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. The small stellated dodecahedron appeared ca. The great stellated dodecahedron was published by Wenzel Jamnitzer in
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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.
The small stellated dodecahedron appeared ca. The great stellated dodecahedron was published by Wenzel Jamnitzer in Kepler rediscovered these two Kepler used the term "urchin" for the small stellated dodecahedron and described them in his work Harmonice Mundi in The two known solids, great dodecahedron , and great icosahedron were subsequently re discovered by Poinsot in 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. These names probably originated with Arthur Cayley, who first used them in Cauchy proved that these four exhaust all possibilities for regular star polyhedra Ball and Coxeter A table listing these solids, their duals , and compounds is given below.
Like the five Platonic solids, duals of the Kepler-Poinsot solids are themselves Kepler-Poinsot solids Wenninger , pp. The polyhedra and fail to satisfy the polyhedral formula. In four dimensions, there are 10 Kepler-Poinsot solids, and in dimensions with , there are none.
In four dimensions, nine of the solids have the same polyhedron vertices as , and the tenth has the same as. Ball, W. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. Cauchy, A. Cayley, A. Coxeter, H. London Ser. A , , Jamnitzer, W. Perspectiva Corporum Regularium.
Muraro, M. Venice, Pappas, T. Quaisser, E. Braunschweig, Germany: Vieweg, pp. Pure Appl. Webb, R. Wells, D. London: Penguin, pp. Wenninger, M. Dual Models. Cambridge, England: Cambridge University Press, pp.
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In geometry , a Kepler—Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron , and differ from these in having regular pentagrammic faces or vertex figures. They can all be seen as three-dimensional analogues of the pentagram in one way or another. These figures have pentagrams star pentagons as faces or vertex figures. The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex polygonal faces, but pentagrammic vertex figures.