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Author Topic: Pickover Popcorn  (Read 4118 times)
Description: Exploration of Pickover Popcorn appears to be neglected
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element90
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« on: January 20, 2012, 05:35:04 PM »

I first came across Pickover Popcorn in a Computer Recreations article in the July 1989 edition of Scientific American, there was a small picture and it didn't show much detail. Looking on the web there doesn't seem to be an abundance of Pickover Popcorn Pictures and there don't seem to be many programs that implement Pickover Popcorn apart from the venerable fractint and vision of chaos so I haven't got much to compare with the pictures produced by my software.

From the pictures I've produced it seems to me that exploration of Pickover Popcorn is worth pursuing.




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DarkBeam
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« Reply #1 on: January 21, 2012, 04:19:43 PM »

Well, afaik it's also in Incendia and Xenodream A Beer Cup - and obviously, UltraFractal. Tried also Gnarl, Vine, Glyph variations? They are lovely. smiley
« Last Edit: January 21, 2012, 04:21:37 PM by DarkBeam » Logged

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element90
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« Reply #2 on: January 21, 2012, 05:43:44 PM »

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.





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huminado
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« Reply #3 on: January 21, 2012, 08:24:34 PM »

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

Can someone please explain "orbits"?  I'm assuming "escape time" refers to julia/mandelbrot how long a point takes to fly off to infinity or fall into a hole.  Are "orbits" splines derived by the points used to determine escape time?  I have this gnawing feeling I will hopelessly not follow any explanation, so I now brace myself.

[edit] - okay - I'm done bracing myself.  Instead I will determine to understand it right off the bat.  smiley

« Last Edit: January 21, 2012, 08:27:02 PM by huminado » Logged
element90
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« Reply #4 on: January 21, 2012, 09:13:02 PM »

The orbit is the set of values where each value is the result of an iteration. It took me a while to discover what orbits were, I found out when implementing orbit traps for escape time fractals. For the Mandelbrot algorithm the orbits all start at 0, the number of values in the orbit is either the number of iterations it takes for the value to meet the bailout condition or it is the maximum number of iterations if the bailout condition is not reached. For escape time fractals the values in the orbit are not plotted, for "orbit" type fractals they are, "orbit" fractals" include Pickover Popcorn and Strange Attractors. 
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huminado
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« Reply #5 on: January 21, 2012, 09:32:33 PM »

okay - if I choose some point on the mandelbrot set, and plug it in to the formula it yields another point, which you are saying is the next point in the orbit.  However the points are usually a long ways apart from each other.

Which is why I was guessing these images were splines which used the points in the orbits.  If they were the orbits themselves, I'd expect straight lines, or maybe moire patterns...  Oh... those ARE moire patterns!  smiley

Thank you!  smiley

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David Makin
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« Reply #6 on: January 22, 2012, 01:09:47 AM »

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

Can someone please explain "orbits"?  I'm assuming "escape time" refers to julia/mandelbrot how long a point takes to fly off to infinity or fall into a hole.  Are "orbits" splines derived by the points used to determine escape time?  I have this gnawing feeling I will hopelessly not follow any explanation, so I now brace myself.

[edit] - okay - I'm done bracing myself.  Instead I will determine to understand it right off the bat.  smiley




Much simpler than that.

In a normal escape-time fractal (Mandelbrot or Julia form) separate calculations are done for each pixel on screen such that the starting values for these calculations for each pixel is based on a location relating to the pixel's position.
So for an individual pixel we have a set of fixed values that the calculations (iterations) to be performed are based one - these iterations continue until either bailout is reached (and the pixel is coloured as "outside") or a specified maximum iteration count is reached (and the pixel is coloured as "inside").
Now during these iterations for the individual pixel there is a calculation repeatedly perfomed such that we have a value, we do a calculation with it to give a new value and that new value is fed back into the same process i.e. the calculation is performed again but using the new value again resulting in another value....and so on until bailout or max iterations.
Now the "orbit" is the path followed by the value returned as each of these repeated calculations is performed and in a normal escape-time fractal *every single pixel on screen* has a separate set of these repeated calculations (iterations) and hence a separate orbit.

When rendering single orbits we simply plot the path followed by a single orbit of this type i.e. instead of calculating separate orbits for every pixel to decide if the points are inside or outside we simply have one set of start values and perform one set of these calculations (iterations) but actually plot the points within the orbit i.e. the value from each repeated step during the calculations.
Often orbits are quite boring - spiraling off to infinity or spiraling to a single point or multipli spiraling to a set of points, but sometimes a single orbit can produce a "strange attractor" which is a defined shape for the orbit itself (of Hausdorff dimension >0).

Rendering IFS fractals using the chaos game is similar except here the calculation applied on each iteration is not fixed but rather taken from a possible set of calculations (usually called transforms or transformations).

I hope that helps....
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iteron
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« Reply #7 on: January 22, 2012, 05:04:22 AM »



An orbit is the trajectory of a point or an object either through physical space (like a planet) or through mathematical space (such as the complex plane).

More specifically you are referring to orbital iterations on a plane.

When you construct iterative procedures on a 2D surface such as the computer screen, this is iteration on a plane.  When you  iterate a single point on a plane you are dealing with a relationship between points to generate a sequence.


It's the same concept as when you iterate a formula and also produce an orbit, that is a sequence of real numbers, for example;

Suppose you have following formula;
 


we start with


and generate a sequence of values


that is called the orbit of 0.1 under the iteration of
  



We generate the orbit by iteration so that;



that is

0.1, 0.18, 0.2952, 0.416114...

a sequence of real numbers.

In your case the iteration will produce an orbit that is a sequence of points (x,y).

As was stated by David usually what we are interested in is the fate of these orbits.


« Last Edit: January 22, 2012, 05:09:17 AM by iteron » Logged
element90
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« Reply #8 on: January 24, 2012, 07:42:34 PM »

Getting back to the exploration of orbit plotted Pickover Popcorn, here are some more:




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element90
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« Reply #9 on: February 05, 2012, 06:36:58 PM »

Here are a couple more:




I've also been experimenting with orbit plotting for escape time fractals here is a Mandelbrot.


A Nova


A Quartic Mandelbrot

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