Isohedral numbers: boundedness and realizable values

Determine which positive integers can occur as the isohedral number of a shape in the Euclidean plane and whether the set of isohedral numbers is bounded above.

Background

The isohedral number of a shape is the minimum number of transitivity classes in any tiling it admits; it quantifies how far a shape is from allowing an isohedral tiling. Empirical computations have found examples up to 10 (with 7 not observed in the reported dataset), but there is no known bound or full characterization.

The author explicitly states that we do not know which integers are achievable or whether a bound exists, paralleling the uncertainty around Heesch numbers.