Golygon

A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) 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).^[1] 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°.^[2]

The smallest golygon has 8 sides. It is the only solution with fewer than 16 sides. It contains two concave corners, and fits on an 8×10 grid. It is also a spirolateral, 8_90°^1,5.

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

${\displaystyle \pm 1\pm 3\pm \cdots \pm (n-1)=0}$ 

${\displaystyle \pm 2\pm 4\pm \cdots \pm n=0.}$ 

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

The number of golygons for a given permissible value of n may be computed efficiently using generating functions (sequence A007219 in the OEIS). The number of golygons for permissible values of n is 4, 112, 8432, 909288, etc.^[3] 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

•  
  16-sided golygon. Spirolateral 16_90°^1,3,6,8,11
•  
  32-sided golygon. Spirolateral 32_90°^{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.^[4] Golygons are a special case of serial-sided isogons.^[5]

A spirolateral is similar construction, notationally n_θ^{i₁,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.

•  
  A serial-sided isogon order 9, internal angle 60°.^[5]
  Spirolateral 60°₉^1,4,7.
•  
  A serial-sided isogon order 11, internal angle 60°.^[5]
  Spirolateral 60°₁₁^4,5,7,8.
•  
  A serial-sided isogon order 12, internal angle 120°.^[5]
  Spirolateral 120°₁₂^1,4,8.
•  
  A serial-sided isogon order 5, internal angles 60° and 120°.^[5]

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.^[6][7]

Golyhedrons have been found with values of n equal to 32, 15, 12, and 11 (the minimum possible).^[8]

References

1. Dewdney, A.K. (1990). "An odd journey along even roads leads to home in Golygon City". Scientific American. 263: 118–121. doi:10.1038/scientificamerican0790-118.
2. Harry J. Smith. . Archived from the original on 2009-10-27.
3. Weisstein, Eric W. "Golygon". MathWorld.
4. Sallows, Lee (1992). "New pathways in serial isogons". The Mathematical Intelligencer. 14 (2): 55–67. doi:10.1007/BF03025216. S2CID 121493484.
5. Sallows, Lee; Gardner, Martin; Guy, Richard K.; Knuth, Donald (1991). "Serial isogons of 90 degrees". Mathematics Magazine. 64 (5): 315–324. doi:10.2307/2690648. JSTOR 2690648.
6. "Can we find lattice polyhedra with faces of area 1,2,3,…?"
7. Golygons and golyhedra
8. Golyhedron update

External links

• Golygons at the On-Line Encyclopedia of Integer Sequences