Golygon

Properties

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 \pm \cdots \pm (n-1) = 0 :\pm 2 \pm 4 \pm \cdots \pm n = 0.

It follows from this that n must be a multiple of 8. For example, in the figure we have -1 + 3 + 5 - 7 = 0 and 2 - 4 - 6 + 8 = 0.

The number of golygons for a given permissible value of n may be computed efficiently using generating functions . The number of golygons for permissible values of n is 4, 112, 8432, 909288, etc. 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.

Examples

Golygon 16-sides.svg|16-sided golygon. Spirolateral 1690°1,3,6,8,11 golygon 32-sides.svg|32-sided golygon. Spirolateral 3290°1,3,5,7,11,12,14,17,19,21,23,26,29,31

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.

A spirolateral is similar construction, notationally nθi1,i2,...,i**k which sequences lengths 1,2,3,...,n with internal angles θ, with option of repeating until it returns to close with the original vertex. The i1,i2,...,i**k superscripts list edges that follow opposite turn directions.

serial isogon_9_120.svg|A serial-sided isogon order 9, internal angle 60°.Spirolateral 60°91,4,7. Serial_isogon_11_60.svg|A serial-sided isogon order 11, internal angle 60°.Spirolateral 60°114,5,7,8. serial isogon 12 60.svg|A serial-sided isogon order 12, internal angle 120°.Spirolateral 120°121,4,8. Serial isogon 5 60 120.svg|A serial-sided isogon order 5, internal angles 60° and 120°.

Golyhedron

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, first introduced in a MathOverflow question.

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

References

References

1. Dewdney, A.K.. (1990). "An odd journey along even roads leads to home in Golygon City". [[Scientific American]].
2. Harry J. Smith. "What is a Golygon?".
3. "Golygon".
4. Sallows, Lee. (1992). "New pathways in serial isogons". [[The Mathematical Intelligencer]].
5. (1991). "Serial isogons of 90 degrees". [[Mathematics Magazine]].
6. [https://mathoverflow.net/q/164588/6094 "Can we find lattice polyhedra with faces of area 1,2,3,…?"]
7. [http://cp4space.wordpress.com/2014/04/30/golygons-and-golyhedra/ Golygons and golyhedra]
8. [http://cp4space.wordpress.com/2014/05/11/golyhedron-update/ Golyhedron update]