Hyperbolization of a Dodecahedron

This Demonstration shows how to "hyperbolize" a dodecahedron (and other Platonic solids). Effectively, hyperbolization with parameter means to apply the map to all points of the surface of a dodecahedron. Positive values of generate sharp 2D cusps along the edges. Negative values of generate round bulbs at the centers of the faces of the original Platonic solid. A decorated hyperbolic dodecahedron has been used as the Mathematica logo since Mathematica Version 2.