About the formulas.
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Colours algorithms:
- NumberSeeker colouring.
This realy is just standart exponent smoothing of first iterations and with "orbit trap" center.
Maybe exponent smoothing is one of the best colour methods as it produces smooth fractals
and don't do same things on all the fractals.
This includes features I found usefull with kali /ducks pattern sets.
Numberseeker becouse if you place number 2, formula marks all -2 it founds with dots.
In first mandelbrot iteration there are one dot, in second - two dots, 4 dots...
Alsou this differentiates negative and positive sides of power 3 mandelbrot.
Calculating colours from just few iterations in mandelbrot formula allows to smooth
out noisy details, but then zoom in will require to increase the iteration value.
- TwinLamps.
Since exponent smoothing is somewhat plain I wanted something more advanced.
Then in mind came the word "Twin Lamps". So lets combine two good methods.
Divided exponent smoothing with fractal dimension from Kerry Mitchells statistics.
After fine tuning exponent smoothing is divided by 1+ statistics, and I implemented few levelling.
Fractal dimension of exp- is almoust sole exponent smoothing,
ln leveled z is stronger and modulus z is of original strenght.
Only default exp- and large bailout value will create very smooth gradient.
Square root a bitt weakens exponents smoothing, tanh weakens it strongly.
This algorithm reveals more details than exp smooth, but
it don't goes well with newton fractals. Works with kali/ducks patterns.
Removed bug creating no colour spots.
Here is the result:
http://edo555.deviantart.com/art/Fractal-cloud-Angel-279321918
- TwinLamps Direct (c) aka Direct Exponent Smoothing (c).
Looking at help files glorifying direct colouring I wanted to make my own;)
Didn't wanted to make something like of others, so I even didn't sneeked into code
of others. Just remembered that NASA coulours astrophotos by low values giving to
red channell, middle to green and high to blue.
Hmm, exponent smoothing could do the same, just e could be changed
with something different. Using just real numbers mandelbrot looks like shining sun,
but including real negative and imaginary numbers it generates enought differences
to make very colourfull "cosmic" image. y=(-2)^x is no more an exponent,
graph of this is a periodic 3 dimensional complex number curve.
Downside is, that the colours are influenced by everything, iteration number,
exponents, formula type e.t.c.
The larger the number of exponent base, the less of that colour is used. 0 means,
that 0 of that colour is used. It is just like 1/2, 1/3, 1/4, 1/5...
Real bases tend to stick to insides, negative to middle out, and imaginary further outside.
Default setting produce white insides, but decreesing colour iteration number to < 15
decreases the shine and reveals much more details. Or does increase of exponent bases.
Colouring method ignores all the colour settings (density, transfer etc) and gradient.
Good for layering with pallete based colourings, coz it generates far more colours than any
gradient (palette),and it can create nice shining effect, but it lacks tonality of gradient.
Removed bug creating no colour spots.
Description: http://www.fractalforums.com/new-theories-and-research/direct-colouring-exponent-smoothing-2d/
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Fractal formulas:
- MalinovskyDecoFractal (c)
Yaeh, I know, the name is pretty lame. It was found many years agou in Fractal Explorer.
I was trying to implement sinh function into the fractals equation,
to produce something mandelbrot like.There are 2 ways to do it,
but only this is bailout stable. It is built in the last version of Fractal Explorer.
Went years, I totally forgotten fractals, but now I re-implenened it back.
Throught it lacked interesting broken satelytes at the corners
as in FE, which produced planetary images. Newer soft uses z=0 as calculation start,
older soft uses z= pixel as start (value of first iteration of mandelbrot),
so I added z= pixel as choice. It have the same spirals as mandelbrot,
exept without elephant valley, but it have hudge number of julia like small satellyte fractals
flying around the main set and each other. Every spiral will contain some satelites,
some of pretty interesting geometry. Each satellyte have a small mandelbrot set
in its core. Second feature is leaves pattern.
Pattern could be replicated by bailout formula: abs(real(z))+abs(imag(z)) <= bailout.
Here is zoom in:
http://www.youtube.com/watch?v=CDlJjy9TybA
- Menorah fractal
It is circle inverted version of modified spider formula from Fractal Explorer.
Fractal Explorer spider formula is a bitt more sophisticated than original from Fractint,
producing julia set looking as Fractints spider formula. Inverted version is a bitt more
interesting as this candlestic fractal is infinite. With large enought maxiter and bailout
you can zoom out for days not having not enought precise numbers limitation of zoom in.
There are interesting spirals zoomed in or in quaternion numbers and in 3D.
Larger powers means more but smaller "candlesticks".
Zoom out:
http://www.youtube.com/watch?v=EU3eEpxISb0
- StarBrot
Alsou an inverted fractal, this produce star made of fractal. IMHO the star looks much
like natural star of see. Star sides are equation power -1.
Coefficient allows to modify star geometry, imaginary factor curves the star,
real streches the fractal. Main interest are shape, as zoom in reveals the
same spirals as countless other mandelbrot like fractals. All julias are pretty the same
and of the same size (inside of radius =1).
Example:
http://edo555.deviantart.com/gallery/34751689#/d4n48di
- Vector Mandelbrot (c)
Unit vector is z/|z| but mathematicaly meaningless functions alsou produces interesting things.
It came to be when I placed equation generating a bitt realistic q surfaces after the mandelbrot.
Originaly it was intended as a way how in every iteration to add 0.1 of the same sign to z.
With positive factor this tweak produced unkempt mandelbrot with grass like stalks.
Negative value produced mandelbrot with rings on its stalks. Putting
abs(real(z))+abs(imag(z)) instead of cabs(z) aka modulus of z turns rings into squares
and triangles. Coresponding julias are very unusual consisting of rings.
Interesting is that cabs function deffined with some larger paired power like 8
insted of square produces 'pillow' squares with round edges.
It looks that this tweak redefines what is power of z.
Before I downloaded UF, this formula nicely implemented as Unit Vector Tweak Mandelbrot by
Kerry Mitchell, and then with more functions as Unit Vector Explorer by Toby Marshall.
Name "Unit Vector" was coined by Kerry Mitchel. Before had no idea, what unit vector is;)
It is interesting to watch, how this idea evolves into many different things.
- Newtonian Nova Moon
Is one more inverted variation of Fractal Explorer formula, something like
Newton/Mandelbrot, a variation on Nova fractal formula. Equation are much the same as
of Nova fractal, and having power 2 it do looks like inverted Nova. But unlike Nova,
larger powers produces more fractals and with different shape and stalks.
So power 5 fractal with some factors are black square with mandelbrot like borders.
Julias are of infinite size, some are pretty nice, but much like a mandelbrot julias.
Here is example pic of moon, but made in FE:
http://edo555.deviantart.com/gallery/34751689#/d4lto7v
- Bumblebrot (c)
Formula have two switchable equations, bumblebrot and multi power tricorn.
Tricorn fractals have interesting shapes but pretty bleak zooms.
But inverted tricorns looks like flowers with petal number = power + 1.
IMHO inverted pentacorn is the most realistic flower.
Multi-power-corns alsou have some julias unlike of multi power mandelbrot.
Bumblebrot are 2 popular formulas, talis and tricorn combined together,
it have unusual somewhat assymetric shape and lots of very complicated julias.
It have properties of both tricorn, talis and mandelbrot, so zoom in do reveals spirals
and it have mandelbrot like valleys. But unlike in 'normal' fractals all it's lines are curved,
and it is not rotated by 'normal' 45 degrees, even if it looks so, and it is assymetrical
even if looking symetrical. + an unit vector for more floral images.
Here is direct coloured julia:
http://edo555.deviantart.com/#/d4nz879
- Zuzubrot (c)
Mandelbrot shaped square pattern set similar jet unlike of kalisets. Like kalisets, details are
revealed by exponantial smoothing and having small iteration numbers, about 10 - 50,
or colouring by first iterations. Having bailout value of 12 shows square flakes and crosses,
bailout value of 1200 with abs function and 0.5 as calculation start reveals intricate
rectangular urban pattern. Julias are like of mandelbrot set, but right angled.
The formula is a result of mistake, made when trying to implement version
of true 3D mandelbrot set based on intriguing work of Francisco de Asis Fernandes Diaz
about cyclic nature of numbers (negative numbers having power cycle of 2, and imaginary of 4).
http://www.fractalforums.com/index.php?topic=9842.0
And by mixing up real negative x part with an imaginary y.
Since at first I had doubts about this formula, I wanted it to be at the end of list, so searched
for a word with z. Couldn't find good one, so named it zuzubrot, coz zuzu sounds pretty charming.
- Walbrot
4 advanced fractal formulas by Fracmonk aka Jeffrey Barthelmes
From this thread:
http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/
First two are even and odd power multibrots having many power shapes in single connected set.
Nice complex julias. Even formula + determinant =2 is very long fractal 'mandelsword'.
Second two are foam like sets with different power shapes and soap bubble julias.
Julia set of soap cubed have small stars mixed in foam.
Probably kalisets are derived / based on soap squared formula.
Extreme complexity of fractal means that just very small change of julia seed can drasticly change julia,
and it is hard to find right colour method. So lower colour density or chose sqrt as colour transfer.
Smooth (mandelbrot) requires exponent beeing set to 8 for even power, 21 for odd power,
2 for soap squared and 3 for soap cubed equations.
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Copyright:
You may redistribute this algorithm, modify, derivate or use comercialy as you wish as long as you give proper credits.
Or simpler: this is result of hours of my intellectual work, it is released to be used, do with code whatever you like, just when modifying put the name of the original author (me).
p.s.
And sorry for my bad English;)