# Tagged Questions

Questions related to polyhedra and their properties.

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### 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 sphere?...
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### 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 ...
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### How many sewings are there on a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings are ...
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### 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 ...
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### 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 ...
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### 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 $S$...
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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 3D picture of the 38-sided Engel space-filling polyhedron

On page 220 of Peter Engel's Geometric Crystallography, he describes a 38-sided convex polyhedron that can fill space. I've seen this this accepted as the record in various places, but I've never ...
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### Chebyshev center = center of mass?

I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of mass/...
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### 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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### $3\mathrm D$ Projection Of $4\mathrm D$ Polyhedron

Can someone identify this shape? I think it is a $3\mathrm D$ projection of $4\mathrm D$ polyhedron. The body in the center seems to be a truncated octahedron, so as the body in the middle. The ...
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### 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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### Biggest Little Polyhedron

The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1. Let's add a dimension and find the largest volumes. What is the biggest little polyhedron ...
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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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### 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 ...
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### How many $A_5$ are there inside $A_6$?

I am reading a paper that says: There are $12$ versions of $A_5$ in $A_6$: $1)$ the permutations that leave one thing unmoved. $2)$ the permutations of the six pairs of antipodal ...
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### 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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### 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.
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### 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 ...
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### Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
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### Tetrahedron inside a sphere

What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
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### Number of distinct nets of dual polyhedra

There are 11 non-congruent nets of a cube as well as 11 distinct nets of an octahedron. Both a dodecahedron and an icosahedron have 43380 distinct nets. Is it true that any pair of dual convex ...
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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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 it possible to inscribe a regular tetrahedron in every convex body?

Is it possible to inscribe at least one regular tetrahedron in every convex body?
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### Maximum area of triangle inside a convex polygon

Prove that within any convex polygon of area $A$, there exists a triangle with area at least $cA$, where $c=\tfrac{3}{8}$. Are there any better constants $c$? I'm not sure how to approach this ...
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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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### Coloring the pentagonal hexecontahedron

So, I'd like to color the pentagonal hexecontahedron in a way that is satisfying aesthetically and mathematically. For me this equates to, in order of priority - 1. No same-colored faces can share an ...
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### Can I specify the edge lengths of a simplicial polyhedron?

Let $X$ be a convex polyhedron in $\mathbb{R}^3$ whose faces are all triangles, and let $\ell$ be a function which assigns a positive real number to each edge of $X$. We say that $\ell$ is realizable ...
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### 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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### Is the adjective 'regular' necessary in the definition of Platonic solids?

The definition I mean can be found in the tag Wiki of Platonic solid tag and also in Wikipedia: Definition 1: A Platonic solid is a regular, convex polyhedron with congruent faces of ...
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### 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$_{12}$, ...