P79672 Cubes in a dodecahedron link reply
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As part of studying group theory I've been playing around with various concrete groups, among them the rotational symmetry groups of the regular polyhedra. And I noticed something cool, although I'm far from the first to notice it.

In a regular dodecahedron, you can connect certain vertices together to form an inscribed cube. There are five different ways of doing this. If you draw out the edges of the five cubes, each cube in a different color, then on each face of the dodecahedron you end up drawing a pentagram with each of the five line segments in a different color. And on each face of the dodecahedron the colors in the pentagram are arranged differently; in other words, each face displays a different circular permutation of the five colors. There are twenty-four possible circular permutations of five objects, and the twelve permutations seen on the dodecahedron are specifically the even permutations, permutations you can reach with an even number of swaps.

This is an easy way to see that the rotational symmetries of the dodecahedron are isomorphic to the even permutations of five objects. Each rotation in the group of symmetries permutes the five colored cubes, and given an even permutation of the five colors, you can figure out the rotation it corresponds to by picking a face, identifying which face will need to rotate to that face's location to get the desired permutation of colors up to rotation, then seeing how that face will need to be rotated about its center to make the permutation match exactly.

Files related are a picture of the cubes and a foldable version from
https://www.chiark.greenend.org.uk/~sgtatham/polypics/dodec-cubes.html
Referenced by: P79683 P82029
P79673 link reply
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Some more nice pictures and links.

Here's a picture showing making a dodecahedron by adding "hats" or "roofs" to the faces of one of the inscribed cubes:
https://www.cutoutfoldup.com/813-dodecahedron-as-a-cube-with-hats.php

This construction is apparently in Euclid's Elements (Book XIII, Proposition 17), so it's a very old observation indeed.

This page has an animation showing how the hats can be folded one way to make a dodecahedron and another way to make a cube:
https://cage.ugent.be/~hs/polyhedra/dodeca.html

(In either configuration, there's an empty space in the middle, so you should not incorrectly take this as implying the two solids have the same volume.)
P79674 link reply
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Related: forming a tetrahedron by connecting vertices of a cube.
P79683 link reply
P79672
What program do u use for this?

Referenced by: P80185
P80185 link reply
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P79683
I didn't make any of these pictures; I just found them on the web. The OP image were apparently generated by the code discussed here:
https://www.chiark.greenend.org.uk/~sgtatham/polyhedra/

When I was confirming it for myself I just drew everything out on a Schlegel diagram (a diagram like the ones in pic related). When I get the time it might be fun to print out a foldable version like the PDF in the OP so I can see it in three dimensions.
Referenced by: P80186
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