# 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 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[edit | edit source]

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 and . 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[edit | edit source]

A **golyhedron** is a non-self-intersecting polyhedron where all n faces are simply connected polyominoes and have areas . 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.

## External links[edit | edit source]

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