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
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