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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 fol

• 6nf 7y ago
  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%285%29%29%29%5E3%29*%28%2815+%2B+7*sqrt%285%29%29%2F4%29 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.
• jjnoakes 7y ago
  Whoops! I was indeed talking about the animation in the video, my mistake.