Infinite dihedral group: Difference between revisions

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[[Image:Golygon.png|thumb|right|upright=1.5|An example of a simple 8-sided golygon]]
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A '''golygon''' (technically referred to as a "serial isogon of 90 degrees") is any [[polygon]] with all [[right angle]]s, 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). 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°.  
 
In any golygon, all horizontal edges have the same [[parity (mathematics)|parity]] as each other, as do all vertical edges. Therefore, the number ''n'' of sides must allow the solution of the system of equations
:<math>\pm 1 \pm 3\cdots \pm (n-1) = 0</math>
:<math>\pm 2 \pm 4\cdots \pm n = 0.</math>
 
It follows from this that ''n'' must be a multiple of 8.
 
The number of solutions to this system of equations may be computed efficiently using generating functions {{OEIS|id=A007219}} but 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 [[tessellation|tile]] the plane by 180-degree rotation using the [[Conway criterion]].
 
==See also==
*[[Rectilinear polygon]]
 
==References==
*{{cite journal |author=Dewdney, A.K. |title=An odd journey along even roads leads to home in Golygon City |journal=[[Scientific American]] |volume=263 |pages=118–121 |year=1990}}
*{{cite journal |author=Sallows, Lee |title=New pathways in serial isogons |journal=[[The Mathematical Intelligencer]] |volume=14 |pages=55–67 |year=1992 |doi=10.1007/BF03025216 |issue=2}}
*{{cite journal |doi=10.2307/2690648 |author=Sallows, Lee; [[Martin Gardner|Gardner, Martin]]; [[Richard K. Guy|Guy, Richard K.]]; [[Donald Knuth|Knuth, Donald]] |title=Serial isogons of 90 degrees |jstor= 2690648 |journal=[[Mathematics Magazine]] |volume=64 |issue=5 |year=1991 |pages=315–324}}
*{{cite web |author=Harry J. Smith |title=What is a Golygon? |url=http://www.geocities.com/hjsmithh/Golygons/GolyWhat.html |archiveurl=http://web.archive.org/web/20091027155427/http://geocities.com/hjsmithh/Golygons/GolyWhat.html |archivedate=2009-10-27}}
*{{mathworld|title=Golygon|urlname=Golygon}}
 
==External links==
*[http://oeis.org/search?q=golygon&language=english Golygons] at the [[On-Line Encyclopedia of Integer Sequences]]
 
[[Category:Polygons]]

Latest revision as of 19:58, 28 November 2014

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