History
An early example of abstract polytopes was the discovery by Coxeter and Petrie of the three infinite structures {4, 6}, {6, 4} and {6, 6}, which they called regular skew polyhedra.
In the 1960s Branko Grünbaum issued a call to the geometric community to consider generalizations of the concept of regular polytopes that he called polystromata. He developed a theory of polystromata, showing examples of new objects including the 11-cell.
Grünbaum also discovered the 11-cell, a self-dual 4-polytope whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the same face (Grünbaum, 1977). A few years after Grünbaum's discovery of the 11-cell, H.S.M. Coxeter discovered a similar polytope, the 57-cell (Coxeter 1982, 1984), and then independently rediscovered the 11-cell.
Egon Schulte defined "regular incidence complexes" and "regular incidence polytopes" in his PhD dissertation in the 1980s - the first time the modern definition was introduced. Subsequently, he and Peter McMullen developed the basics of the theory in a series of research articles that were later collected into a book. Numerous other researchers have since made their own contributions, and the early pioneers (including Grünbaum) had also accepted Schulte's definition as the "correct" one.
Read more about this topic: Abstract Polytope
Famous quotes containing the word history:
“Certainly there is not the fight recorded in Concord history, at least, if in the history of America, that will bear a moment’s comparison with this, whether for the numbers engaged in it, or for the patriotism and heroism displayed.”
—Henry David Thoreau (1817–1862)
“When the history of this period is written, [William Jennings] Bryan will stand out as one of the most remarkable men of his generation and one of the biggest political men of our country.”
—William Howard Taft (1857–1930)
“History has neither the venerableness of antiquity, nor the freshness of the modern. It does as if it would go to the beginning of things, which natural history might with reason assume to do; but consider the Universal History, and then tell us,—when did burdock and plantain sprout first?”
—Henry David Thoreau (1817–1862)