Teruhisa Sugimoto has a draft paper that enumerates 17 types of convex, finitely surroundable pentagons! I hope to be able to read an English translation at some point. http://tilingpackingcovering.web.fc2.com/abstract-e.html#heeschCP
And finally, the dramatic conclusion to my series on Heesch numbers: does there exist a convex pentagon that can be surrounded, but doesn't tile the plane, and admits a surround that's edge-to-edge? Heesch's original 1968 shape does all of the above except the edge-to-edge part. http://isohedral.ca/heesch-numbers-part-4-edge-to-edge-pentagons/ https://mathstodon.xyz/media/a0bUlDZY_utSfkXzkbs
The story of Heesch numbers continues in two posts. In http://isohedral.ca/heesch-numbers-part-2-polyforms/, I compute Heesch numbers of polyominoes and polyiamonds in search of interesting new examples; in http://isohedral.ca/heesch-numbers-part-3-bamboo-shoots-and-ice-cream-cones/, I present a new family of simple polygons, all with Heesch number 1.
Time to justify my presence here...
The Heesch number of a shape is the maximum number of layers of copies of that shape by which you can surround it. Heesch's Problem asks which positive integers can be Heesch numbers. I'll show a few fun new results over a series of blog posts; today, I offer a basic introduction to the topic. http://isohedral.ca/heesch-numbers-part-1/
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