21 research outputs found
Translational tilings by a polytope, with multiplicity
We study the problem of covering R^d by overlapping translates of a convex
body P, such that almost every point of R^d is covered exactly k times. Such a
covering of Euclidean space by translations is called a k-tiling. The
investigation of tilings (i.e. 1-tilings in this context) by translations began
with the work of Fedorov and Minkowski. Here we extend the investigations of
Minkowski to k-tilings by proving that if a convex body k-tiles R^d by
translations, then it is centrally symmetric, and its facets are also centrally
symmetric. These are the analogues of Minkowski's conditions for 1-tiling
polytopes. Conversely, in the case that P is a rational polytope, we also prove
that if P is centrally symmetric and has centrally symmetric facets, then P
must k-tile R^d for some positive integer k
Concrete polytopes may not tile the space
Brandolini et al. conjectured that all concrete lattice polytopes can
multitile the space. We disprove this conjecture in a strong form, by
constructing an infinite family of counterexamples in .Comment: 6 page
CONCRETE POLYTOPES MAY NOT TILE THE SPACE
Brandolini et al. conjectured in (Preprint, 2019) that all concrete lattice polytopes can multitile the space. We disprove this conjecture in a strong form, by constructing an infinite family of counterexamples in β3