Michel Deza and Mikhail Shtogrin - Embedding of regular tilings and star-honeycombs

We review the regular tilings of d-sphere, Euclidean d-space, hyperbolic d-space and Coxeter's regular hyperbolic honeycombs (with infinite or star-shaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a m-cube or m-dimensional cubic lattice.

The last remaining 2-dimensional case is decided: for any odd , star-honeycombs are embeddable while are not (unique case of non-embedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate 36 Riemann surfaces representing all nine regular polyhedra on the sphere. Also non-embeddability of all remaining star-honeycombs (on 3-sphere and hyperbolic 4-space) is proved.

Finally, all cases of embedding for dimension d > 2 are identified. Besides hyper-simplices and hyper-octahedra, they are exactly those with bipartite skeleton: hyper-cubes, cubic lattices and 8, 2, 1tilings of hyperbolic 3-, 4-, 5-space (only two, 435 and 4335, of those 11 are compact).

absRobert Erdahl Queen's University, Kingston, Ontario  K7L 3N6, Canada Voronoi's hypothesis on perfect domains

It was implicit in Voronoi's famous last two geometrical memoirs that he felt the partition of the cone of metrical forms into lattice type domains is a refinement of the partition into perfect domains. This property of perfect domains is now referred to as Voronoi's Hypothesis. The first of these memoirs was on perfect forms and their associated domains, and in the second he introduced his famous theory of lattice types; so this hypothesis links these last two memoirs. Voronoi's hypothesis has played an important role in the programs to classify higher dimensional lattices that followed. More precisely, this hypothesis played a decisive role in the fundamental work on classification of Delaunay, and more recently of Ryshkov and Baranovski. Ryshkov and Baranovsky were able to show that Voronoi's hypothesis holds up to dimension 5. I will show that this convenient relationship does not hold in general by demonstrating that the L-type partition in 6 dimensions is not a refinement of the perfect partition. This work is joint with Konstantin Rybnikov.