Shown below are the snub dodecahedron and its dual, the pentagonal hexecontahedron.
Seeking a way to make a “new” polyhedron (one never seen before), I augmented each face of the orange dual, above, with prisms. These prisms have a height equal to twice the average edge length of their bases.
Next, I used the software I use to manipulate polyhedra (Stella 4d, available here) to create the convex hull of this augmented pentagonal hexecontahedron.
Finally, I used Stella’s “try to make faces regular” function, and obtained this result, which I liked enough to stop here. There’s no way for me to know with certainty that this polyhedron has never been seen before, of course, but that didn’t stop me from having fun making it.
RobertLovesPi.net 
What would you be willing to bet that the rectangles in the convex shell are not a regular expression of tau?
In other words, would you rule out [1+ squrt(5)]/2 as being involved. Until you do, the shell can be considered
“regular” as is. I’m suggesting that the Golden Ratio would legitimize a claim of regularity; the final funtional
operation’s result sure as hell doesn’t. Those of us with a knowledge of basic polyhedral geometry have no
need for Stella 4d!
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I don’t “see” tau in the next-to-last polyhedron, but a careful choice of prism-height in the previous step could certainly make tau appear.
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