Generalized Straightedge and Compass Fractal

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Define "the circles of a triangle" to mean the three circles centered at the vertices of the triangle with a radius equal to the "length multiplier" times the edge length of the triangle (it is most natural to let the length multiplier equal 1).


Given a triangle , the vertices of its midpoint triangle are the midpoints of the sides of . A fractal is made by starting with an equilateral triangle and at each iteration adding vertices of any midpoint triangle and circles at these vertices, as well as at any vertex generated at a prior iteration, with the radius of the circles added at an iteration equal to the product of the length multiplier and the edge length of the midpoint triangles added at that iteration.

Alternatively, apply the replacement rules shown in white at the top wherever possible.


Contributed by: Richard Southwell (June 2014)
Open content licensed under CC BY-NC-SA



An exploration of the dynamics of this system is given in my YouTube video [1], where I describe how to make this fractal using only straightedge and compass.


[1] R. Southwell. The Generalized Circle Fractal: Exploration & Straightedge-Compass Construction [Video]. (Jun 2, 2014)

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