80 Stellated dodecahedron I 81 Stellated icosahedron I 82 Stellated dodecahedron II 83 Stellated icosahedron II 84 Stellated icosahedron III 85 Stellated Octahedron

To understand the stellated polyhedra, you need to look under the surface. The faces of these polyhedra are not the external facets, but rather larger polygons that extend through the middle of the figure, intersecting each other. This is easiest to see with the stellated octahedron (85). The large triangular faces cut through each other, forming two intersecting tetrahedra. One can obtain this figure by starting with an octahedron (which would be visible from the inside of this model), and allowing the faces to expand symmetrically until they intersect each other in the edges of a new polyhedron. This process is called stellation. Performing this process of stellation on a dodecahedron yields the small stellated dodecahedron (80), whose faces are pentagrams (five-pointed stars). Allowing these faces to expand even further (stellating the stellation) yields the great dodecahedron (not shown), and stellating this figure in turn yields the great stellated dodecahedron (82). Continuing the process yields nothing new, thus the dodecahedron has three stellations. The icosahedron has 59 stellations, of which three are shown here: 81, 83, and 84. Four of the stellated polyhedra are regular. These are the Kepler-Poinsot polyhedra: the small stellated dodecahedron; its dual, the great dodecahedron; the great stellated dodecahedron; and its dual, the great icosahedron.