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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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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) =...
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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 $\textit{nice}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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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 ...
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1answer
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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 ...
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2answers
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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 $(1,1,0)$ ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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Convex Hulls and maximizing volume

I thought of a function (recreational mathematics) and wonder if there is any existing math about it. Google searching did not turn anything up. Let $n\in \mathbb{N}$ be the dimension, and $x\in \...
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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 ...
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1answer
142 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 ...
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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 ...
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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 ...
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1answer
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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 ...
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146 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 ...
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1answer
260 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 $\frac{\sqrt 6}{4}a$?
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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 ...
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1answer
146 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'...
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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 ...
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2answers
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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 ...
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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 ...
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2answers
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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 ...
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1answer
46 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? ...
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1answer
207 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 ...
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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.
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1answer
249 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, \...
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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 ...
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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 ...
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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 ...
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1answer
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computing the coordinates of vertices of convex regular polyhedra and 4-polytopes

I am considering what I understand to be generalizations of the platonic solids. In the plane one can easily obtain the vertices of a convex regular k-gon by computing roots of unity, and this can be ...
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3answers
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Platonic solids: duality. What is meant by “reversing inclusion”?

Cube $C$ and Octahedron $O$ are dual Platonic solids in the sense that the the faces and the vertices are interchanged. Often this is expressed like this: There is an bijection $B\colon C\to O$ which ...
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Exactly 5 Platonic solids: Where in the proof do we need convexity and regularity?

The famous statement that Only five convex regular polyhedra exist. is usually proven as follows: Let $P$ be a convex regular polyhedron with V vertices, E edges and F faces. Moreoever, ...
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1answer
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Conjugacy in rotational symmetries of tetrahedron and icosahedron

Let $T$, resp. $I$, denote the group of rotational symmetries of tetrahedreon, resp. icosahedron. Fix a face of each of these polyhedrons; let centers of these polyhedrons be $(0,0,0)$ in $\mathbb{R}^...
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2answers
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Tetrahedron version of Pythagorean theorem [closed]

Consider a tetrahedron with an equilateral base and two of the other three faces being right triangles with their right angle points meeting. This leaves the last face to be some isosceles triangle. ...
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1answer
344 views

Can platonic solids be constructed using compass and straightedge?

By moving the concept of geometric construction into three dimensions, could one trace the 3D wireframe of any of the five platonic solids using only a compass and straightedge? If not, what ...
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1answer
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What is the angle of a line through the interior and two vertices to one face of an icosahedron?

The following diagram represents three faces of a regular icosahedron, flattened out for ease of illustration. If they are folded appropriately, line $ab$, in blue, passes through the interior of the ...
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1answer
58 views

Is there any mathematical formula to find the coordinates of equidistant poins on the surface of a sphere?t

The general form of the coordinates is: coords = r $\{$$\cos\theta$ $\sin\phi$, $\sin\theta$ $\sin\phi$, $\cos\phi$} I've considered the radius $r=1$. Now varying the angles $\theta$ and $\phi$, I ...
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1answer
274 views

Determine the length of a side of an octahedron from the volume

I am looking for a formula that can convert the volume of an octahedron to the length of an edge. So far, I have come across $\frac{1.442\cdot3\sqrt{v}}{1.122}$, but I am not sure if this equation is ...
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1answer
198 views

what abstract regular polyhedra exist?

It is well-known since Plato that there are only 5 regular polyhedra which live in 3D Euclidean space. However abstract polytopes are defined solely by their incidences, and are not confined by the ...
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Suggested name for “inflated” tetrahedron

What's the name or class of the following tetrahedron-like shape? Sketchup Model WebGL 3D-viewer It's apparently some sort of (not strictly convex) shell of a tetrahedron and it's scaled spherical ...
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1answer
131 views

Proof of the existance of Platonic Solids other than in Euclids 13 Elements

Does anyone know where there is a complete proof of the existence of the Platonic solids, particularly the Dodecahedron and the Icosahedron (other than amongst Euclids 13 elements)? I do not mean a ...
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1answer
439 views

Relationship between circumscribed sphere radius and edge length of a dodecahedron? [duplicate]

Hello and I'm a secondary student doing a math exploration, but I'm currently stuck with this problem... Can anyone kind enough to show me the derivation of the relationship between the circumscribed ...
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483 views

Stacking the dual tetrahedron in an ordinary tetrahedron. Is it possible?

I'm thinking about the following problem. Introduction First let me introduce the problem with a 2D example. The area of the triangle constructed by connecting the midpoints of a triangle is 1/4 of ...