### The Kepler-Poinsot solids

The four Kepler-Poinsot solids have been known for many centuries. Similar to the five Platonic solids they are each constructed from multiple copies of completely symmetric vertex stars. But unlike the Platonic solids, the faces of the polyhedra intersect with themselves; thus the Kepler-Poinsot solids are no longer convex. Just as with any other isogonal polyhedron the vertices can be labeled so as to remove various degrees of their symmetries and these labeled vertex stars can also be used to construct isogonal polyhedra. Below are listed the different labeled versions of the four Kepler-Poinsot solids along with VRML models of them and their vertex stars. But because these models are constructed by assembling multiple copies of the vertex stars and because the stars are also constructed by assembling copies of the polygons, depending on your display software their non-convexity may cause them not to display well on your computer. (For example, the first two below may end up looking just like the second two.) However many other locations on the Web have pictures of them.

The Schläfli symbol for these vertex stars needs some extra explanation. A 5-gon, or pentagon, contains five vertices with each one connected to its nearest neighbor and creating a cyclic ordering among the vertices. This could be represented as a 5/1-gon, since each vertex is connected to the next one in the cycle. But if the connections are instead made to the second vertex in the cycle then a star shape, or pentagram, is created and this is referred to as a 5/2-gon. (A 5/3-gon would be the same as a 5/2-gon, but in reverse. Similarly, a 5/4-gon would be the same as a 5/1-gon or 5-gon in reverse.) Examples of this are the small stellated dodecahedron (symbol {5/2,5}) and the great stellated dodecahedron (symbol {5/2,3}).

On the other hand, the symbol {5,3}, meaning 3 copies of 5-gons, represents pentagons attached in such a way that they return to the first one in the sequence using angles appropriate to a third of their total angle. Extending this, the symbol {5, 5/2} represents five pentagons attached so that they use the angles more appropriate to that of a pentagram; that is, the polygons cross over each other as they attach to each other but still return to the original polygon in the sequence. The great dodecahedron (symbol {5,5/2}) and great icosahedron (symbol {3,5/2}) are examples of this type of non-convexity. As you look at the VRML pictures of these you should be able to see these patterns.

The small stellated dodecahedron and the great dodecahedron have exactly the same labeled variations. The great stellated dodecahedron and the Platonic dodecahedron also share the same labelings: both of them require some degree of symmetry. This is not all that surprising since they both have the same set of 20 vertices. (The other three Kepler-Poinsot solids all have 12 vertices, the same as the icosahedron.)

#### The 4 Kepler-Poinsot solids and their isogonal labelings

• Small stellated dodecahedron   (5/2.5/2.5/2.5/2.5/2)
1. [a a a a a;  a]
2. [a+ a+ a+ a+ a+;  a+]
3. [a b+ c+ c- b-;  a c- b-]
4. [a+ b+ c+ d+ e+;  a+ d+ e+ b+ c+]
5. [a+ b+ c+ d+ e+;  c+ d+ a+ b+ e+]
6. [a+ b+ c+ d+ e+;  c+ e+ a+ d+ b+]
7. [a+ b+ c+ d+ e+;  d+ b+ e+ a+ c+]
8. [a+ b+ c+ d+ e+;  d+ e+ c+ a+ b+]

• Great stellated dodecahedron   (5/2.5/2.5/2)
1. [a a a;  a]
2. [a+ a+ a+;  a+]

• Great dodecahedron   (5.5.5.5.5)
1. [a a a a a;  a]
2. [a+ a+ a+ a+ a+;  a+]
3. [a b+ c+ c- b-;  a c- b-]
4. [a+ b+ c+ d+ e+;  a+ d+ e+ b+ c+]
5. [a+ b+ c+ d+ e+;  c+ d+ a+ b+ e+]
6. [a+ b+ c+ d+ e+;  c+ e+ a+ d+ b+]
7. [a+ b+ c+ d+ e+;  d+ b+ e+ a+ c+]
8. [a+ b+ c+ d+ e+;  d+ e+ c+ a+ b+]

• Great icosahedron   (3.3.3.3.3)
1. [a a a a a;  a]
2. [a+ a+ a+ a+ a+;  a+]
3. [a b+ c+ c- b-;  a c+ b+]
4. [a+ b+ c+ d+ e+;  a+ c+ b+ e+ d+]
5. [a+ b+ c+ d+ e+;  b+ a+ c+ e+ d+]
6. [a+ b+ c+ d+ e+;  b+ a+ d+ c+ e+]
7. [a+ b+ c+ d+ e+;  e+ b+ d+ c+ a+]
8. [a+ b+ c+ d+ e+;  e+ c+ b+ d+ a+]

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Last updated: April 22, 2019