Kepler-Poinsot Solid
The Kepler-Poinsot solids are the four regular Concave Polyhedra with intersecting facialplanes. They are composed of regular Concave Polygons and were unknown to the ancients.Kepler 
discovered two and described them about 1619. These two were subsequently rediscovered by Poinsot, who also discoveredthe other two, in 1809. As shown by Cauchy, they are stellated forms of the Dodecahedron andIcosahedron.
The Kepler-Poinsot solids, illustrated above, are known as the GreatDodecahedron, Great Icosahedron, Great Stellated Dodecahedron, and Small Stellated Dodecahedron. Cauchy (1813) proved that these four exhaust all possibilities for regular star polyhedra (Ball and Coxeter 1987).
A table listing these solids, their Duals, and Compounds is givenbelow.
| Polyhedron | Dual Polyhedron | Compound |
| Great Dodecahedron | Small Stellated Dodecahedron | Great Dodecahedron-Small Stellated Dodecahedron Compound |
| Great Icosahedron | Great Stellated Dodecahedron | Great Icosahedron-Great Stellated Dodecahedron Compound |
| Great Stellated Dodecahedron | Great Icosahedron | Great Icosahedron-Great Stellated Dodecahedron Compound |
| Small Stellated Dodecahedron | Great Dodecahedron | Great Dodecahedron-Small Stellated Dodecahedron Compound |
The polyhedra 
and 
fail to satisfy the Polyhedral Formula
where
is the number of faces, 
the number of edges, and 
the number of faces, despite the fact that theformula holds for all ordinary polyhedra (Ball and Coxeter 1987). This unexpected result led none less than Schläfli(1860) to conclude that they could not exist.In 4-D, there are 10 Kepler-Poinsot solids, and in 
-D with 
, there are none. In 4-D, nine of the solids havethe same Vertices as 
, and the tenth has the same as 
. TheirSchläfli Symbols are 
, 
, 
,
, 
, 
, 
, 
,
, and 
.
Coxeter et al. (1954) have investigated star ``Archimedean'' polyhedra.
See also Archimedean Solid, Deltahedron, Johnson Solid, Platonic Solid, PolyhedronCompound, Uniform Polyhedron
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 144-146, 1987. Cauchy, A. L. ``Recherches sur les polyèdres.'' J. de l'École Polytechnique 9, 68-86, 1813. Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.'' Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954. Pappas, T. ``The Kepler-Poinsot Solids.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989. Quaisser, E. ``Regular Star-Polyhedra.'' Ch. 5 in Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 56-62, 1986. Schläfli. Quart. J. Math. 3, 66-67, 1860.
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