The Kepler-Poinsot solids are the four regular Concave Polyhedra with intersecting facial planes. 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 discovered the other two, in 1809. As shown by Cauchy, they are stellated forms of the Dodecahedron and Icosahedron.

The Kepler-Poinsot solids, illustrated above, are known as the Great Dodecahedron, 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 given below.

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 the formula 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 have the same Vertices as , and the tenth has the same as . Their Schläfli Symbols are , , , , , , , , , and .

Coxeter *et al. *(1954) have investigated star ``Archimedean'' polyhedra.

**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.

© 1996-9

1999-05-26