Yes and yes then, thanks to Dan and Scott. The webpage http://en.wikipedia.org/wiki/Permutohedron and Dan remark on a simple and elegant coordinatisation for the d-space tiling, embedded in the hyperplane x_1 + ... + x_(d+1) = 0 . I failed to find this, having been wickedly sidetracked by another d = 3 embedding, into 3-space, which (multi-dimensionally speaking) plainly leads nowhere :--- (x,y,z) is a vertex of the 3-space tiling just when these 6 congruences all fail: x +/- y , y +/- z , z +/- x <> 0 (mod 4) . Dan: << (I became interested in P(n) because any two of these tiles gP(n), hP(n) are either disjoint or intersect on an (n-1)-face, making this tiling a good setting for n-dimensional Hex* on, when n is even.) >> Alright, I'll buy it --- why only when n is even? (or should that have read "odd"?) Fred Lunnon On 2/8/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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