I don't recall if we ever discussed the 11-cell, one of only two amazing 4-dimensional "abstract regular polytopes" (ARP) built from projective planes. See < http://en.wikipedia.org/wiki/11-cell > and < http://en.wikipedia.org/wiki/Abstract_polytope > and < http://en.wikipedia.org/wiki/Abstract_polytope#Regular_abstract_polytopes > for the precise definitions of ARP and 11-cell. But roughly, an ARP is a combinatorial generalization of a usual regular polytope that may seem too abstract at first but can grow on you. Anyhow, the 11-cell was discovered (apparently) independently by both Branko Grunbaum in 1977 and H.S.M. Coxeter in 1984. It consists of 11 projective planes P^2, represented as hemi-icosahedra -- the quotient of a regular icosahedron by its antipodal map -- so each P^2 is 10 equilateral triangles. (In fact, as a combinatorial object, metric properties are irrelevant to the 11-cell, but I like to make everything as regular as possible.) Each triangular face of each P^2 is identified with another such face of another P^2, so the 11-cell has 55 triangular faces in all. When this is done right, the 11-cell has a combinatorial automorphism group that is transitive on flags. A flag is a quadruple (F, T, E, V) where F is any of the 11 P^2's (so-called "3-dimensional faces") of the 11-cell, T is any triangle of F, E is any edge of T, V is any vertex of E. So there are 11x10x3x2 = 660 automorphisms in all. Also, this bizarrely symmetrical object is self-dual. I mean, who would've thought there could be such a thing with *11* faces. (The only other such thing has 55 faces -- it's 55 hemi-dodecahedra.) I'd like to know more about the 11-cell as a topological object: What is its universal covering space, its homology and its homotopy? --Dan Sometimes the brain has a mind of its own.