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# Self-Avoiding Polygon

A lattice polygon consisting of a closed self-avoiding walk on a square lattice. The perimeter, horizontal perimeter, vertical perimeter, and area are all well-defined for self-avoiding polygons. Special classes of self-avoiding polygons include the bar graph polygon, convex polygon, Ferrers graph polygon, stack polyomino, and staircase polygon. Self-avoiding polygon are used in physics to model crystal growth and polymers (Bousquet-Mélou 1992).

Enumerating self-avoiding polygons according to perimeter or area is an unsolved problem (Bousquet-Mélou et al. 1999).

Polyomino, Self-Avoiding Walk, Staircase Polygon

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## References

Bousquet-Mélou, M. "Convex Polyominoes and Heaps of Segments." J. Phys. A: Math. Gen. 25, 1925-1934, 1992.Bousquet-Mélou, M.; Guttmann, A. J.; Orrick, W. P.; and Rechnitzer, A. "Inversion Relations, Reciprocity and Polyominoes." 23 Aug 1999. http://arxiv.org/abs/math.CO/9908123.Janssens, P. "Counting Closed Self-avoiding Walks (CSAW) in the Square Lattice up to Direct Isometry." http://www.afront.be/polydoc/paper.html.

## Referenced on Wolfram|Alpha

Self-Avoiding Polygon

## Cite this as:

Weisstein, Eric W. "Self-Avoiding Polygon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Self-AvoidingPolygon.html