Welcome to Fractal Forums

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/lQRktmPZR0v0 (https://copy.com/lQRktmPZR0v0)
(https://copy.com/uAgL8vg637Gn)
https://copy.com/uAgL8vg637Gn (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/GQySigC9QZLz (https://copy.com/GQySigC9QZLz)
(https://copy.com/gQ26VcXovsxt)
https://copy.com/gQ26VcXovsxt (https://copy.com/gQ26VcXovsxt)
(https://copy.com/TIUsX7xMyd68)
https://copy.com/TIUsX7xMyd68 (https://copy.com/TIUsX7xMyd68)
(https://copy.com/mIW9VvlBM3G0)
https://copy.com/mIW9VvlBM3G0 (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)
https://copy.com/QgIf0yXQLPeU (https://copy.com/QgIf0yXQLPeU)

A very interesting interpretation of this classic fractal type.  I like the way it came out.     :D
 

Here's an other one:

(https://copy.com/xEFuHDuhpV1Z)
https://copy.com/xEFuHDuhpV1Z (https://copy.com/xEFuHDuhpV1Z)

cool, I like those, 'cause they were pretty fast to create on my 8MHz Atari and still looking nice.
Did you notice this one? http://www.fractalforums.com/fractal-website-of-the-month/hopalong-tunnel/msg63954/#msg63954 (http://www.fractalforums.com/fractal-website-of-the-month/hopalong-tunnel/msg63954/#msg63954)

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)
https://copy.com/3NL3ZLxzWiHw (https://copy.com/3NL3ZLxzWiHw)

I assumed that. They are denser than everything else I saw.

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 (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)
https://copy.com/UMhtKFJSjpqe (https://copy.com/UMhtKFJSjpqe)

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

The Hopalong formula produced as if were a Mandelbrot.

(https://copy.com/mH9Ci1bztohv)
https://copy.com/mH9Ci1bztohv (https://copy.com/mH9Ci1bztohv)
(https://copy.com/EgHAAO9mDsdv)
https://copy.com/EgHAAO9mDsdv (https://copy.com/EgHAAO9mDsdv)
(https://copy.com/rn1kV3ej1owk)
https://copy.com/rn1kV3ej1owk (https://copy.com/rn1kV3ej1owk)
(https://copy.com/VNCsVfbRAjyT)
https://copy.com/VNCsVfbRAjyT (https://copy.com/VNCsVfbRAjyT)

Expanded using Titan to 12000x12000 and scaled to 1000x1000 for fractal forums produces this:

(https://copy.com/oBDGah0OqINZ)
https://copy.com/oBDGah0OqINZ (https://copy.com/oBDGah0OqINZ)

 :o :o :o MIND BLOWING! Some zooms? :dink:

i am unaware about the "hopalong" method, do you have a description what this method does ???

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.

so, it is like a buddhabrot ?!
dont confuse mandelbrot/julia :D ;) the mandelbrot set is all points at (0,0) of julia sets ;)
so, you do a julia orbit graphing ;)

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 (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)
https://copy.com/8QA0ypsALZ9k (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.

No, it is just a single orbit, There is nothing stopping anybody using it with Buddhabrot method.


I not not, I'm referring to the general algorithms used to produce the Mandelbrot and its Julia sets, they can be used with any formula.

 
Yes, also Mandelbrot orbit graphing, which produce Buddhabrot and anti-Budhabrot without the use of random numbers. The disadvantage of the method is that it commonly produces gridlines in the resulting image. I use the same method for producing orbit plotted fractals based on strange attractors see http://element90.wordpress.com/2013/05/17/strange-attractors-an-alternative-approach/ (http://element90.wordpress.com/2013/05/17/strange-attractors-an-alternative-approach/)

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 (http://www.Nahee.com/spanky/www/fractint/fractint.html) documentation, or start at the following web page:

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

sure paul  :secret:, i see, its a formula with a function for each axis -   
regarding the description from encrypted i was thinking it was plotting just the orbits of mandelbrot function starting at (0,0)
which basically describes the "mandelbrot" method, which is as we all know examining the julia set for that seed :D i knew that fractal from old fractint, sure, but i was unaware of its method

Here are couple of zooms:

(https://copy.com/EOJ29262E46D)
https://copy.com/EOJ29262E46D (https://copy.com/EOJ29262E46D)
(https://copy.com/EXOip9ctdTF9)
https://copy.com/EXOip9ctdTF9 (https://copy.com/EXOip9ctdTF9)

If you do orbit plot, maybe this is usefull (or maybe there are other ways to do orbit plot).

http://www.fractalforums.com/general-discussion-b77/not-all-random-numbers-are-equaly-random/ (http://www.fractalforums.com/general-discussion-b77/not-all-random-numbers-are-equaly-random/)