable that $g$ < $F$ i.e. the genus of a polehydron is less than the number ...
2
votes
0answers
47 views

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 ...
7
votes
2answers
867 views

Name of this convex polyhedron?

Does anyone recognize / know the name of the convex polyhedron depicted below as the intersection of a Cuboctahedron and a Rhombicdodecahedron? Please note you have to interpret this picture and ...
5
votes
3answers
115 views

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 ...
5
votes
0answers
128 views

Is Paley-13 graph a unit distance graph in 3D space?

The 13-node Paley graph has vertices 1 to 13 that are connected by an edge when their difference is one of the values $(1,3,4,9,10,12)$ Can this graph be put into 3D space so that all edges have ...
5
votes
3answers
206 views

Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

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 ...
4
votes
3answers
437 views

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 ...
3
votes
0answers
33 views

Asymptotic bounds on the number of faces needed to construct a polyhedron of a certain genus

Let a polyhedron be a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices, where moreover we require that every edge touches exactly two faces, every ...
3
votes
1answer
392 views

Tetrahedron volume

How to calculate volume of tetrahedron given lengths of all it's edges?
1
vote
1answer
43 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 ...
1
vote
3answers
200 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 ...
6
votes
1answer
190 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 ...
4
votes
2answers
196 views

Why is Octahemioctahedron topologically a torus?

I'm afraid, that I have a very bad space vision, because I don't see, that Octahemioctahedron is topologically a torus. Could somebody explain it for me, why is it? Scene 2. @aes: Finally, ...
4
votes
1answer
189 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 ...
3
votes
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 ...
3
votes
1answer
65 views

What does an Euler characteristic of a topological space greater than 2 topologically mean?

Recently I've found a polyhedron with Euler characteristic $\chi=9$. This is made from the Octahemioctahedron with adding the intersections of the hexagon faces as vertices. It has $V=13, E=36, ...
2
votes
1answer
33 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 ...
2
votes
1answer
256 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 ...
2
votes
1answer
158 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, ...
2
votes
1answer
147 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.
1
vote
0answers
86 views

Stereographic projection of the icosahedron and snub cube?

Using a steoreographic projection, the three equations associated with the icosahedron with unit circumradius, inradius, and midradius (respectively) are, $$f=z^{20} - 228z^{15} + 494z^{10} + 228z^5 ...