Asymptotical Unboundedness of the Heesch Number in Ed for d→∞

Abstract

We solve d-dimensional Heesch’s problem in the asymptotic sense. Namely, we show that, if $d\to \infty$, then there is no uniform upper bound on the set of all possible finite Heesch numbers in the space ${\mathbb{E}}^d$; in other words, given any nonnegative integer n, we can find a dimension d (depending on n) in which there exists a hypersolid whose Heesch number is finite and greater than n.