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I've now started the development of version 4.1.0 of Saturn and Titan. This will be a minor feature release and will mostly provide more fractals to play with. The first fractal type to be added is the Hopalong fractal, here it is in xfractint.


https://copy.com/lQRktmPZR0v0

https://copy.com/uAgL8vg637Gn

Hopalong is an orbit plotted fractal starting at (0, 0) and there is usual just one orbit plotted. The formula is added to Saturn and Titan so that it conforms to all the other fractals (except Lyapunov) and as such it can be calculated using the Julia algorithm and a multitude of orbits plotted each with a different starting point. The best results are using the "average" colouring method.

This is Saturn and Titan's version of Hopalong:


https://copy.com/GQySigC9QZLz

https://copy.com/gQ26VcXovsxt

https://copy.com/TIUsX7xMyd68

https://copy.com/mIW9VvlBM3G0

The image was fed into Titan and expanded to 12000x8000 and this is a scaled version (1200x800):


https://copy.com/QgIf0yXQLPeU

Here's an other one:


https://copy.com/xEFuHDuhpV1Z

Yes I did. These take a little longer as I use a large number of orbits, here's an other one.


https://copy.com/3NL3ZLxzWiHw

I tried the same technique with quadruptwo (xfractint/fractint name) but I'm not happy with the results. I'm going to add a new class of fractals to Saturn and Titan called "Single Orbit" fractals. I may retain Quadrup 2 even though I not happy with the results for the multi-orbit plotted fractal, it depends on what I get for the Mandelbrot algorithm version and the ordinary escape time versions (both Mandelbrot and Julia algorithms).

I've also found more variations of "Hop fractals" to play with: http://www.fluxury.com/a5x/hopfrac1.htm

I'll post an orbit plot of Hopalong using the Mandelbrot algorithm soon.

Here's an other multi-orbit plotted Hopalong:


https://copy.com/UMhtKFJSjpqe

The size of fractal is controlled by using a bailout limit.

  MIND BLOWING! Some zooms?

Hopalong isn't a method it the name of an old an old fractal type which I found in xfractint and like strange attractors it is simply a plot a single orbit starting at (0,0). For my program I used the same formula and instead of using a single orbit plot I used it with Mandelbrot and Julia algorithms, Saturn & Titan can also produce orbit plots of the Mandelbot and Julia algorithm calculations.

Not yet, I've been generalising the Hopalong formula so that it can handle all the formula variations listed here: http://www.fluxury.com/a5x/hopfrac1.htm. In the process I discovered that the Hopalong formula I'd actually implemented was:

x(n+1) = y(n) - signum(x(n)) * sqrt(abs((b * x(n)) + c ))
y(n+1) = a - x(n)

and not

x(n+1) = y(n) - signum(x(n)) * sqrt(abs((b * x(n)) - c ))
y(n+1 = a - x(n)

The generalised formula is:


https://copy.com/8QA0ypsALZ9k

One thing to note is the signum function, the function is called sign on the Hop-Fractals web site (see link above) and according to its definition returns 1 for +ve values otherwise it returns -1, I can only assume that it treats 0 as positive. The signum function differs in that it returns 0 if the input value is zero.

It takes a long time fractaler, and one very experienced (like me  ), to know about Barry Martin's "Hopalong", which is an "orbit" type fractal like Lorenz.  It has been around for many years.  And you can find out its details by reading the FractInt documentation, or start at the following web page:

    http://www.Nahee.com/spanky/www/fractint/martin_hop_type.html

As usual, I believe it is you, Chris that is confused, but that is probably only from lack of knowledge, being too impetuous, and speaking before really thinking things through.