A plane path on a set of equally spaced lattice points, starting at the origin, where the first step
is one unit to the north or south, the second step is two units to the east or west,
the third is three units to the north or south, etc., and continuing until the origin is again reached. No crossing or backtracking is allowed.
The simplest golygon is (0, 0), (0, 1), (2, 1), (2, 
), (
,
), (
, 
),
(
, 
), (
,
0), (0, 0).
A golygon can be formed if there exists an even integer 
such that
(Vardi 1991). Gardner proved that all golygons are of the form 
.
The number of golygons of length 
(even), with each initial direction
counted separately, is the product of the coefficient
of 
in
 |
(3) |
with the coefficient of 
in
 |
(4) |
The number of golygons 
of length 
for the first few 
is therefore 4, 112, 8432, 909288, ... (OEIS A006718),
and is asymptotic to
 |
(5) |
(Sallows et al. 1991, Vardi 1991).
See also
Canonical Polygon, Lattice Path, Lattice Polygon
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References
Dudeney, A. K. "An Odd Journey Along Even Roads Leads to Home in Golygon City." Sci. Amer. 263, 118-121, July 1990.Sallows, L. C. F. "New Pathways in Serial Isogons." Math. Intell. 14, 55-67, 1992.Sallows, L.; Gardner, M.; Guy, R. K.; and Knuth, D. "Serial Isogons of 90 Degrees." Math Mag. 64, 315-324, 1991.Sloane, N. J. A. Sequence A006718/M3707 in "The On-Line Encyclopedia of Integer Sequences."Smith, H. J. "Golygons." http://www.geocities.com/hjsmithh/Golygons.html.Vardi, I. "American Science." §5.3 in Computational Recreations in Mathematica. Redwood City, CA: Addison-Wesley, pp. 90-96, 1991.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Golygon." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Golygon.html