Kepler-Poinsot Solid

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.

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 $\{{\textstyle{5\over 2}}, 5\}$ and $\{5, {\textstyle{5\over 2}}\}$ fail to satisfy the Polyhedral Formula

$$V-E+F=2,$$

where $V$ is the number of faces, $E$ the number of edges, and $F$ 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 $n$-D with $n\geq 5$, there are none. In 4-D, nine of the solids have the same Vertices as $\{3, 3, 5\}$, and the tenth has the same as $\{5, 3, 3\}$. Their Schläfli Symbols are $\{{\textstyle{5\over 2}}, 5, 3\}$, $\{3, 5, {\textstyle{5\over 2}}\}$, $\{5, {\textstyle{5\over 2}}, 5\}$, $\{{\textstyle{5\over 2}}, 3, 5\}$, $\{5, 3, {\textstyle{5\over 2}}\}$, $\{{\textstyle{5\over 2}}, 5, {\textstyle{5\over 2}}\}$, $\{5, {\textstyle{5\over 2}}, 3\}$, $\{3, {\textstyle{5\over 2}}, 5\}$, $\{{\textstyle{5\over 2}}, 3, 3\}$, and $\{3, 3, {\textstyle{5\over 2}}\}$.

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

See also Archimedean Solid, Deltahedron, Johnson Solid, Platonic Solid, Polyhedron Compound, 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.