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Author Topic: Hopalong  (Read 3519 times)
Description: Addition of the Hopalong fractal to Saturn and Titan
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element90
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« on: September 13, 2013, 11:46:17 AM »

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
« Last Edit: May 21, 2014, 05:24:27 PM by element90, Reason: Changed file links. » Logged

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« Reply #1 on: September 13, 2013, 12:08:51 PM »

    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.

    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.

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

A very interesting interpretation of this classic fractal type.  I like the way it came out.     cheesy
 
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element90
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« Reply #2 on: September 13, 2013, 12:43:22 PM »

Here's an other one:


https://copy.com/xEFuHDuhpV1Z
« Last Edit: May 21, 2014, 05:27:07 PM by element90, Reason: Changed file link. » Logged

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« Reply #3 on: September 13, 2013, 09:44:35 PM »

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
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element90
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« Reply #4 on: September 13, 2013, 10:01:23 PM »

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
« Last Edit: May 21, 2014, 05:28:40 PM by element90 » Logged

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« Reply #5 on: September 13, 2013, 11:38:16 PM »


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


I assumed that. They are denser than everything else I saw.
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element90
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« Reply #6 on: September 14, 2013, 12:12:36 PM »

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.
« Last Edit: May 21, 2014, 05:30:25 PM by element90 » Logged

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« Reply #7 on: September 16, 2013, 01:34:32 PM »

The Hopalong formula produced as if were a Mandelbrot.


https://copy.com/mH9Ci1bztohv

https://copy.com/EgHAAO9mDsdv

https://copy.com/rn1kV3ej1owk

https://copy.com/VNCsVfbRAjyT

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


https://copy.com/oBDGah0OqINZ
« Last Edit: May 21, 2014, 05:40:32 PM by element90, Reason: Replace file links. » Logged

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« Reply #8 on: September 16, 2013, 04:07:39 PM »

 shocked shocked shocked MIND BLOWING! Some zooms? wink
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« Reply #9 on: September 16, 2013, 04:49:39 PM »

i am unaware about the "hopalong" method, do you have a description what this method does huh?
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element90
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« Reply #10 on: September 16, 2013, 05:33:26 PM »

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

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.
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« Reply #11 on: September 16, 2013, 05:36:53 PM »

so, it is like a buddhabrot ?!
dont confuse mandelbrot/julia cheesy wink the mandelbrot set is all points at (0,0) of julia sets wink
so, you do a julia orbit graphing wink
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element90
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« Reply #12 on: September 16, 2013, 05:49:45 PM »

Quote
MIND BLOWING! Some zooms?

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.

« Last Edit: May 21, 2014, 05:42:05 PM by element90 » Logged

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element90
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« Reply #13 on: September 16, 2013, 06:09:18 PM »

Quote
so, it is like a buddhabrot ?!

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

Quote
dont confuse mandelbrot/julia cheesy wink the mandelbrot set is all points at (0,0) of julia sets

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.

 
Quote
so, you do a julia orbit graphing

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/
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« Reply #14 on: September 17, 2013, 04:35:12 AM »

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

It takes a long time fractaler, and one very experienced (like me  smiley), 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

    so, it is like a buddhabrot ?!
    dont confuse mandelbrot/julia cheesy wink  the mandelbrot set is all points at (0,0) of julia sets  wink
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.     sticking out tongue
 
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