Hyperbolization of a Dodecahedron

Platonic solid

tetrahedron

octahedron

cube

dodecahedron

icosahedron

hyperbolicity

This Demonstration shows how to "hyperbolize" a dodecahedron (and other Platonic solids). Effectively, hyperbolization with parameter

α

means to apply the map

rr

α

(r.r)

to all points

r

of the surface of a dodecahedron. Positive values of

α

generate sharp 2D cusps along the edges. Negative values of

α

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.