Calcyman's Idea (2D)

[stigomaster]

Warning: Horrible pun ahead!

I guess the Mandelbrot set got a little makeover... It's such a nice tan.

[kram1032]

First fractal joke being funny so far

[LesPaul]

Haha, stigoman. 

Bib, where did you find those images?  Are you saying that they're not z*tan(z)?

[bib]

lol

I did these images last year, with different formulae: Supernova, Herman Ring, Phoenix-Halley-Nova, so not z*tan(z)

[stigomaster]

This is the bifurcation diagram of x_n+1 = x_n * tan(x_n) + x₀. It features a moderately sized logistic map-ish thing in the middle along with many others of different sizes spread around the entire set. View size (-5, -3.75)-(5, 3.75)

[kram1032]

strange bifurcation diagram
How wide is that range that is clearly not chaotic, in the middle?

[stigomaster]

It goes from about -1.11 to about 0.24, a width of a tiny bit less than 1.35.

[Timeroot]

Interesting diagram; rather than "bands" of chaos or periodic "windows", it looks more like chaotic "curtains" and periodic "brdiges". 

For those of you on the UF mailing list, someone recently posted that he had seen the "mandelbfields" - those weird, flying Mandelbrots we saw at 1:16 in Reply #37 - in the phoenixDoubleNova fractal. Oh dear god, fractals are much, much too weird for us mortals... 

[kram1032]

interesting that the biggest "island of order" lies on the lower site of the diagram rather than on the higher one

[Nahee_Enterprises]

For those of you on the UF mailing list, someone recently posted that
he had seen the "mandelbfields" - those weird, flying Mandelbrots....

Actually, I believe that Daniel J. Wills (aka. inkling, enkling, enk, gdanzo) stated the following:

• Turn up the 'Relaxation' (eg > 2.5), a noisy lake will form in the 'inside' area..  this is interesting to explore in itself, it's full of 'mandelfields' (those fuzzy mandelbrot-shaped things - 'mandelfield' is my name for them.  They also contain the 'micro-lakes' - areas where the detail flattens out.)
• Play with the number of iterations - very low numbers will reveal the large structures more, and hide the mandelfields.

[lkmitch]

Here's another view of the same bifurcation diagram--I find it easier to see as black on white.

[bib]

Hi

I post here as an answer to Fractex, in case it might interest anybody.

Fractex asked if I could provide the UF file for the animations I did. I have to apologize because I lost my parameters. As I was playing with the script (which I don't do often) I messed up with the versions and names of the formulae and lost my work . So I did another one quickly. It's not exactly the same and you don't have the animations, but it's something to start again with.

There are 2 layers.
Some suggestions to play and get more familiar with UF (if needed)
- Change the layering mode
- Change and play with the parameters of the the inside/outside coloring of each layer
- change the bailout (NB : to change a parameter value at once for multiple layers, simply multi-select the layers with the shift key before editing the value)
- change the powers (there are two parameters to do z^p1*tan(z)^p2, so for example one could do an animated transition from z²+c to z*tan(z)+c... )
- Pan, zoom, Explore !!!

I probably said it in a previous post, but there are 2 particular things to me in this z*tan(z)+c fractal (by particular I mean very unusual compared to the Mandelbrot set despite the graphical similarities)
1- increasing the bailout has a very original effect, it create new shapes (due to the periodically divergent nature of f(x)=tan(x) I guess)
2- The inside is really full of surprises (these so-called Mandelfields in particular, I have no idea what they are)

Code:

ztanzForFractex {
fractal:
  title="ztanz for fractex" width=1024 height=768 layers=2
  credits="arsene;3/16/2010"
layer:
  caption="Layer 1" opacity=100 mergemode=screen transparent=yes
mapping:
  center=0/0 magn=.1
formula:
  maxiter=100 filename="Standard.ufm" entry="Calcymansidea"
  p_bailout=400000 f_function=tan p_power=1/0 p_power2=1/0 p_start=0/0
inside:
  density=10 transfer=sqrt filename="Standard.ucl"
  entry="GenericDirectColoring"
  p_coloringClass="Standard.ulb:Standard_Smooth"
  p_coloringClass.v_generic=100 p_coloringClass.v_coloring=100
  p_coloringClass.v_gradientcoloring=100 p_coloringClass.power=2/0
  p_coloringClass.bailout=128.0
outside:
  density=2 transfer=cuberoot filename="Standard.ucl"
  entry="OrbitTraps" p_trapshape=point p_diameter=1.0 p_traporder=4.0
  p_trapfreq=1.0 p_trapcolor=distance p_traptype=closest
  p_threshold=0.25 p_trapcenter=0/0 p_aspect=1.0 p_angle=0.0
  p_solidcolor=no
gradient:
  smooth=yes rotation=-28 index=54 color=16711713 index=213 color=0
  index=215 color=16777215 index=222 color=16313190 index=238
  color=11075600 index=361 color=13403696
opacity:
  smooth=no index=0 opacity=255
layer:
  caption="Background" opacity=100
mapping:
  center=0/0 magn=.1
formula:
  maxiter=100 filename="Standard.ufm" entry="Calcymansidea"
  p_bailout=400000 f_function=tan p_power=1/0 p_power2=1/0 p_start=0/0
inside:
  transfer=cuberoot filename="Standard.ucl" entry="Decomposition"
outside:
  transfer=cuberoot filename="Standard.ucl" entry="Smooth" p_power=2/0
  p_bailout=128.0
gradient:
  smooth=yes rotation=-7 index=29 color=6211054 index=148 color=0
  index=225 color=943060
opacity:
  smooth=no index=0 opacity=255
}

And Calcymansidea formula:

Code:

Calcymansidea {

init:
  z = @start
loop:

z=z^@power*(@function(z))^@power2 + #pixel

bailout:
  |z| <= @bailout
default:
  title = "Calcyman's Idea"
  center = (-0.5, 0)

$IFDEF VER50
  rating = recommended
$ENDIF

  param start
    caption = "Starting point"
    default = (0,0)
    hint = "The starting point parameter can be used to distort the Mandelbrot \
            set. Use (0, 0) for the standard Mandelbrot set."
  endparam

  param power
    caption = "Power"
    default = (1,0)

  endparam
  param power2
    caption = "Power2"
    default = (1,0)
    hint = "z=(z^powerbizarre*f(z))^power2"
  endparam
  float param bailout
    caption = "Bailout value"
    default = 4.0
    min = 1.0
$IFDEF VER40
    exponential = true
$ENDIF
    hint = "Try very large values"
  endparam
switch:
  type = "Julia"
  seed = #pixel
  power = power
  bailout = bailout
}

[kram1032]

I just did an Antibuddhabrot style render of the set. It doesn't really do it justice - I'd love to see a highquality one - but it looks great and has a lot of interesting details

If you go directly to dA, you can find a slightly higher resolution version



http://kram1032.deviantart.com/art/A-nice-tan-157868619

[Fractex]

Interesting! Did you try the regular Buddhabrot style render of z*tan(z)+c?

[stardust4ever]

I'd post it but I used an evaluation version of UltraFractal and the whole AVI is stamped with "Evaluation Copy."  I guess it's time to shell out the hundred dollars.  

Those videos are really great, bib.

Just a tip: While UltraFractal is generally more user-friendly and what not and has a better GUI and stuff, ChaosPro can load UF formulas, animate them, and it's free. I just use it when I have a good fractal form UF and want to render it. :-P

Can't wait to see vid!

Hey, thanks for the tip. I've been playing around with ChaosPro for a little over a day now and it is awesome. For a while, I had been mulling over whether or not to chuck out the cash to buy UltraFractal, seeing all the great stuff it can do, until I read your post. Now I know I don't have to. ChaosPro runs very much like a well-polished commercial app, but it's completely free, and by free I mean really 100% free, not "pirate" free.