Gruber P. Convex and Discrete Geometry

Convex Polytopes
The early history of convex polytopes is lost. About 2000 BC convex polytopes
appeared in a mathematical context in the Sumerian civilization, in Babylonia and in
Egypt. Sources are the Moscow papyrus and the Rhind papyrus. Some of the regular
polytopes were already known by then. A basic problem was to calculate the volumes
of truncated pyramids. This was needed to determine the number of bricks for
fortifications and buildings. Babylonians sometimes did the calculations correctly,
sometimes not, while Egyptians used the right formula. For pertinent information the
author is obliged to the assyriologist Hermann Hunger [531]. In the fifth century BC
Democritos also discovered this formula and Eudoxos proved it, using the method
of exhaustion. Theaitetos developed a theory of regular polytopes, later treated by
Plato in the dialogue Timaios. Euclid, around 300 BC, considered metric properties
of polytopes, the volume problem, including the exhaustion method, and the
five regular polytopes, the Platonic solids. Zenodoros, who lived sometime between
200 BC and 90 AD, studied the isoperimetric problem for polygons and polytopes
and Pappos, about 300 AD, dealt with the semi-regular polytopes of Archimedes.
In the renaissance the study of convex polytopes was in the hands of artists such
as Uccello, Pacioli, da Vinci, Dürer, and Jamnitzer. Then it went back to mathematics.
Kepler investigated the regular and the semi-regular polytopes and planar
tilings. Descartes considered convex polytopes from a metric point of view, almost
arriving at Euler’s polytope formula, discovered by Euler only hundred years later.
Contributions to polytope theory in the late eighteenth and the nineteenth century
are due to Legendre, Cauchy, Steiner, Schläfli and others. At the turn of the nineteenth
and in the twentieth century important results were given by Minkowski,
Dehn, Sommerville, Steinitz, Coxeter and numerous contemporaries. At present,
emphasis is on the combinatorial, algorithmic, and algebraic aspects. Modern relations
to other areas date back to Newton (polynomials), Fourier (linear optimization),
Dirichlet, Minkowski and Voronoĭ (quadratic forms) and Fedorov (crystallography).
In recent decades polytope theory was strongly stimulated and, in part, re-oriented
by linear optimization, computer science and algebraic geometry. Polytope theory,
in turn, had a certain impact on these areas. For the history, see Federico [318] and
Malkevitch [681].