Kepler–Poinsot polyhedron

The Kepler-Poinsot solids are the regular star polyhedra. Each has faces which are congruent regular convex polygons or star polygons and has the same number of faces meeting at each vertex (compare to Platonic solids).

There are four Kepler-Poinsot solids:

It is unfortunate that these figures have come to be known as 'solids' because they are better understood as surfaces.

Geometry

Name	Schläfli Symbol {p,q}	Faces {p}	Edges	Vertices {q} verf.	χ	Symmetry	Dual
Small stellated dodecahedron	{5/2,5}	12 {5/2}	30	12 {5}	-6	I_h	Great dodecahedron
Great dodecahedron	{5,5/2}	12 {5}	30	12 {5/2}	-6	I_h	Small stellated dodecahedron
Great stellated dodecahedron	{5/2,3}	12 {5/2}	30	20 {3}	2	I_h	Great icosahedron
Great icosahedron	{3,5/2}	20 {3}	30	12 {5/2}	2	I_h	Great stellated dodecahedron

These figures are deceptive for including pentagrams as faces and vertex figures. Where two faces intersect along a line that is not an edge of either face, these lines are false edges and are not counted. Likewise where three faces intersect at a point that is not a corner of any face, these points are false vertices and are not counted. The images above show golden balls at the actual vertices, and silver rods along the actual edges.

The small and great stellated dodecahedron have nonconvex regular pentagram faces. The great dodecahedron and great icosahedron have convex pentagon faces, but pentagrammic vertex figures. The first pair and second pair are duals of each other.

The small stellated dodecahedron and great icosahedron share the same vertices and edges. The icosahedron and great dodecahedron also share the same vertices and edges.

The Kepler-Poinsot polyhedra exist in dual pairs:

The three dodecahedra are all stellations of the regular convex dodecahedron, and the great icosahedron is a stellation of the regular convex icosahedron. The small stellated dodecahedron and the great icosahedron are facettings of the convex dodecahedron, while the two great dodecahedra are facettings of the regular convex icosahedron.

If the intersections are computed as new edges and vertices, the figures obtained will not be regular, but they can still be considered stellations. (See also List of Wenninger polyhedron models)

History

A small stellated dodecahedron appears in a marble tarsia (inlay panel) ont he floor of St. Mark's Basilica, Venice, Italy. It dates from the 1400s and is sometimes attributed to Paolo Uccello.

In his Perspectiva corporum regularium (Perspectives of the regular solids) [1], a book of woodcuts published in the 1500s, Wenzel Jamnitzer depicts the great dodecahedron. It is clear from the general arrangement of the book that he regards only the five Platonic solids as regular, and does not understand the regular nature of his great dodecahedron. He also depicts a figure often mistaken for the great stellated dodecahedron, though the triangular surfaces of the arms are not quite coplanar, so it actually has 60 triangular faces.

The Kepler solids were discovered by Johannes Kepler in 1619. He obtained them by stellating the regular convex dodecahedron, for the first time treating it as a surface rather than a solid. He noticed that by extending the edges or faces of the convex dodecahedron until they met again, he could obtain star pentagons. Further, he recognised that these star pentagons are also regular. In this way he found two stellated dodecahedra, the small and the great. Each has the cetntral convex region of each face "hidden" within the interior, with only the triangular arms visible. Kepler's final step was to recognize that these polyhedra fit the definition of regular solids, even though they were not convex, as the traditional Platonic solids were.

In 1809, Louis Poinsot rediscovered these two figures. He also considered star vertices as well as star faces, and so discovered two more regular stars, the great icosahedron and great dodecahedron. Some people call these two the Poinsot solids. Poinsot did not know if he had discovered all the regular star polyhedra.

Three years later, Augustin Cauchy was to prove the list complete, and almost half a century later Bertrand provided a more elegant proof by facetting the Platonic solids.

The Kepler-Poinsot solids were given their English names in the following year, 1859, by Arthur Cayley.

The Euler characteristic

A Kepler-Poinsot solid covers its circumscribed sphere more than once, with the centers of faces acting as winding points in the solids with pentagrammic faces and the vertices in the others. Because of this, they are not necessarily topologically equivalent to the sphere as Platonic solids are, and in particular the Euler relation

V − E + F = 2

doesn't always hold.

The value of the Euler characteristic χ depends on how we look upon the polyhedral form. Consider for example the small stellated dodecahedron [2]. It consists of a dodecahedron with a pentagonal pyramid on each of its 12 faces. Each of the 12 faces is a pentagram with the central pentagonal part hidden inside the solid. The outside part of each face consists of five triangles which only touch at five points. Alternatively we might we count these triangles as separate faces - there are 60 of them (but they are only isosceles triangles, not regular polygons). Similarly each edge line would now be divided into three edges (but then they are of two kinds). Also the "five points" just mentioned, together are 20 points that now form additional vertices, so that we have a total of 32 vertices (again two kinds). The hidden inner pentagons are no longer needed to form the polyhedral surface, and can disappear. Now the Euler relation holds: 60 - 90 + 32 = 2. However this polyhedron is no longer the one described by the Schläfli symbol {5/2,5}, and so can not be a Kelper-Poinsot solid even though it still looks like one from outside.

Trivia

A dissection of the great dodecahedron was used for the 1980s puzzle Alexander's Star.
Artist M.C. Escher's interest in geometric forms often led to works based on or including regular solids; Gravitation is based on a small stellated dodecahedron.

References

J. Bertrand, Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46 (1858), pp. 79-82, 117.
Augustin Louis Cauchy, Recherches sur les polyèdres. J. de l'École Polytechnique 9, 68-86, 1813.
Arthur Cayley, On Poinsot's Four New Regular Solids. Philos. Mag. 17, pp. 123-127 and 209, 1859.
P. Cromwell, Polyhedra, Cabridgre University Press, Hbk. 1997, Ppk. 1999.
Theoni Pappas, (The Kepler-Poinsot Solids) The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 113, 1989.
Louis Poinsot, Memoire sur les polygones et polyèdres. J. de l'École Polytechnique 9, pp. 16-48, 1810.
Magnus Wenninger Dual Models Cambridge, England: Cambridge University Press, pp. 39-41, 1983.

External links