# Heesch number

The Heesch number is the maximum number of times that a closed plane figure can be completely
surrounded by copies of itself. The Heesch number of a triangle, quadrilateral,
regular hexagon, or any other single shape that can completely tile the
plane (see tiling), is infinity. *Heesch's
problem* is to find the largest possible finite Heesch number, or, more
generally, what values other than zero and infinity can occur as Heesch
numbers. In considering this problem, it's helpful to define the Heesch
number more precisely. In a tiling, the *first corona* of a tile is
the set of all tiles that have a common boundary point with the tile, including
the original tile itself. The *second corona* is the set of tiles that
share a point with anything in the first corona; and so on. The Heesch number
of a shape is the maximum value of *k* for which all tiles in the *k*-th
corona of any tiling are congruent to that shape. For a long time the record
holder for the largest finite value of *k* was a shape found by the
American computer scientist Robert Ammann, which consisted of a regular
hexagon with small projections on two sides and matching indentations on
three sides. This was thought to have a Heesch number of three; however,
in 2000, Alex Day argued that the Ammann hexagon actually has a Heesch number
of four, though there is a dispute about whether the difference has to do
with a definition of tiling. In any event, it has since been shown by Casey
Mann, of the University of Arkansas that there exists an infinite family
of tiles (consisting of indented and outdented pentahex) with Heesch number
five (or six by Day's reckoning) – the largest finite value currently
known.

It remains an open issue whether any polygon has higher Heesch number, but these new techniques seem very powerful and likely to lead to further results. Mann believes that more rounded polyominos than the long skinny ones he's been using may have a better chance of giving unbounded Heesch numbers.

The Heesch number question seems closely connected to two other famous open
tiling problems: does there exist an algorithm for determining whether a shape can tile, and does there exist a shape that
can only tile aperiodically? Aperiodic tiling seems to act as a barrier
to the existence of tiling algorithms, so it isn't expected that both of
these problems have the same answer. On the other hand, if no finite Heesch
number is larger than some *k*, then it seems that this could be used
as the basis of an algorithm to test whether a shape tiles: simply attempt
to fill out a tiling to the (*k* + 1)st corona; if successful, the
shape must tile the plane, and if not, the shape does not tile. Similar
questions can also be asked about Heesch numbers for tilings in higher dimensions.