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.

Filter by
Sorted by
Tagged with
1
vote
0answers
23 views

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)...
0
votes
2answers
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, ...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
3
votes
2answers
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 ...
5
votes
2answers
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 ...
1
vote
1answer
26 views

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 ...
3
votes
1answer
96 views

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 ...
0
votes
1answer
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. ...
1
vote
0answers
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. ...
0
votes
2answers
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 ...
1
vote
1answer
41 views

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 ...
2
votes
0answers
54 views

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 ...
1
vote
0answers
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 ...
0
votes
1answer
947 views

How can I prove Euler's formula using mathematical induction

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 ...
2
votes
0answers
55 views

Why are the 1-skeleton graphs of the Regular Polytopes distance transitive?

A graph $G$ is distance transitive if for all vertices $u,v,w,x$ of $G$ such that $D_G(u,v) = D_G(w,x) $ implies that there exists a graph automorphism of $G$, $\psi \in \Gamma(G)$ such that $\psi(u) =...
2
votes
0answers
33 views

Polyhedra with coplanar non-adjacent faces

Two non-adjacent faces of a polyhedron are called $\textit{buddies}$ if they lie on the same plane. Call a polyhedron $\textit{nice}$ if every face has a buddy. What is the smallest $\textit{nice}$ ...
0
votes
1answer
43 views

Can higher dimensional spheres be regularly partitioned/discretized?

A circle can be partitioned into $n\in\mathbb{N}$ congruent 1-spherical line segments similar to the regular polygons. A sphere can be partitioned into $n\in\{4,6,20\}$ congruent 2-spherical ...
3
votes
1answer
88 views

All the 2-d polygons that can be used to form 3-d polyhedrons

The requirement is that we start with a 2-d polygon and using just copies of this polygon, cobble them together into a closed 3-d polyhedron (no gaps). I want to find a way to identify all such 2-d ...
1
vote
0answers
215 views

Electron Groups and Platonic Solids

In my Chemistry class, lately I have been learning about Lewis Structures for molecules, and how the arrangements of groups of electrons on each molecule repel each other to form the molecule into a ...
1
vote
0answers
29 views

Which subgroup of a symmetric group is isomorphic to the symmetry group of a platonic solid?

Is there a way to directly determine to which subgroup a symmetric group is the symmetry group of a polyhedron isomorphic to? In example, I know that the symmetry group of a tetrahedron is isomorphic ...
1
vote
0answers
38 views

Construct a Trigonal Trapezahedron from another platonic solid

Looking into the Tetartoid, which is a version of the Dodecahedron where all pentagons are not regular, described in the answer by Aretino here is a way to construct it from a Tetrahedron. Similarly,...
4
votes
5answers
878 views

Angles subtended by an edge in a regular dodecahedron?

If I have a regular dodecahedron and construct lines between the center of the dodecahedron and its vertices. How do I calculate the angle between such lines, subtended by an edge? This picture can ...
2
votes
1answer
102 views

Stuck on a proof about a tetrahedron's medians intersecting each other in the ratio 1:3

I am reading a proof about the above that goes like this: "Let $(A, B, C, D)$ be a tetrahedron and let $X ∈ (A, B, C, D)$. Let $m_X$ be the median line going from $X$ to the centroid of the opposite ...
1
vote
2answers
95 views

Number of Regular tetrahedron from Unit cube

Can we count number of Regular tetrahedrons formed out of Unit cube? If vertices of Unit cube are taken as $(0,0,0,)$, $(0,0,1)$, $(0,1,1)$, $(0,1,0)$, $(1,0,0)$, $(1,0,1)$, $(1,1,1)$ and $(1,1,0)$ ...
1
vote
1answer
629 views

What does “n-fold axis” mean in symmetry groups?

Reading about the symmetries of a cube, here, they talk about (in the section "Details") "3 x rotation about a four fold axis" etc. I'm not quite sure what "four fold axis" means in the context. Can ...
0
votes
1answer
139 views

What are the symmetries of a Trigonal trapezohedron?

The asymmetric version of a Trigonal Trapezohedron is supposed to be a fair die just like a cube, meaning I can start with one face and rotate it about the center of the solid to get each of the other ...
3
votes
2answers
430 views

Solids that are platonic apart from faces being irregular polyhedra.

Scouring through Wikipedia, I've found the following analogs to platonic solids that are composed of irregular faces. Cube = Trigonal Trapezohedron Dodecahedron = Tetartoid Tetrahedron = Disphenoid ...
3
votes
3answers
183 views

Irregular analogue of cube and octahedron.

If we take a Dodecahedron and remove the constraint that the pentagonal faces have to be regular pentagons, we get a solid called a Tetartoid. If we take the dual of that, we will end up with the ...
10
votes
4answers
588 views

How does this proof of the regular dodecahedron's existence fail?

On Tim Gowers' webpage he has an example "proof" of the regular dodecahedron's existence which he claims contains a flaw. He writes Of course, I have not written the above proof in a totally ...
0
votes
2answers
48 views

How to tell which faces of a convex solid are visible.

I have a series of faces that describe a 3-d solid. If I draw these faces, I've drawn the solid (my light source is at infinity). Except, for most of the solids I'm drawing (platonic solids and ...
0
votes
1answer
345 views

Coordinates of vertices of an icosahedron sitting on a face

I want to 3D print a wireframe of a icosahedron. For that I need the coordinates of the vertices so that one triangle lies flat on the $z=0$ plane. All I've found are descriptions for an icosahedron ...
9
votes
3answers
493 views

Making a regular tetrahedron out of concrete

I'm trying to make the following tetrahedron made of concrete just for fun: Each edge is a beam with a triangular cross section. I imagine the easiest way is to make 6 identical truncated triangular ...
52
votes
1answer
1k views

Why do all the Platonic Solids exist?

In three dimensions it is quite easy to prove that there exist at most five Platonic Solids. Each has to have at least three polygons meeting at each vertex, and the angles of these polygons have to ...
2
votes
1answer
139 views

Dichoral angle in 4D platonic solid from schlafli symbol

Is there a way to find the dichoral angle between two cells of a 4D platonic solids solely from its schlafli symbol ? I'm thinking of a trigonometric identity similar to the one for 3D platonic ...
1
vote
2answers
318 views

The order of the Symmetry Group of Platonic Solids

From Paolo Aluffi's "Algebra: Chapter 0", question II.2.8: Calculate the order of the symmetry groups for platonic solids. I can easily look this up, and some tutorials give the actual groups ...
0
votes
1answer
573 views

Finding the circumradius of a regular tetrahedron

Given a regular tetrahedron of edge length $a$, how do I prove that the circumradius of the tetrahedron is equal to $\frac{\sqrt 6}{4}a$?
1
vote
2answers
2k views

Compute the dihedral angle of a regular pyramid

Given a regular pyramid, defined as a right pyramid with a base which is a regular polygon, with the vertex above the centroid of the base, I would like to compute the dihedral angle between adjacent ...
0
votes
1answer
276 views

Angle between vertex and center of face in Platonic solids?

I'm trying to find the angle between a vertex and the center of one of the nearest faces in a dodecahedron. This would be nice to know the formula and/or number for all the Platonic solids though. I'...
3
votes
2answers
93 views

Joining the centers of the edges of Platonic solids

We know that taking the centers of the faces of any 3d polyhedron (say, the Platonic solids) produces the dual solid. And repeating this operation gives us back the original solid. Another possible ...
1
vote
2answers
82 views

Group of order $3$ acting on the tetrahedron

It is well-known that the group of (orientation-preserving) symmetries of the tetrahedron is isomorphic to $A_4$. Since $\mathbb{Z}/3$ is a quotient of $A_4$, $\mathbb{Z}/3$ also acts on the ...
50
votes
9answers
9k views

How many fair dice exist?

We know a coin is a fair die with a 50-50 probability for two alternatives. Similarly, all five Platonic solids are fair dice. That makes six solids that can be fair dice, but can there be more? One ...
3
votes
2answers
88 views

Construct the vertices of a cube from $ \vec{v} = \frac{1}{\sqrt{10}} (3,1,0) $

The exercise is to construct a cube (inscribed in the unit spher) with one of the corners at: $$ \vec{v} = \frac{1}{\sqrt{10}} (3,1,0) \in S^2 $$ I'm a bit stuck constructing the other seven ...
0
votes
1answer
65 views

Loops in a Platonic solid

For a given Platonic solid, how many closed paths are there on the edges of the solid if each edge can only be traversed once, and paths related by a rotation of the solid are considered the same? ...
2
votes
2answers
353 views

Do non-convex platonic solids exist?

Consider a solid with the following properties - It is composed of congruent, regular polygons. At each vertex, the same number of edges and faces meet. This is the same as the requirement for the ...
12
votes
4answers
2k views

How many $6$-sided and $8$-sided standard dice exist? [closed]

Standard means that the sum of the opposite sides is $7$ for $6$-sided die and $9$ for $8$-sided die.
2
votes
1answer
512 views

Equations of facelets of dodecahedron

I'm referring to the article on dodecahedrons from Wikipedia - https://en.wikipedia.org/wiki/Regular_dodecahedron It says that the vertices are given by - $$(\pm 1, \pm 1, \pm 1)$$ $$(0, \pm \phi, \...
3
votes
3answers
2k views

Symmetry group of Tetrahedron

Exercise : Find the Symmetry Group of : The Tetrahedron The Cube The sphere with radius $r=1$ on $\mathbb R^3$ Discussion : I am having a hard time understanding and solving exercises like this on ...
5
votes
0answers
155 views

How many platonic solids do exist in non-euclidean space?

The proof that there exists only five platonic solids assumes that the angle between the adjacent sides must be less than 360°, because otherwise the surfaces would be flat or even overlap. However ...
1
vote
0answers
194 views

Coordinates of Circles in a Pattern of a Truncated Icosahedron on a Sphere (soccer ball)

Given are circles of a certain radius $r_c$. These are arranged on a sphere with the radius $r_s$ in a pattern of hexagons and pentagons similar to a soccer ball (truncated icosahedron). Five circles ...