Pickover Popcorn

Pickover Popcorn fractals are different to most of the fractals implemented in Saturn and Titan as they are ‘orbit’ fractals rather than escape time fractals. The fractals are constructed by plotting a point for each iteration of a pair of equations, the set of points plotted for a number of iterations is called the ‘orbit’ hence the term ‘orbit’ fractals. The images produced by Saturn are formed not by a single orbit but by a multitude of orbits of a set length each with an initial value determined by the position in a a grid super-imposed onto a rectangle, the orbits are plotted by starting in the top left hand corner and processing a row of points (corresponding to the initial value of an orbit) before starting on the next row and continuing raster fashion until all the orbit have been plotted.

Here is typical pair of equations for Pickover Popcorn:

xn+1 = xn - hsin(yn + tan(ayn))
yn+1 = yn - hsin(xn + tan(bxn))

Pickover Popcorn h = 0.05 a = 3 b = 3 orbit length = 35

The typical pair of equations is implemented as “Pickover Popcorn” in Saturn and Titan, in addition there are three more variants containing four trigonometric functions:

xn+1 = xn - hsin(yn + sin(ayn))
yn+1 = yn - hsin(xn + sin(bxn))

xn+1 = xn - hsin(yn + cos(ayn))
yn+1 = yn - hsin(xn + cos(bxn))

xn+1 = xn - hcos(yn + cos(ayn))
yn+1 = yn - hsin(xn + sin(bxn))

these pairs of equations are implemented in Saturn and Titan as “Pickover Popcorn 2”, “Pickover Popcorn 3” and “Pickover Popcorn 4”, in addition there is also “Pickover Popcorn 5” which uses six trigonometric functions:

xn+1 = xn - hsin(yn + sin(ayn + sin(ayn)))
yn+1 = yn - hsin(xn + sin(bxn + sin(bxn)))

It is clear that the general form of these pairs of equations can be generalised:

xn+1 = xn - hf1(yn + f2(ayn))
yn+1 = yn - hf3(xn + f4(bxn))

xn+1 = xn - hf1(yn + f2(ayn + f3(ayn)))
yn+1 = yn - hf4(xn + f5(bxn + f6(bxn)))

The functions f1 to f6 can each be assigned a function, either sin, cos or tan so the 5 “Pickover Popcorn” fractals implemented in Saturn & Titan versions 1.0 can be replaced with two generalised fractals “Pickover Popcorn 4 Function” and “Pickover Popcorn 6 Function” in version 1.1 which will require function parameters to be implemented. Instead of being restricted to the just 5 types of “Pickover Popcorn” version 1.1 will have 81 variations of the four function version and 729 variations of the six function version.

To reveal structure in Pickover Popcorn fractals some means of assigning colour is required as using a single colour will result in areas coalescing together in uniform coloured blobs and in most cases a rectangle of a single colour. Saturn & Titan provide three colouring methods for these fractals, the method used in the first picture is “accumulation”, the colour is selected based on the number of times the same point has been plotted i.e. where orbits intersect. The colour is periodically changed in the other two methods “last” and “average”, for “last” the colour used is the colour current when a point was last plotted, “average” averages all the colours plotted for a given point. The period between colour changes is specified by the colour interval which is a count of iterations, the default is the orbit length multiplied by the number of points in a row. The best results are obtained using “accumulation” and “average”, the results for “last” are lackluster so it may be replaced if I come across a better method. Note: changing the colour map will restart fractal calculation when the “average” and “last” methods are used, also when zooming in or out the colours used in the resulting picture will be different, this is not the case for “accumulation”.

1. Pickover Popcorn

2. Pickover Popcorn

3. Pickover Popcorn 2

4. Pickover Popcorn 3

5. Pickover Popcorn 3

6. Pickover Popcorn 4

7. Pickover Popcorn 4

8. Pickover Popcorn 5

9. Pickover Popcorn 5

1. h = 0.05 a = 3 b = 3 orbit length = 35 average colouring
2. h = 0.05 a = 3 b = 3 orbit length = 35 accumulation colouring
3. h = 0.05 a = 7 b = 5 orbit length = 80 average colouring
4. h = -0.225 a = 6.7 b = 3.75 orbit length = 50 accumulation colouring
5. h = -0.25 a = 6.7 b = 3.75 orbit length = 50 accumulation colouring
6. h = 0.05 a = -2.25 b = 3 orbit length = 121 average colouring
7. h = 0.05 a = -2.25 b = 3 orbit length = 121 accumulation colouring
8. h = 0.75 a = 2 b = 2 orbit length = 50 accumulation colouring
9. h = 0.75 a = 2 b = 2 orbit length = 50 average colouring

Saturn & Titan versions 1.0 also contain escape time fractal versions of the five Pickover Popcorn orbit fractals called “PP Mandelbrot” and “PP Julia”, for version 1.1 these fractals will also be generalised in the same way as the orbit fractal versions.
 

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