There's a Wiki article about the permutohedron: < http://en.wikipedia.org/wiki/Permutohedron >. I think you can think of this as the convex hull of its vertices -- e.g., the ordered n-tuples comprising all permutations of {1, 2, 3, . . ., n} in R^n. Call this convex hull, the n-permutohedron, P(n). The symmetry in its definition proves that all n! permutations of the vertices extend to isometries of the polytope. According to Wikipedia, the n-permutohedron tiles n-space for all n: Each permutohedron P(n) lies in the affine hyperplane Q of R^n defined as Q := {(x_1,...,x_n) in R^n | x_1 + ... + x_n = n}. Let lattice L in R^n be defined by L := {(k_1,...,k_n) in Z^n | k_1+...+k_n = 0 AND all k_i-k_j == 0 mod n} Then L acts as a group of translations on Q, and the translations {gP(n) | g in L} of P(n) tile R^n. (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.) --Dan ____________________________________________________________________ * A game I invented in grad school. I tried to market it, but there were manufacturing problems.
On Feb 8, 2015, at 12:42 PM, 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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