# 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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### How Much To "Twist" A Polygon To Match A Face On A Regular Polyhedron

Suppose I want to create an icosahedron by building a set of twenty triangular pyramids (aka tetrahedrons, but see below) of an appropriate size and then rotating each into position. I need to do ...
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### Enumeration of uniform polyhedra

It is known that there are two infinite classes of polyhedra (prisms and antiprisms) together with 75 uniform polyhedra that do not belong to these classes. For regular convex polyhedra (i.e., ...
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### What is analogous to the set of four main diagonals in Rotational Symmetry of Cube, if we want to find the total symmetry of cube?

I am working on a project on applications of group theory, starting from point groups and molecular symmetry. Coming up on representations of each point groups, I started studying Representations of ...
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### Artist needing to determine geometric angle for sculpture based on platonic solid

Dear Mathematicians I need your help for a new sculpture! I will attach images but first imagine 2 hexagons - where one is rotated 30 deg. They are separated by 12 equilateral triangles. I need to ...
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### Why do we get equiangular lines from Platonic solids that have triangles as faces.

Consider placing $n$ lines in $d$ dimensional space in a way that the angles between any two pairs of lines is always the same (and they all pass through the origin). When $d=3$, we get configurations ...
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### Number of distinct configurations of equiangular lines passing through origin

We have a $d$ dimensional space and want to place $n$ lines passing through the origin in such a way that the angles between any two pairs of lines is the same. Such a placement results in a ...
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### Interpreting the isometries of a tetrahedron

A tetrahedron has $12$ rotational symmetries and $24$ isometries in total. This means that the group of isometries is isomorphic to $S_4.$ If we denote the vertices of a tetrahedron as $1$, $2$, $3$, ...
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### Why doesn't an inscribed cube perfectly sample the surface of a sphere?

I was curious about ways to sample perfectly dispersed points on the surface of the sphere. This question had some interesting info: Is the Fibonacci lattice the very best way to evenly distribute N ...
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### Dodecahedron faces are buttons, vertices have counters that track the use of the buttons

As a follow up to this question, I'm trying to teach invariants by creating a game. The idea is to start with a dodecahedron where each of the 20 vertices has a counter on it and each of the 12 faces ...
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### Isometry of cube in $\mathbb R^4$?

Find the Symmetry Group of : Tetrahedron and Cube. I know that there is a duplicate question Symmetry group of Tetrahedron but my professor suggested us to write coordinates of both cube and ...
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### Algebraic argument for why any $A_5$ in $S_6$ can be extended to an $S_5$ in $S_6$

It is well known that $S_5$ is a subgroup of $S_6$ in a way that acts transitively on $6$ points, a surprising fact related to the outer automorphism of $S_6$. One can see this using the icosahedron: ...
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### Does Euclid's demonstration that there are only five Platonic solids need to assume convexity?

At the end of the Elements Book XIII Euclid gives a demonstration that there are only five regular solids, i.e. 'no other figure, besides the said five figures, can be constructed by equilateral and ...
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### Symmetric distribution of points on a sphere

Background: The VSEPR theory provides a useful guide to predicting molecular geometries. One implication of the theory is that the orientation of ligands in three-dimensional space is such that the ...
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The following appear as exercises in Dummit and Foote's Algebra (Section $1.2$ - Dihedral Groups): Let $G$ be the group of rigid motions in $\mathbb{R}^3$ of a tetrahedron. Show that $|G| = 12$ Let $... • 14.2k 13 votes 3 answers 515 views ### Why does a convex polyhedron being vertex-, edge-, and face-transitive imply that it is a Platonic solid? Suppose that we have a convex polyhedron$P$, such that the symmetry group of$P$acts transitively on its vertices, edges, and faces (that is, it is isogonal, isotoxal, and isohedral). It then ... • 17.7k 0 votes 1 answer 121 views ### Is it possible to tile dodecahedrons in a 3D grid without any spaces between them? [closed] Is it possible to tile dodecahedrons in a 3D grid without any spaces between them? Feel free to send me web links about that. 0 votes 1 answer 324 views ### Uniform distribution of points on a sphere: only Platonic solids? I'm quite sure the only way to uniformly distribute$n$points on the sphere$S^2$is by inscribing one of the 5 Platonic solids, thus there only exists a solution for$n=4,6,8,12,20$. But am I right? ... 3 votes 1 answer 200 views ### Can an irregular tetrahedron be inscribed in a cuboid or prism? A regular tetrahedron can be inscribed in a cube in a way that the tetrahedron edges are diagonals of the cube faces. Is it possible to similarly inscribe any irregular tetrahedron in a cuboid or ... 1 vote 0 answers 176 views ### Koch-like fractal of the dodecahedron: Does this object already exist? Platonic solids are made of squares, triangles and one of them is made of pentagons. If the faces of the cube are taken while preserving its vertices, each face can be divided into smaller squares ... • 11 9 votes 4 answers 689 views ### How to show that if two Platonic solids have the same number of edges, vertices, and faces, then they are similar in$\mathbb{R}^{3}\$?

Note: It appears that some of the terms here do not have standardized definitions, so some sources may give conflicting info. I was looking into the proof that there are only five Platonic solids in ...
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