The above polyhedron was first described in Professor Bonnie Stewart's "Adventures Among the Toroids" (2nd Edition page 157), where it receives the name m* , although Professor Stewart was not specifically searching for Acrohedra. It was rediscovered by Professor John Conway as part of his specific search for a solution for the 5-4-3 acron.  In having only 12 faces it edged out the previous contender, the Stewart G3, by just one face.  See here for more on the Stewart G3.

The minimal non-self-intersecting case can be improved upon if we permit self-intersection.  Dr Richard Klitzing has studied the facets of the small ditrigonal icosidodecahedron amongst which was the above 9-faced figure.  Dr Klitzing informs me that he has to date only studied facets with a three-fold or five-fold symmetry, so this result may still be improved upon.  

The full results of Dr Klitzing's work on faceting a number of the uniform polyhedra are hosted by Ulrich Mikloweit here.

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