How would you make a (physical) dodecahedron with edges instead of faces?

Call this a "math problem disguised as a woodworking problem" or vice versa.

Background: You can construct a dodecahedron by cutting 12 identical, regular pentagon faces, beveling all the edges of these to half the dihedral angle and gluing them all together. The joints become the edges. Where 3 faces meet, you have the vertex. Simple enough.

Now let's say you want to build a dodecahedron, but instead of boards for the faces, you want the boards to be the edges. Three edges come together at each vertex, etc.

Relative to some dimension of the board itself, how would you determine/describe the two planes you'd cut/mill at the end of each board in order to allow the three boards to meet and be correctly oriented in three dimensions? Let's say you want one surface of the board to split the dihedral. (i.e. boards are halfway between being coplanar with the 2 faces on either side.)

I've been over at Wolfram banging my head on this for way too long. It's time to ask smarter folks.

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2022-07-25 20:44:28
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