Polytope - History

History

The concept of a polytope originally began with polygons and polyhedra, both of which have been known since ancient times:

It was not until the 19th century that higher dimensions were discovered and geometers learned to construct analogues of polygons and polyhedra in them. The first hint of higher dimensions seems to have come in 1827, with Möbius' discovery that two mirror-image solids can be superimposed by rotating one of them through a fourth dimension. By the 1850s, a handful of other mathematicians such as Cayley and Grassman had considered higher dimensions. Ludwig Schläfli was the first of these to consider analogues of polygons and polyhedra in such higher spaces. In 1852 he described the six convex regular 4-polytopes, but his work was not published until 1901, six years after his death. By 1854, Bernhard Riemann's Habilitationsschrift had firmly established the geometry of higher dimensions, and thus the concept of n-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime.

In 1882 Hoppe, writing in German, coined the word polytop to refer to this more general concept of polygons and polyhedra. In due course, Alicia Boole Stott introduced polytope into the English language.

In 1895, Thorold Gosset not only rediscovered Schläfli's regular polytopes, but also investigated the ideas of semiregular polytopes and space-filling tessellations in higher dimensions. Polytopes were also studied in non-Euclidean spaces such as hyperbolic space.

During the early part of the 20th century, higher-dimensional spaces became fashionable, and together with the idea of higher polytopes, inspired artists such as Picasso to create the movement known as cubism.

An important milestone was reached in 1948 with H. S. M. Coxeter's book Regular Polytopes, summarizing work to date and adding findings of his own. Branko Grünbaum published his influential work on Convex Polytopes in 1967.

More recently, the concept of a polytope has been further generalized. In 1952 Shephard developed the idea of complex polytopes in complex space, where each real dimension has an imaginary one associated with it. Coxeter went on to publish his book, Regular Complex Polytopes, in 1974. Complex polytopes do not have closed surfaces in the usual way, and are better understood as configurations. This kind of conceptual issue led to the more general idea of incidence complexes and the study of abstract combinatorial properties relating vertices, edges, faces and so on. This in turn led to the theory of abstract polytopes as partially ordered sets, or posets, of such elements. McMullen and Schulte published their book Abstract Regular Polytopes in 2002.

Enumerating the uniform polytopes, convex and nonconvex, in four or more dimensions remains an outstanding problem.

In modern times, polytopes and related concepts have found many important applications in fields as diverse as computer graphics, optimization, search engines, cosmology and numerous other fields.