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About this paper
Dmitry Kamenetsky, Tristrom Cooke
The research defines "holey polyominoes," a new class of geometric shapes incorporating visible and transparent squares, and systematically categorizes their ability to tile rectangular regions. Utilizing computational methods and theoretical proofs, it classified 107 such polyominoes with up to 5 visible squares into rectifiable, unrectifiable, or unknown categories.
Key insights
- Holey polyominoes introduce a structural extension to traditional polyominoes by explicitly including transparent squares, enriching the combinatorial geometry landscape.
- Computational algorithms, enhanced with heuristics like prioritizing constrained grid locations, are effective for navigating large search spaces in complex geometric tiling problems.
- The rectifiability of a polyomino is not always an absolute property, as an unrectifiable piece can be embedded within a larger, rectifiable polyomino.
Summary by alphaXiv.
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