As Dan observed, the set of points I was clumsily attempting to define is simply the "checkerboard" lattice D_n = { (x_1, ..., x_n) in |Z^n | x_1 + ... + x_n == 0 mod 2 } , which is congruent to its complementary D'_n = { (x_1, ..., x_n) in |Z^n | x_1 + ... + x_n == 1 mod 2 } . Now define 2^(n-1) pencils of parallel hyperplanes x_1 +/- ... +/- x_n = 2 k , one for each possible choice of signs: these hyperplanes bisect vertex diagonals of lattice hypercubes odd vertices, satisfying x_1, ..., x_n == 1 mod 2 . [ Notice that hypercube centres reside always in D_n , while the vertices reside in D_n , D'_n for n even, odd respectively. For example when n = 3 , bisector planes meet a cube in hexagons with vertices at mid-points of cube edges, which together with cube centres constitute D_3 ; while cube vertices and face centres constitute D'_3 . ] Each point of D_n lies on a unique member of every pencil: the combined configuration constitutes the universal cover of Dan's partitioned n-tore; its points are the vertices of "piece" polytopes, whose facets tile the hyperplanes. The piece tessellation must just be the Delaunay triangulation Del(D_n) mentioned by Neil. Quotienting out the even lattice yields a periodic n-tore with content 2^n and 2^(n-1) distinct congruent vertices; we require to enumerate the types and frequencies of pieces occurring per period. My previous assertion concerning these was over-impetuous: it seems safe to assert that one type comprises 2^(n-1) regular cross-polytopes centred on points of D'_n ; however occupancy of the voids in between remains obscure to me for n > 4 . Case n = 3 : pieces comprise 4 octahedra and 8 tetrahedra, vertices 4 cuboctahedra, from tessellation h{4,3,4} : see https://en.wikipedia.org/wiki/Tetrahedral-octahedral_honeycomb Case n = 4 : pieces comprise 24 cross-polytopes (dual tesseracts), vertices 8 icositetratopes, from tessellation {3,3,4,3} : see https://en.wikipedia.org/wiki/24-cell_honeycomb This topic must surely have already been explored by others --- Conway & Sloane "Sphere Packings, Lattices, and Groups"; Coxeter "Regular Polytopes"; or perhaps the Rostock group involved in "Delaunay Tessellations of Point Lattices" (ERC Workshop Oct 2013) http://www.mi.fu-berlin.de/en/math/groups/discgeom/dates/2nd_ERC_Workshop/sl... Fred Lunnon