Golygon

A golygon with 8 sides.

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 $1,2,\dots ,n$ 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 $\pm 1\pm 3\pm \cdots \pm (n-1)=0$ and $\pm 2\pm 4\pm \cdots \pm n=0$. 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

An 11-sided golyhedron discovered by Alexey Nigin.

A golyhedron is a non-self-intersecting polyhedron where all n  faces are simply connected polyominoes and have areas $1,2,\dots ,n$. 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.

External links

• Wikipedia contributors. "Golygon".
• Weisstein, Eric W. "Golygon" at MathWorld.
• OEIS, A006718