Kepler–Poinsot Polyhedron - History

History

Most, if not all, of the Kepler-Poinsot polyhedra were known of in some form or other before Kepler. A small stellated dodecahedron appears in a marble tarsia (inlay panel) on the floor of St. Mark's Basilica, Venice, Italy. It dates from the 15th century and is sometimes attributed to Paolo Uccello. In his Perspectiva corporum regularium (Perspectives of the regular solids), a book of woodcuts published in the 16th century, Wenzel Jamnitzer depicts the great dodecahedron and the great stellated dodecahedron. It is clear from the general arrangement of the book that he regarded only the five Platonic solids as regular, and did not understand the regular nature of his great dodecahedra.

The small and great stellated dodecahedra, sometimes called the Kepler polyhedra, were first recognized as regular 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 recognized that these star pentagons are also regular. In this way he constructed the two stellated dodecahedra. Each has the central 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 regularity, even though they were not convex, as the traditional Platonic solids were.

In 1809, Louis Poinsot rediscovered Kepler's figures, by assembling star pentagons around each vertex. He also assembled convex polygons around star vertices to discover two more regular stars, the great icosahedron and great dodecahedron. Some people call these two the Poinsot polyhedra. Poinsot did not know if he had discovered all the regular star polyhedra.

Three years later, Augustin Cauchy proved the list complete by stellating the Platonic solids, and almost half a century after that, in 1858, Bertrand provided a more elegant proof by facetting them.

The following year, Arthur Cayley gave the Kepler–Poinsot polyhedra the names by which they are generally known today.

A hundred years later, John Conway developed a systematic terminology for stellations in up to four dimensions. Within this scheme, he suggested slightly modified names for two of the regular star polyhedra:

Conway's names have seen some use but have not been widely adopted.