# Golygon

An example of a simple 8-sided golygon

A golygon is any polygon with all right angles, whose 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).[1] 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°.[2]

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. Thus the number of golygons for n = 1, 2, 3, 4, ... is 4, 112, 8432, 909288, etc.[3]

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.

## Generalizations

A serial-sided isogon of order n is a closed polygon with a constant angle at each vertex and having consecutive sides of length 1, 2, ..., n units. The polygon may be self-crossing.[4] Golygons are a special case of Serial-sided isogons.[5]

The three-dimensional generalization of a golygon is called a golyhedron–a closed simply-connected solid figure confined to the faces of a cubical lattice and having face areas in the sequence 1, 2, ..., n, for some integer n.[6] Golyhedrons have been found with values of n equal to 32, 15, 12, and 11 (the mininum possible).[7]