Kepler-Poinsot Polyhedron


The Kepler-Poinsot polyhedra are four regular polyhedra which, unlike the Platonic solids, contain intersecting facial planes. In addition, two of the four Kepler-Poinsot polyhedra are constructed using regular polygrammic as opposed to regular polygonal faces. The following table summarizes these polyhedra the regular polygons (or polygrams) from which they are constructed.

A list of the Kepler-Poinsot polyhedra as implemented in the Wolfram Language can be given by PolyhedronData["KeplerPoinsot"].

While the external appearances of the Kepler-Poinsot polyhedra are visually indistinguishable from solids created by the augmentation of dodecahedra and icosahedra, they are actually icosahedron and dodecahedron stellations that contain portions of faces internally. As a result, the term "Kepler-Poinsot polyhedra" is preferable to the more common term "Kepler-Poinsot solids."

The names of the polyhedra probably originated with Arthur Cayley, who first used them in 1859. Cauchy (1813) proved that these four exhaust all possibilities for regular star polyhedra (Ball and Coxeter 1987). While these polyhedra were unknown to the ancients, the small stellated dodecahedron appeared ca. 1430 as a mosaic by Paolo Uccello on the floor of San Marco cathedral, Venice (Muraro 1955). The great stellated dodecahedron was published by Wenzel Jamnitzer in 1568. Kepler rediscovered these two (Kepler used the term "urchin" for the small stellated dodecahedron) and described them in his work Harmonice Mundi in 1619. The two known polyhedra, great dodecahedron, and great icosahedron were subsequently (re)discovered by Poinsot in 1809. As shown by Cauchy, they are stellated forms of the dodecahedron and icosahedron.

A table listing these solids, their duals, and compounds is given below. Like the five Platonic solids, duals of the Kepler-Poinsot polyhedra are themselves Kepler-Poinsot polyhedra (Wenninger 1983, pp. 39 and 43-45).

The polyhedra {5/2,5} and {5,5/2} fail to satisfy the polyhedral formula


where V is the number of vertices, 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 erroneously conclude that they could not exist.

In four dimensions, there are 10 Kepler-Poinsot polyhedra, and in n dimensions with n>=5, there are none. In four dimensions, nine of the polyhedra have the same polyhedron vertices as {3,3,5}, and the tenth has the same as {5,3,3}. Their Schläfli symbols are {5/2,5,3}, {3,5,5/2}, {5,5/2,5}, {5/2,3,5}, {5,3,5/2}, {5/2,5,5/2}, {5,5/2,3}, {3,5/2,5}, {5/2,3,3}, and {3,3,5/2}.

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

See also

Archimedean Solid, Deltahedron, Equilateral Polyhedron, Johnson Solid, Platonic Solid, Polyhedron Compound, Star Polyhedron, Uniform Polyhedron

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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.Cayley, A. "On Poinsot's Four New Regular Solids." Philos. Mag. 17, 123-127 and 209, 1859.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.Jamnitzer, W. Perspectiva Corporum Regularium. Nürnberg, Germany, 1568 Reprinted Frankfurt, 1972.Muraro, M. "L'esperianza Veneziana di Paolo Uccello." Atti del XVIII congresso internaz. di storia dell'arte. Venice, 1955.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, L. "On The Multiple Integral whose Limits Are p_1=a_1x+b_1y+...+h_1z>0, p_2>0, ..., p_n>0 and x^2+y^2+...+z^2<1." Quart. J. Pure Appl. Math. 3, 54-68 and 97-108, 1860.Webb, R. "Kepler-Poinsot Solids.", D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 130-131, 1991.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, pp. 39-41, 1983.

Cite this as:

Weisstein, Eric W. "Kepler-Poinsot Polyhedron." From MathWorld--A Wolfram Web Resource.

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