Questions related to polyhedra and their properties.

learn more… | top users | synonyms

18
votes
3answers
6k 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 ...
13
votes
3answers
887 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 ...
12
votes
1answer
530 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 ...
11
votes
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 ...
11
votes
2answers
244 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 ...
11
votes
2answers
445 views

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 ...
10
votes
1answer
340 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 ...
9
votes
1answer
164 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 ...
9
votes
1answer
457 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 ...
9
votes
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 ...
8
votes
4answers
6k 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 ...
8
votes
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 ...
8
votes
1answer
272 views

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 ...
8
votes
2answers
477 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 ...
7
votes
1answer
417 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 ...
7
votes
2answers
256 views

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 ...
7
votes
1answer
143 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 ...
6
votes
3answers
2k 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.
6
votes
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 $\mathbb{R}^n$, ...
6
votes
2answers
2k views

Tetrahedron inside a sphere

What's the largest regular tetrahedron (side length x) you can fit inside a sphere with a radius of one?
6
votes
3answers
175 views

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?
6
votes
2answers
216 views

Is there such a thing as the “edge-face dual” of a polyhedron, and is the “edge-face dual” of a cube a rhombic dodecahedron?

The dual of a polyhedron is a polyhedron where the vertices of one correspond to the faces of the other, and vice versa. Is there always a similar correspondence between a pair of polyhedra where the ...
6
votes
1answer
89 views

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 ...
6
votes
0answers
106 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 ...
5
votes
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 ...
5
votes
1answer
117 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 ...
5
votes
0answers
85 views

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 ...
5
votes
0answers
67 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
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$_{12}$, ...
4
votes
2answers
353 views

Which unfolding of an icosahedron has the least number of edges to be glued?

Does every unfolding of an icosahedron has the same number of edges to be glued to construct it back to the solid? If yes, what are those numbers for Platonic solids? If no, which unfoldings have the ...
4
votes
3answers
127 views

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 ...
4
votes
3answers
13k views

Height of a tetrahedron

How do I calculate: The height of a regular tetrahedron, 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 be from ...
4
votes
1answer
285 views

Tetrahedron volume

How to calculate volume of tetrahedron given lengths of all it's edges?
4
votes
2answers
150 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 ...
4
votes
3answers
198 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 ...
4
votes
1answer
83 views

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 ...
4
votes
2answers
72 views

Is projection of a convex polyhedron on a plane a convex polygon?

If we have a convex polyhedron with vertices $\mathbf{V}$ and project it on a plane $\mathbf{P}$, is this procedure equivalent to projecting points in $\mathbf{V}$ on the plane $\mathbf{P}$ and then ...
4
votes
1answer
76 views

Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
4
votes
1answer
141 views

An unbounded convex polyhedron realizing the primes?

Does there exist an unbounded convex polyhedron with faces that have 3, 5, 7, 11, 13, ... edges, i.e., such that the number of edges of each face realize exactly the odd primes, with each prime ...
4
votes
1answer
479 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: ...
3
votes
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 ...
3
votes
3answers
121 views

Intuition about the faces in the connected planar graphs

In the Euler formula, for counting the number of faces, we count the regions bounded by edges, including the outer, infinitely-large region, so in the graph $K_1$ there is only one face which is outer ...
3
votes
3answers
218 views

What shape is this?

im doing a question that involves a shape with 8 faces, 10 vertices and 16 edges. Can anyone enlighten me as to what this shape is called? Many Thanks
3
votes
1answer
363 views

Angles for a great dodecahedron

Could someone describe to me how to find the angle between two intersecting pentagonal faces on a great dodecahedron? Thanks
3
votes
2answers
76 views

Forming a polyhedron from concave polygonal faces.

A polyhedron is a convex, three dimensional region bounded by a finite number of polygonal faces. So is it possible that some of those polygonal faces be concave ? Can concave polygons be used in the ...
3
votes
1answer
80 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
1answer
149 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)? ...
3
votes
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. ...
3
votes
2answers
485 views

Radius of a Sphere inscribed in a Convex Polyhedron

My teacher gave me this problem in class as a challenge. It has stumped me for days, yet he refuses to give me the answer! Let $PQRSTU$ and $PQR'S'T 'U'$ be two regular planar ...
3
votes
0answers
41 views

Regular apeirohedra?

Have been toying with structures that I think are best describe as unbounded regular polyhedra. More specifically I arrived at non-convex polyhedra that are unbounded in one direction: Alternate ...