After making the above polyhedron using Stella 4d (a program you can try for free at www.software3d.com/Stella.php), I checked its dual, which is shown below. I was surprised at its appearance, for it resembles a stellated polyhedron, even though it was created by a completely different process.
The 72 faces of this polyhedron are all pentagons. Twelve are regular, while the other sixty are not. Something I do not know, but would like to find out: is 72 the maximum number of faces a convex polyhedron can have, if all its faces must be convex pentagons?
[Update: 72 is not the maximum, as was explained to me on Facebook. I’ll try to find ways to generate images of more all-pentagon polyhedra, with more faces, for future posts here.]
Side note: this is the 1000th post on this blog. To all who follow and comment, thank you.
The program I use for these polyhedral investigations is Stella 4d, available at www.software3d.com/Stella.php.
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