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# Canonical Polygon

As defined by Kyrmse, a canonical polygon is a closed polygon whose vertices lie on a point lattice and whose edges consist of vertical and horizontal steps of unit length or diagonal steps (at angles which are multiples of with respect to the lattice axes) of length . In addition, no two steps may be taken in the same direction, no edge intersections are allowed, and no point may be a vertex of two edges. The numbers of distinct canonical polygons of , 2, ... sides are 0, 0, 1, 3, 3, 9, 13, 52, 140, 501, 1763, 6786, 25571, ... (OEIS A052436).

There are exactly eight distinct convex canonical polygons, illustrated above.

The concept can also be generalized to diagonals rotated with respect to the lattice axes.

Note that this mathematically recreational use of the term "canonical" is completely unrelated to the mathematically established concept of canonical polyhedra as applied to dual polyhedra and midspheres.

Canonical Polyhedron, Golygon, Lattice Polygon

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## References

Kyrmse, R. E. "Canonical Polygons." http://www.geocities.com/kyrmse/canonic-e.htm.Sloane, N. J. A. Sequence A052436 in "The On-Line Encyclopedia of Integer Sequences."

## Referenced on Wolfram|Alpha

Canonical Polygon

## Cite this as:

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