There are twelve regular decagons in this polyhedron, and sixty irregular pentagons. If the pentagons were closer to regularity, this would qualify as a near-miss to the ninety-two Johnson Solids. It is not known how many of these “near-misses” exist — primarily because this group of polyhedra lacks a precise definition.
This polyhedron was discovered with the aid of Stella 4d, software you can try for yourself at http://www.software3d.com/stella.php.
RobertLovesPi.net 
Hi, I would be very interested in how to make that Polyhedron – I guess it’s derived from a Dodecahedron or Icosahedron, but how exactly?
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After seeing your question, I tried to recreate this polyhedron using Stella 4d, and got very close to it. I started with the truncated dodecahedron, and augmented its twelve decagonal faces with antiprisms. Next, I formed the convex hull, and stellated it once. Finally, I tried to use Stella’s “try to make faces regular” function to finish the job, but this didn’t work. My other recourse would be to adjust the height of the antiprisms by trial and error, and that would take a while.
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