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Golygon

An example of a simple 8-sided golygon

A golygon (technically referred to as a 'serial isogon of 90 degrees') is any polygon with all right angles both internally and externally. (Polygons cannot have all concave angles and quadrilaterals are never golygons.) Its sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A. K. Dewdney in a 1990 Scientific American column (Smith). Variations on the definition of golygons involve allowing edges to cross, using sequences of edge lengths other than the consecutive integers, and considering turn angles other than 90°0°.

In any golygon, all horizontal edges have the same parity as each other, as do all vertical edges. Therefore, the number n of sides must allow the solution of the system of equations

\pm 1 \pm 3\cdots \pm (n-1) = 0
\pm 2 \pm 4\cdots \pm n = 0.

It follows from this that n must be a multiple of 8.

The number of solutions to this system of equations may be computed efficiently using generating functions (sequence A007219 in OEIS) but finding the number of solutions that correspond to non-crossing golygons seems to be significantly more difficult.

There is a unique eight-sided golygon (shown in the figure); it can tile the plane by 180-degree rotation using the Conway criterion.

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