Uniform Polyhedra - History

History

• The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid.
• Johannes Kepler (1571–1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.

Regular star polyhedra:

• Kepler (1619) discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot (1809) discovered the other two.

Other 53 nonregular star polyhedra:

• Of the remaining 53, Albert Badoureau (1881) discovered 36. Edmund Hess (1878) discovered 2 more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered.
• The geometer H.S.M. Coxeter discovered the remaining twelve in collaboration with J. C. P. Miller (1930–1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.
• Coxeter, Longuet-Higgins & Miller (1954) published the list of uniform polyhedra.
• Sopov (1970) proved their conjecture that the list was complete.
• In 1974, Magnus Wenninger published his book Polyhedron models, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.
• Skilling (1975) independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility.
• In 1993, Zvi Har'El produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper Uniform Solution for Uniform Polyhedra, counting figures 1-80.
• Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.
• In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.