These are plots of orbits, these are not the escape time variants.

I have two versions of the Pickover Popcorn formula, 4 function and 6 function.

x(n+1) = x(n) - A*f1(y(n) + f2(B*y(n)))

y(n+1) = y(n) - C*f3(x(n) + f4(D*x(n)))

x(n+1) = x(n) - A*f1(y(n) + f2(B*y(n) + f3(C*y(n))))

y(n+1) = y(n) - D*f4(x(n) + f5(E*x(n) + f6(F*x(n))))

where x(n) is the value of x at iteration n, y(n) is the value of y at iteration n, A to F are real numbers, f1 to f6 are one of sin, cos or tan.

x(0), y(0) is the starting point of the orbit which is a location on a grid, the points x(1), y(1) to x(N), y(N) are plotted if the co-ordinates lie with in the display area, N is the length of the orbit (typically 50 iterations). Orbits are calculated for all the locations in the grid. There are two good methods of colouring these images:

i) accumulation - based on the number of times a location in the display area is visited by an orbit.

ii) average - the colour to plotted is periodically changed, the colour for each point is accumulated for that point as it is visited by orbits, the final colour is the accumulated value divided by the number of visits.

The biggest problem with plotting these 'fractals' is that zooming into the images results in loss of detail when the display area and the calculation area are the same as orbits that would have passed through the reduced area aren't calculated as the starting point of the orbit is outside the display area. To reduce this problem I use a calculation area that is nine times the size of the display area, this results in much better images when zooming in but it still suffers from data loss. To get good images a deeper zooms the calculation area should be increased in relation to the display area which of course greatly increases the length of time required to generate the picture.

Here are some more examples, the first two use accumulation colouring and the second pair use average colouring.