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:
“I believe that history might be, and ought to be, taught in a new fashion so as to make the meaning of it as a process of evolution intelligible to the young.”
—Thomas Henry Huxley (1825–95)
“Every generation rewrites the past. In easy times history is more or less of an ornamental art, but in times of danger we are driven to the written record by a pressing need to find answers to the riddles of today.... In times of change and danger when there is a quicksand of fear under men’s reasoning, a sense of continuity with generations gone before can stretch like a lifeline across the scary present and get us past that idiot delusion of the exceptional Now that blocks good thinking.”
—John Dos Passos (1896–1970)
“The second day of July 1776, will be the most memorable epoch in the history of America. I am apt to believe that it will be celebrated by succeeding generations as the great anniversary festival. It ought to be commemorated, as the day of deliverance, by solemn acts of devotion to God Almighty. It ought to be solemnized with pomp and parade, with shows, games, sports, guns, bells, bonfires and illuminations, from one end of this continent to the other, from this time forward forever more”
—John Adams (1735–1826)