This reminds me of Kepler’s idea that nested Platonic solids describe the relative orbit diameters of the (known) planets:

https://en.m.wikipedia.org/wiki/Mysterium_Cosmographicum

John Conway had a lecture where he discusses this (among other things):

https://www.youtube.com/watch?v=W63uFjlgNds

Wow, I had never watched one of his lectures before, that was a treat. Thank you!

>He found that each of the five Platonic solids could be uniquely inscribed and circumscribed by spherical orbs; nesting these solids, each encased in a sphere, within one another would produce six layers, corresponding to the six known planets—Mercury, Venus, Earth, Mars, Jupiter, and Saturn. By ordering the solids correctly—octahedron, icosahedron, dodecahedron, tetrahedron, and cube—Kepler found that the spheres correspond to the relative sizes of each planet's path around the Sun, generally varying from astronomical observations by less than 10%.

See also the Titius–Bode law [1]

[1] https://en.wikipedia.org/wiki/Titius%E2%80%93Bode_law

When you fold a dodecahedron into a cube as in the animation, are there gaps inside, or is it perfectly solid? My geometric intuition is failing me here.

There is a link close to the animated gif to another post (https://mikesmathpage.wordpress.com/2016/01/16/can-you-belie...) where this is shown with more details.

When you fold the dodecahedron into a cube, inside the cube there is an empty space having the form of a stellated polyhedron, which has 8 pointed vertices on the 8 vertices of the cube. In the center of the stellated polyhedron there is an empty icosahedron.

There's a solid dodecahedron with a cube-shaped hole removed from it, and unfolding that dodecahedron to remove the cube from the cube-sized hole gives 6 "flaps" but none of them are flat, each is a piece of the solid part of the dodecahedron that did not intersect the cube-sized hole.

Are we talking about the same animation? It sounds like you're describing the embedded YouTube video at the top. I'm asking about the dodecahedron-to-cube folding below (dodecahedron-fold.gif), next to "Can you believe that a dodecahderon folds into a cube?"

I understand your question. You want to know a solid dodecahedron can be chopped into pieces and folded like that animation and leave behind a solid cube (no empty gaps inside)

Assume the cube has volume 1 and thus each side is length 1. In the animation we start with a dodecahedron, we remove a cube from the interior, we fold it over along the cuts and end up (hopefully) with a solid cube again. Thus the volume of the dodecahedron must be 2.

Two opposite corners of the pentagon faces make up the edges of the cube, which we know are length 1. A regular pentagon's ratio of a diagonal to a side is the golden ratio (!) so we know the pentagons have sides equal to 2/(1+sqrt5)

Volume of a dodecahedron is the constant (15 + 7sqrt(5))/4 multiplied by the cube of the sides.

Wolfram Alpha says https://www.wolframalpha.com/input/?i=%28%282%2F%281%2Bsqrt%... that the volume of the dodecahedron is only about 1.8. We needed it to be 2 so there will in fact be gaps inside the cube as folded in that animation.

Whoops! I was indeed talking about the animation in the video, my mistake.