The Heesch number of a closed plane figure is the maximum number of times that figure can be completely surrounded by copies of itself. The determination of the maximum
possible (finite) Heesch number is known as Heesch's
problem. The Heesch number of a triangle, quadrilateral,
regular hexagon, or any other shape that can tile
or tessellate the plane, is infinity. Conversely,
any shape with infinite Heesch number must tile the plane (Eppstein).
A tile invented by R. Ammann has Heesch number three (Senechal 1995), and Mann has found an infinite family of tiles with Heesch number five (illustrated above), the largest (finite) number known.
A database of Heesch tilings is maintained by Mann (2008).