Golygon

File:Golygon.png

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). 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°.

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.

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. Golygons are a special case of Serial-sided isogons.

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. Golyhedrons have been found with values of n equal to 32, 15, 12, and 11 (the minimum possible).

See also

• Rectilinear polygon

External links

• Golygons at the On-Line Encyclopedia of Integer Sequences

• Weisstein, Eric W., "Golygon", MathWorld.
• Golygons and golyhedra
• Golyhedron update