The Heesch number in ${\mathbb{E}}^3$ is unbounded

The Heesch number of a figure is the largest number of times that the figure can be surrounded by layers of its congruent copies. Heesch’s problem asks whether finite Heesch numbers are bounded. We show that in the three-dimensional Euclidean space, and thus also in higher-dimensional Euclidean spaces, they are not: for every positive integer $h$, we construct a shape whose Heesch number is finite and at least $h$. The construction is explicit, and it starts from a four-point set of integers that does not tile the line, but packs long intervals.