
Nesting Platonic Solids - colinprince
https://mikesmathpage.wordpress.com/2016/05/28/nesting-platonic-solids/
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glial
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](https://en.m.wikipedia.org/wiki/Mysterium_Cosmographicum)

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

[https://www.youtube.com/watch?v=W63uFjlgNds](https://www.youtube.com/watch?v=W63uFjlgNds)

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

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ken
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.

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jjnoakes
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.

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ken
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?"

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6nf
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%...](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)
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.

