That’s a mouthful, for sure. It identifies a 3-D geometric shape, formed of 12 pentagonal pyramids, each made up of 5 identical triangles. Thus, it has 60 (12 * 5) identical pieces.
In this case, we’re talking about an assembly of sixty 3/4″ plywood pieces, whose dimensions and angles permit them to fit together to form form 12 identical pentagonal pyramids (“pentamids”), which in turn fit together to form the pentakis dodecahedron. Once assembled, this was put on my lathe, and turned to a sphere.
Here is an ordinary dodecahedron, which has 12 identical hexagonal faces:
And here is a pentakis dodecahedron, with 12 identical penta-pyramid faces:
I obtained guidance for the dimensions and angles for the pieces in this assembly from an online calculator for the pentakis dodecahedron, at https://rechneronline.de/pi/pentakis-dodecahedron.php
I built jigs for the purpose of cutting plywood triangles of the correct dimensions and angles, and cut out 60 pieces (plus some additional ‘just-in-case’ pieces).
I constructed a jig for assembling the pentamids consistently. The jig perimeter pieces were ‘hinged’ on one screw, so they could be opened to insert pieces into the jig, and then closed to tighten the pieces into the proper configuration. The completed pentamids at the top of the photo show the beveled edges which permit the 5 pieces to fit snugly together with each other as a pentamid, and later with other pentamids, as a complete pentakis dodecahedron assembly.
Once the 12 pentamids had been assembled, and the glue set up, I assembled the pentakis dodecahedron, gluing it together.
Once the entire assembly had been glued and cured, I put it on the lathe, between centers. The first placement was between two points at opposite (“polar”) ends of the assembly. Then, once the assembly had been rounded with the exception of the North and South poles, I rotated it 90 degrees, and again put it between centers, to complete the rounding, as shown in this last photo.
