Editing Abstract polytope (section)

== History ==
In the 1960s [[Branko Grünbaum]] issued a call to the geometric community to consider generalizations of the concept of [[regular polytope]]s that he called ''polystromata''. He developed a theory of polystromata, showing examples of new objects including the [[11-cell]].

The [[11-cell]] is a [[Dual polytope|self-dual]] [[4-polytope]] whose [[Facet (geometry)|facets]] are not [[Icosahedron|icosahedra]], but are "[[hemi-icosahedron|hemi-icosahedra]]" &mdash; 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.

With the earlier work by [[Branko Grünbaum]], [[H. S. M. Coxeter]] and [[Jacques Tits]] having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by [[Egon Schulte]] in his 1980 PhD dissertation. In it he defined "regular incidence complexes" and "regular incidence polytopes". 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) have also accepted Schulte's definition as the "correct" one.

Since then, research in the theory of abstract polytopes has focused mostly on ''regular'' polytopes, that is, those whose [[automorphism]] [[group (mathematics)|groups]] [[Group action (mathematics)|act]] [[Group action (mathematics)#Types of actions|transitively]] on the set of flags of the polytope.