Wednesday, January 25, 2012

Dynamic Spirograph: Circle Rolling around Rolling Circle (4)

Circle 1 rolls around outside of the fixed base circle.
Circle 2 rolls around outside of circle 1.

( Mathematical software used: GeoGebra )

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  1. Have you tried arriving at the mandelbrot set using the fractal spirograph method? I'm curious what would happen when experimenting with the GoldenRatio as the ratio of circle radii. Also, do you have parametric equations for the circle-within-a-circle fractal spirographs? I assume it's just epitrochoid or hypotrochoid equations nested several times, but I had trouble figuring it out.

  2. Hi BryDude

    The form of the parametric equation is:
    x(t) = Σ(k=1 to ∞) R(k)*cos(a(k)*t)
    y(t) = Σ(k=1 to ∞) R(k)*sin(a(k)*t)

    The golden ratio is an irrational number. If the ratio of the radii
    of the circles is irrational, then the curve never closes.

    I haven't found the Mandelbrot set using this method.