Sounds like the D_n root lattices. See Ch 4 of Sphere Packings, Lattices and Groups. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, Feb 14, 2015 at 11:11 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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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