As a schoolboy (I know --- it's difficult to imagine!) I became interested in a Euclidean d-space polytope I'll call TKD_d : the convex hull of points assigned the same value of multinomial coefficient, in some layer of the obvious generalisation of Pascal's triangle from binomials to d+1 variables (assuming most general position). Another way to envisage its vertices and edges is the Cayley graph of permutations on n+1 symbols, generated by adjacent transpositions. TKD_d has (d+1)! vertices, 2^(d+1) - 2 facets (if d > 1 , at any rate). TKD_1 is a just line segment; TKD_2 is a regular hexagon; TKD_3 is an Archimedean tetrakaidecahedron (truncated octahedron) with 6 square and 8 hexagonal faces; TKD_4 has 120 equal tetrahedral vertices, 180 edges, 90 faces, 30 cells comprising 10 tetrakaidecahedra and 20 hexagonal prisms. Two questions: For what d does TKD_d tile d-space? (It surely does so for d = 1,2,3 --- what about d = 4 ?) Is (the edge-skeleton of) such a tiling O'Keeffe's "d-dimensional sodalite" net, described cryptically as "holes in the A_d^* lattice" on page 2387 of J. H. Conway, N. J. A. Sloane (1997) "Low–dimensional lattices. VII. Coordination sequences" http://neilsloane.com/doc/Me220.pdf ? Fred Lunnon