Let’s say I want to make a dodecahedron. How would I determine the shape and size of the facets? Let’s also assume I want the finished shape to be 12" in diameter.
That’s a good question. For simplicity’s sake, I’ll look into a regular dodecahedron, which has 12 regular pentagons as its faces. More later.
Across its widest point or its narrowest? Or average?
Widest.
From MathWorld, the circumradius of a dodecahedron with edge length e is
1/4 * (sqrt(15 - sqrt(3)) * e.
You’re looking for the diameter of a circumscribed sphere to be 12 inches, so set 1/2 * (sqrt(15 - sqrt(3)) * e = 12, and solve for e.
If you have access to a university library, you might try to look up the book “Regular Polytopes” by H.S.M. Coxeter, which should contain (among other things) the ratios of the edge lengths to the inradii/outradii/midradii of all five regular Platonic solids.
Tyrrell, I think you mean
1/4 * (sqrt(15)-sqrt(3))*e.
Wait a cotton-pickin’ minute here.
That MathWorld link says that the circumradius is
1/4 * (sqrt(15) + sqrt(3))*e
(Not “-” as I originally typed…sorry…)
But it also says that you can construct a dodecahedron such that two adjacent vertices will have coordinates (1,1,1) and (1/phi, phi, 0) where phi=(sqrt(5)+1)/2. In which case the edge length is sqrt(5)-1 and the circumradius is sqrt(3), giving a ratio of (sqrt(15)-sqrt(3))/3, not (sqrt(15)+sqrt(3))/4!
I think MathWorld is smoking something here…when I worked out the ratio myself by a different technique, I got (sqrt(15)-sqrt(3))/3 for the circumradius/edge-length ratio as well. I think the correct answer to the OP should be 2*(sqrt(15)-sqrt(3)) inches.
Correction: I’m the one who’s smoking something. I divided when I should have multiplied.
(sqrt(15)-sqrt(3))/3 is the ratio of edge length to circumradius. The ratio of circumradius to edge length is one over that, or (sqrt(15)+sqrt(3))/4, which is exactly what MathWorld says.
I’ll shut up now…
More generally, how does one extend what we were taught in plane geometry to three dimensions, to work out what these ratios are?
Very carefully. 