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.
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.
