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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.

There are only five Platonic solids: the regular tetrahedron, the cube, the regular octahedron, the regular dodecahedron, and the regular icosahedron.

185 questions
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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 add up to less than $2\pi$. This narrows down the…
Oscar Cunningham
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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 example could be a two tetrahedra pasted together…
Rohit Pandey
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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 sphere? Is this special, or are there an arbitrary…
Phrogz
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Cleverest construction of a dodecahedron / icosahedron?

One can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedron, cube, and octahedron all admit very intuitive constructions. The cube is a cube, the octahedron is its…
Parker Glynn-Adey
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5 answers

How many faces of a solid can one "see"?

What is the maximum number of faces of totally convex solid that one can "see" from a point? ...and, more importantly, how can I ask this question better? (I'm a college student with little experience in asking well formed questions, much less…
jhch
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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 polygons themselves allow them to faces of regular…
Justin L.
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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 follows that $P$ is one of the Platonic solids, which…
RavenclawPrefect
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Fitting a dodecahedron inside a cube

I'm afraid I'm not fantastic at maths and am struggling with a problem. I am a woodworker and have been asked to cut a solid dodecahedron from a 3 inch cube of wood. I am struggling to figure out what the maximum sized dodecahedron I could make…
Sam Oldfield
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Platonic solids but dropping convexity: Can we still show that two Platonic solids of the same $V, E, F$ are similar in $\mathbb{R}^{3}$?

Note. This question was borne out of a slight ambiguity in the definition of Platonic solids (whether it assumes convexity). When I first made this question I did not think convexity was part of the definition of Platonic solids. Since then, I am…
MaximusIdeal
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Inscribing Platonic solids in each other: why can't you put a dodecahedron in an octahedron?

Given convex polyhedra $P, Q$, say that one can inscribe $P$ in $Q$ if we can find points on the surface of $Q$ whose convex hull is similar to $P$. If we restrict $P, Q$ to be Platonic solids, we can achieve every case except inscribing the…
RavenclawPrefect
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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 formal way. My question is, where would the difficulty…
Oscar Cunningham
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4 answers

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

Standard means that the sum of the opposite sides is $7$ for $6$-sided die and $9$ for $8$-sided die.
Miokloń
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Five Cubes in Dodecahedron

I will demonstrate why the group of rotational symmetries of a Dodecahedron is $A_5$. For that, we have to find five nice objects, on which the group of symmetries acts. One object is "Cubes" inscribed in Dodecahedron. But since my demonstration…
Groups
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The vertices of a tetrahedron lie on a sphere

I am struggling a bit with the following (elementary) question: How to prove that every regular tetrahedron admits a circumsphere, i.e. there exist a sphere on which all four vertices lie. I would like to find a slick elegant proof, which is…
Asaf Shachar
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4 answers

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 principally group theory. Here´s the definition of…
Pilot
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