# Questions tagged [platonic-solids]

A Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.

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### What shapes can be made with 144 tetrahedron? Or multiples 288, 432?

I am wondering if there are any shapes that appear with this number of tetrahedron, similar to the 64 tetrahedron grid with 64. I am looking at the number 144 (12^2)...
59 views

### What does “orientation” of a platonic solid really mean?

Is there any rigorous definition of "orientation" of a platonic solid? Lots of books mention that the whole group of symmetries of platonic solids consists of rotations composed with reflections, ...
81 views

### Quotient of binary icosahedral group by its center, i.e., $2I/\{\pm 1\}$ is isomorphic to $A_5$

I know that the inner automorphisms of the binary icosahedral group is given by the quotient $2I/\{\pm 1\}$ where $\{\pm 1\}$ is the center of $2I$. I also know the elements of the binary icosahedral ...
67 views

### Is this solid (gyrate deltoidal icositetrahedron) a fair die?

I was having some fun looking at different solids to make a die out of (see here ) and came across the gyrate deltoidal icositetrahedron. This solid is not a Catalan solid, but nevertheless all ...
58 views

### Why is a positive angle defect sufficient for the existence of convex regular polytopes?

If we want to build a convex regular n-polytope, we can start with a regular (n-1)-polytope, arrange $k$ copies around each (n-3)-dimensional ridge, and fold into n-space. This gives an easy necessary ...
194 views

### Volume of regular octahedron

According to several sources on the internet, the volume of a regular octahedron with unit edge lengths is approximately 0.47. I haven't seen an explanation for why this is so yet. When I tried to ...
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### Formulae/strategy for performance of deductive reasoning on a Platonic solid?

I have a Platonic solid (in this case, a dodecahedron), and I have twelve names that I must put on this dodecahedron. I have an incomplete list of who is next to who, and I'm trying to fill in the ...
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### Probability that the orthographic projection of a randomly oriented regular tetrahedron is a triangle

I wanted to find the probability that, given a uniformly sampled rotation matrix, when applied to a regular tetrahedron, its orthographic projection is a triangle (instead of a quadrilateral) when ...
44 views

### Graphical relationship of Surface Area against Volume with Platonic solids

What would be the best way to find the function of the surface area based on the volume for platonic solids, so that the Surface Area to Volume ratio would be comparable as shown in the graph below. ...
133 views

### Regular Icosahedron - How much to rotate X, Y, Z to get from face to face?

I have a regular icosahedron shape in Adobe After Effects. I want to start out with the one side flush with the camera, then have the rotations end with another face (adjacent) flush with the camera. ...
51 views

### How to solve the system of inequalities when using topological proof to show that there are exactly 5 platonic solids? (Beginner)

I have seen other answers explaining the topological proof up until the point of $1/p + 1/q > 1/2$ and $p$, $q$ are greater than or equal to three Then they proceed to say that the 5 platonic ...
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### For certain pairs of Platonic Solids, are the edge-centers and face-centers equivalent?

Notice that a Tetrahedron has 6 edges, and a Cube has 6 faces. So lets draw points at the center of those 6 edges, and at the center of the 6 faces. If we project these points onto a unit sphere, do ...
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### Geometrical shape of the solutions of the quintic equation in 3d

I understand the geometric meaning of the solutions of the cubics and quartics in the plane using combinations of equilateral stars with 3 and 4 arms, respectively. I heard that quintic equations can ...
73 views

### Method to build a polyhedral die with given probabilities

Let's define a die as a polyhedron that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The die has $N$ states. Each state $n$ has a ...
Using Euler's formula in graph theory where $r - e + v = 2$ I can simply do induction on the edges where the base case is a single edge and the result will be 2 vertices. A single edge also has ...