Despite my earlier casual dismissal of the alternative construction for the 3-space honeycomb of truncated octahedra, more leisurely contemplation suggests I may have been a little hard on myself (such modesty!). In fact, "omni-truncating" the hypercube tiling of d-space leads to another sequence, which coincides with tiling permutohedra at d = 3 . For d = 2,3,4, these are resp. the square tiling, sodalite, and (I think) the regular honeycomb of 24-cells meeting in tesseract vertices. Do this sequence or some associated lattice have recognised names? Have the coordination sequences been conjectured or proved? Fred Lunnon On 2/8/15, Scott Huddleston <c.scott.huddleston@gmail.com> wrote:
A.k.a. permutahedron. It tiles d-space for all d. It's the Voronoi cell of either the A_d lattice or its dual. Scott
On Sun, Feb 8, 2015 at 12:11 PM, 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
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