Fractals – Legal Circles

Legal Circle Acrylic OrnamentFractals are amazing.  Their beauty can be created by following a few simple rules that anyone can do.  Many of the shapes in nature can be categorized as fractals.  Defined, a fractal is a geometric shape that is created by subdividing a shape according to some iterative process.  We see fractals in snowflakes, ferns, mountains, erosion patterns, and other places in nature.  There are many types of fractals, including legal circles, dragon curveSierpinski carpet, and Sierpinski triangle to name a few.  In a previous post, I described the Dragon Fractal.  In this post, I will discuss a Legal Circle fractal.

A Legal Circle fractal is a very simple fractal to construct and can become complex quickly.  To begin, draw a circle.  Next, using that same radius, draw another circle with the center of the new circle on any point on the first circle.  Those first two circles are defined as legal circles.  To continue the construction, follow these two rules:

  1. Wherever two legal circles of equal radius touch or intersect, draw another legal circle of equal radius with the intersection point as its center.
  2. Within every legal circle with center point p and radius r, draw another legal circle whose center point is also p but whose radius is r/2.

Constructing the fractal

These instructions sound simple and in the early stages are very easy to complete.  As the fractal becomes more complex, it becomes more difficult to move forward.  Here is a diagram of the first nine steps in creating a Legal Circle fractal (the figures in red show the current step).

Diagram on the construction of a Legal Circle Fractal

As you can see, the design gets quite complex very quickly but the results are stunning.  For a very complex construction of a Legal Circle fractal, check out this page.

I hope you have learned something new about fractals today.  As technology gets better and better, it is becoming easier and easier to construct your own fractals and experiment.  If you haven’t made one yet, get busy and create a simple Legal Circle fractal.  Your friends and family will be impressed…

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