Heesch's Problem

Heesch's problem asks how many times a shape can be completely surrounded by copies of itself without being able to tile the entire plane. Equivalently, it asks for the maximum finite Heesch number.

The higher-dimensional version has a different answer. Bašić (2026) proved that finite Heesch numbers are unbounded in every fixed dimension . The question in the plane remains open.

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References

Bašić, B. "The Heesch Number in

Is Unbounded." 16 Oct 2026. https://arxiv.org/abs/2610.24281.Eppstein, D. "Heesch's Problem." https://ics.uci.edu/~eppstein/junkyard/heesch/.

Referenced on Wolfram|Alpha

Heesch's Problem

Cite this as:

Weisstein, Eric W. "Heesch's Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HeeschsProblem.html

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