# Platonic Solids

29 28 the KePler Polyhedra the stellated great stellated dodecahedron The sides of some polygons can be extended until they meet again, for example the regular pentagon extends to form a five pointed star, or pentagram below. This process is known as stellation. Johannes Kepler 15711630 proposed its application to polyhedra, observing the two possibilities of stellation by extending edges, and stellation by extending face planes. Applying the first of these below to the dodecahedron and icosahedron he discovered the two polyhedra illustrated opposite and named them the larger and smaller icosahedral hedgehogs Their modern names, the stellated dodecahedron opposite top and the great stellated dodecahedron lower opposite, reveal that these polyhedra are also two of the face stellations of the dodecahedron. Each is made of twelve pentagram faces, one with five, the other with three to every vertex. They have icosahedral symmetry. Although its five sides intersect each other, the pentagram has equal edges and equal angles at its vertices and so can be considered a nonconvex regular polygon. Likewise, these polyhedra can be regarded as nonconvex regular polyhedra.