Dodecahedra have icosahedral (also called icosidodecahedral) symmetry. In the figure above, this symmetry is changed to tetrahedral, by truncation of four vertices with positions corresponding to the vertices (or, instead, faces) of a tetrahedron. The interchangeability of vertices and faces for the tetrahedron is related to the fact that the tetrahedron is self-dual.
[Image created using Stella 4d, available here.]
RobertLovesPi.net 
(For some reason, I don’t see the image in the WP Reader…)
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But I do see it when I visit your site.
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This polyhedron is related to the self-dual “tetrahedrally diminished dodecahedron” where the truncation completely removes the original edges, and reduces the pentagons into trapezoids.
https://en.wikipedia.org/wiki/Tetrahedrally_diminished_dodecahedron
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Tom, do you know if anyone has compiled a list of known polyhedra which are self-dual?
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