The figure below is not a polygon. It was constructed from a line segment by a precisely defined algorithm: make a copy of the segment, rotate it through an angle α, and scale its length by a factor k (here less than 1).
Repeating this process produces this attractive structure, which converges to a particular point in the plane.
Note that for k = 1 and α = π – 2π/n, the construction would produce a convex regular polygon with n sides. Certain angles also produce star polygons.
Apart from these special cases, the result is a jolygon, a term coined in the magazine Le petit Archimède in 1975.
The jolygon shown above has a special feature: at the third iteration, it returns to A on the original segment. This jolygon is called self-tangent. It was constructed using α = 65° and k = 0.9. Complex numbers can be used to study jolygons. If a segment is represented by a complex number z, the next segment is represented by the number zke*i*α.
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](http://www.lepetitarchimede.fr/pa/PA14.pdf)
Reference: Le petit Archimède, no. 14, 1975. Available online.
