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.
RobertLovesPi.net 
where does the “kai” in the name come from?
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It’s Greek for “and.”
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OK, disregard previous question is it not heptacontrakaidohedron?
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I’ve seen both systems used. As far as I know, there’s no universally agreed-upon naming system for polyhedra by number of faces. There are a lot of names out there, and sometimes they “bump into” each other.
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It’s not a polyhedron, but only because the faces are curved.
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I guess you missed this part from the question
“note: This question is about the shape without the spherical caps, the way that the underlying shape of a football is a truncated icosahedron. In this case though I am not sure if this shape will be a polyhedron or not as the five vertices of the pentagonal faces may not lay in a plane. “
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Sorry about that! I think there’s a polyhedron in there, right below the curved surfaces, but I’m not 100% sure.
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