Heinrich Heesch1906–1995
Posed the problem, 1968
German mathematician who in 1968 asked how many times a shape that cannot tile the plane can be surrounded by copies of itself, the question now answered for each shape by its Heesch number.
Heesch Numbers
A shape that cannot tile the plane, surrounded by rings of its own copies until no further ring will close. The count of rings is its Heesch number.
About this pattern
Heinrich Heesch, 1968 · Craig S. Kaplan, 2022
Some shapes tile the plane forever, and most cannot be surrounded even once. Between them are shapes that can be ringed by copies of themselves a few times and then no more: Heinrich Heesch asked in 1968 how many rings, or coronas, are possible, and the answer for a given shape is its Heesch number.
Each ring here covers every square that touches the rings inside it, at an edge or a corner, with nothing overlapping and no gap left inside. Casey Mann found shapes that take five; Bojan Bašić one that takes six. Craig Kaplan turned the question into a satisfiability problem and computed the Heesch number of every polyomino up to nineteen squares.
The shapes and their Heesch numbers here come from Kaplan's tables, where each is proven; the rings drawn around them were found by GART's own search. In motion the rings are laid down one at a time.
Behind the mathematics
Craig S. Kaplan
Computed Heesch numbers of polyforms with a SAT solver, 2022
Canadian computer scientist whose research joins geometry, computation and ornamental design, including the proof and visualisation of the hat monotile.
Casey Mann
Shapes with Heesch number 5
American mathematician who found a shape that can be surrounded five times and no more, a record for Heesch’s problem that stood for nearly two decades until 2021.
Bojan Bašić
The record, Heesch number 6, 2021
Serbian mathematician who in 2021 found a shape that can be surrounded six times and no more, the current record for Heesch's problem.