# Golygon

A golygon is a non-self-intersecting n -gon with all vertices in a square lattice and all angles 90 degrees (a polyomino) such that all edge lengths are ${\displaystyle 1,2,\dots ,n}$ in circular order. Golygons were first named and investigated by Lee Sallows.

They are a special case of spirolaterals, which permit angles other than 90 degrees.

## Properties

By walking along the boundary of a golygon clockwise and observing the directions of the vertical and horizontal edges, it can be seen that every golygon assigns signs to the two equations ${\displaystyle \pm 1\pm 3\pm \cdots \pm (n-1)=0}$ and ${\displaystyle \pm 2\pm 4\pm \cdots \pm n=0}$. These equations can only be satisfied if n  is a multiple of 8. Although enumerating solutions to the equations is computationally straightforward, many solutions to these equations will produce self-intersecting polygons.

The smallest golygon has eight sides, and can tile the plane.

## Golyhedra

A golyhedron is a non-self-intersecting polyhedron where all n  faces are simply connected polyominoes and have areas ${\displaystyle 1,2,\dots ,n}$. Additionally, the polyhedron's boundary must be topologically equivalent to a sphere. The first golyhedron, at 32 faces, was discovered by Goucher. The smallest golyhedron has 11 faces, a result that has been proven optimal.