Then, repeat with all the edges until the edges all come together to create pentagonal pyramids above each face. To form the small stellated dodecahedron through edge stellation, one simply has to extend the edges of each pentagonal face until they intersect to form a pentagram. The information below is, for the most part, taken from Kavitha d/o Krishnan's paper on Polyhedra: with the right-hand side images from Tom Gettys' page on Kepler-Poinsot Solids These bounded cells can also be stuck together to form new polyhedra with faces that lie in the same plane as the original polyhedron.Ĭubes and Tetrahedrons, however, do not produce any new stellated forms when attempting face-stellation because the extended faces will never intersect. These bounded cells surround the polyhedron that acts as the core of the new stellated polyhedron. The second method, the three-dimensional apprach, involves considering the stellations as being built upon layers of solid or bounded cells. Through the information gathered at this one plane, we can deduce the possible faces for the stellated forms of these polyhedra. The first, the two-dimensional approach, involves selecting a face-plane and observing how the other planes intersect it. There are two ways to approach face stellation. This is the only type of stellation polygons experience, but it does not apply to cubes, tetrahedrons and octahedrons, all of which do not intersect again after their original vertices. This is the simple approach of extending the edges of the polyhedron until they intersect at a later point. Īdditionally, all four Kepler-Poinsot Solids have 30 edges. Upon closer observation of the Schläfli symbols on each of the solids, it should be noted that the Great Dodecahedron and the Small Stellated Dodecahedron do not fulfill Euler's formula of Vertices - Edges + Faces = 2, which led to Schläfli originally assuming that the polyhedrons with these numbers could not exist. Below is a chart that provides the Schläfli symbols for each of the Kepler-Poinsot Solids:

The Schläfli symbol, given in the form of, denotes the number of edges on each face of the polygon (P) as well as the number of faces that meet at each vertex of polyhedron (Q). This is done by extending the edges of each of faces of the polyhedron, or the faces themselves, until they meet again. The four Kepler-Poinsot solids are traditionally created via the stellation of a either a dodecahedron or icosahedron. Interestingly enough, these civilizations also assigned certain elements to the shapes, just as the Greeks had done later on. Īdditionally, although the ancient Greeks are credited with the Platonic Solids, stone carvings of shapes bearing heavy resemblance to the solids were dated to the period from 3000 to 1000 B.C to Indian civilizations. Further research by Cauchy in 1813 proved that these four polyhedrons exhaust all possibilities for regular star polyhedra.

They gained their current names in 1859 from Arthur Cayley. Furthermore, Poinsot rediscovered Kepler's solids and went on to discover the great dodecahedron and great icosahedron in 1809.

These two polyhedra were then later rediscovered and described by Kepler in his 1619 work, Harmonice Mundi. The small stellated dodecahedron was first displayed by Paolo Uccello in 1430 and the great stellated dodecahedron was later published in 1568 by Wenzel Jamnitzer. There are four Kepler-Poinsot solids: the great dodecahedron, the great icosahedron, the great stellated dodecahedron and the small stellated dodecahedron.

Keeplar photox