Golygon (original) (raw)

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Polygon with right angles and consecutive lengths

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, 890°1,5.

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]

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

± 1 ± 3 ± ⋯ ± ( n − 1 ) = 0 {\displaystyle \pm 1\pm 3\pm \cdots \pm (n-1)=0} {\displaystyle \pm 1\pm 3\pm \cdots \pm (n-1)=0}

± 2 ± 4 ± ⋯ ± n = 0. {\displaystyle \pm 2\pm 4\pm \cdots \pm n=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 − 1 + 3 + 5 − 7 = 0 {\displaystyle -1+3+5-7=0} {\displaystyle -1+3+5-7=0} and 2 − 4 − 6 + 8 = 0 {\displaystyle 2-4-6+8=0} {\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.

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_i_1,_i_2,...,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 _i_1,_i_2,...,i k superscripts list edges that follow opposite turn directions.

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]

  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. "What is a Golygon?". 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. ^ a b c d e 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