A Convex Pentagon with Heesch Number Two
Year: 2026
Type: preprint
Abstract: We present an explicit convex pentagon with Heesch number exactly two. An exact arrangement of 27 congruent copies forms two complete coronas around the central tile, with seven tiles in the first and nineteen in the second. The polygon is Sugimoto’s Category 16, n = 2 pentagon, previously assigned Heesch number one; our construction establishes its greater surroundability. A partial-edge contact is essential to the displayed arrangement. To exclude every possible third corona, we prove a local short-side pairing rule and reduce necessary geometric conditions to a finite Boolean contradiction, certified by an independently checkable computational proof. A small companion package checks the construction, regenerates the local placement libraries, and replays the contradiction using only the Python standard library and exact arithmetic in ℚ(√3). To our knowledge, this is the first convex polygon in the Euclidean plane with a positive finite Heesch number greater than one.
Author: Seamons, John
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225778789
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