Pickover Popcorn

Nice Image Pickover Popcorn

Here is another Pickover idea. This one computes and plots the orbits of the dynamic system defined by:
      x(n+1) = x(n) - real(h * fn1( y(n) + fn2(C * y(n) ))
                    - imag(h * fn3( x(n) + fn4(C * x(n) ))
      y(n+1) = y(n) - real(h * fn3( x(n) + fn4(C * x(n) ))
                    - imag(h * fn1( y(n) + fn2(C * y(n) ))

In the original the functions were: sin, tan, sin, tan, and C was 3.

The the initializers x(0) and y(0) equal to ALL the complex values within the "corners" values, and h=.01. ALL these orbits are superimposed, resulting in "popcorn" effect. You may want to use a maxiter value less than normal - Pickover recommends a value of 50. Although you can zoom and rotate popcorn, the results may not be what you'd expect, due to the superimposing of orbits and arbitrary use of color. The orbits frequently occur outside of the screen boundaries. To view the fractal in its entirety, set the preview display to "yes" using the "V" command.

As a bonus, type=popcornjul shows the Julia set generated by these same equations with the usual escape-time coloring. Turn on orbit viewing with the "O" command, and as you watch the orbit pattern you may get some insight as to where the popcorn comes from.

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Noel Giffin,