What are the simple Heesch-2 polyforms? ![Almost3]( At the Tiling Database: There are 3, 20, 198, 1390 non-tiling polyominoes of order 7 to 10. There are 4, 37, 381, 2717 non-tiling polyhexes of order 6 to 9. There are 1, 0, 20, 103, 594, 1192, 6290 non-tiling polyiamonds of order 7 to 13. Have these non-tilers had their Heesch numbers determined? A non-tiler that can surround itself is Heesch-1. If a second complete ring can be made, that's Heesch-2. What simple polyforms are Heesch-2? ![enter image description here](

Heesch numbers of some polyominoes and polyiamonds (but not polyhexes) were determined here: <

Minimal Heesch 2 polyomino has 9 (not 11 as on your picture) squares:

![the only 9-mino with Heesch number 2](

Minimal polyamond with Heesch number 3 has 10 triangles:

![10-iamond with Heesch number 3](

Heesch numbers of edge-marked polyforms (including polyhexes) were studied here: <

Since this article is not available for free, here are 2 examples from it:

5-hex with marked edges and Heesch number 3:

![5-hex with marked edges and Heesch number 3](

13-iamond with marked edges and Heesch number 4:

![13-iamond with marked edges and Heesch number 4](