Your Friendly Neighborhood Voderberg Tile

摘要

In 1934, K. Reinhardt asked if any tile exists such that two copies can completely enclose a third copy [4]. In 1936, Reinhardt’s student Heinz Voderberg answered this question in the affirmative by inventing a special shape, now known as the Voderberg tile, shown in Figure 1. In fact, the two copies of the Voderberg tile can enclose~? a third and a fourth copy (Figure 1)! More generally, a tile T has the r-enclosure property if two copies T_1 and T_2 of T can be arranged so that the complement of T_1 ∪ T_2 has a bounded component, the closure of which is the union of r non-overlapping copies of T (as discussed, for example, by Grünbaum and Shepherd [3, 4]). In Figure 1, we see that the Voderberg tile has the 1- and 2-enclosure property.