About the formulas.

1.Colour algorithms
2. 2D fractal formulas.
3. 3D fractal formulas.

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Colour algorithms:


- NumberSeeker colouring.
This realy is just standart exponent smoothing of first iterations (orbits) and with "orbit trap" center.
Exponent smoothing is one of the best colour methods as it produces smooth fractals and
 don't do same things on all the fractals. Especialy with pattern formulas, like kali set. 
Included features I found very usefull with kali like pattern sets. Coded having 3D in mind.
Adding some number to z creates nice dots on mandelbulbs. Numberseeker becouse if one 
places number 2, formula marks all -2 Zs it founds with dots.
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. 
Goes nicely with mandel- everything - brots, patterns and with most of 3D including mandelbox.
Modulus of Z is the same as quaternion orbital colouring.


- TwinLamps. 
Since exponent smoothing is somewhat plain I wanted something more advanced. Then in mind came 
the word "Twin Lamps". So lets combine two 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 for both methods. 

Fractal dimension of -exp is almoust sole exponent smoothing, ln leveled z is stronger and 
modulus z is of original strenght. Square root a bitt weakens exponents smoothing, tanh weakens 
it strongly. This algorithm reveals more details than exp smooth, jet are able to be pretty smooth, but
it don't goes well with newton fractals, not so great with kali patterns, and useless for mandelbox. 
Here is the result: 
http://edo555.deviantart.com/art/Fractal-cloud-Angel-279321918 


- TwinLamps Direct (c) aka Direct Exponent Smoothing (c).
Looking at UF 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. 
Exponent smoothing can do the same, just e must be changed to something different. Using both 
real, 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 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. But will 0 mean, 
that 0 of that colour is used. It's 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 more details. Or does increase of exponent bases.

Colouring method ignores all the colour settings (density, transfer etc) and gradient.
This creates very smooth all posible colours included buddhabrot like outsides, but mostly white 
insides with just no green or no blue stripes. In 3D RGB colours must be checked, but it makes very
colourfull mandelbulbs, but pretty bleak quaternions since they are outsides, and extremely dark
 but with all posible colours present mandelboxes.
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. 
 
Description: 
http://www.fractalforums.com/new-theories-and-research/direct-colouring-exponent-smoothing-2d/  


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2D 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, 
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 leave like pattern of outsides. 

Have hypercomplex 3D version, but it is pretty bleakand slow.
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
is a bitt more sophisticated than original from Fractint, producing julia set looking as Fractints
formula. This inverted version is a 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 spirals zoomed in. Larger powers adds more "candlesticks", 
but requires coefficent to be set to -5 or more. 

Coded having 3D in mind, so each power versions have different equation, hence switchable powers.
3D of this fractal is infinite saturnian rings.
Zoom out: 
http://www.youtube.com/watch?v=EU3eEpxISb0 


- Tricorn 
Formula have switchable equations, multi power tricorn fractals and bumblebrot.
Tricorn fractals have interesting shapes but pretty bleak zooms.
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 (c) are talis and tricorn combined together,
it have unusual somewhat assymetric shape and lots of very complicated julias. 
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, and it is assymetrical

Coded having 3D in mind, so each power versions have different equation, hence switchable powers.
http://en.wikipedia.org/wiki/Tricorn_(mathematics)
 + an unit vector for more floral images.
Here is direct coloured julia: 
http://edo555.deviantart.com/#/d4nz879 


- Burning ship fractal.
Mathematicaly meaningless mandelbrot variation producing elaborate pattern. Having larger power
fractal features deetiorates. Some nice tree like julias.
Coded having 3D in mind, so each power versions have different equation, hence switchable powers.
http://en.wikipedia.org/wiki/Burning_Ship_fractal


- 2D formula colection:
Bunch of formulas, some pretty nice but aren't even in very large UF database.

- Fractovia. Part of this fractal was suggested as flag of "Fractovia". Arctanh function creates nice
angular dendritic pattern. Alsou nice patterned julias.

- Kalisets. Pattern formulas found by Pablo Roman Andrioli, Kaliset z=abs(z)/c+c goes as julia set,
variation z=sinh(abs(z^-1))+c as mandelbrot set, Kaliset z=abs(zC+D)+1/abs(zC+D)\nz=z^z-z^5+c 
should go with both. First formula is better, much easyer to found some patterns and create more diverse
ones, but second formula reveals an interesting fractal formation "Entangled Trees". Third is somewhere
between the two. Pattern julias appear with julia seed where in mandelbrot set are patterns.
Paterns are revealed by exponent smoothing or alike and having maximum iterations
set to some 10-40 (or setting colour iterations of twin lamps and numberseeker).


- Advanced fractmonk formulas. z=(((((z^2c)+D)^2)-D)^2)-D , z=(((((z^3)c+D)^3)+D)^3)+D and 
Fm z=(z*z*C+D)+1/(z*z*C+D)
by Fracmonk aka Jeffrey Barthelmes. More about different power shapes in same brot are in thread:
http://www.fractalforums.com/new-theories-and-research/is-there-anything-novel-left-to-do-in-m-like-escape-time-fractals-in-2d/
And the posts
http://www.fractalforums.com/index.php?topic=4881.msg27524#msg27524
http://www.fractalforums.com/index.php?topic=4881.msg25998#msg25998 


- Talis. Implementation of Fractal Explorer formula z=z*z/(K+z)+c. Talis are from tails, resembling
 sheep fat tails. (Wikipedia sayes fat tail distribution was researched by Mandelbrot themself.)
This is one of the most popular Fractal Explorer formulas, jet this implementation don't have so much
possibilities as Fractal Explorer implementation. Very long julia sets.

- MandelTeleBrot, MandelGrassBrot, CosineMandelGrass is the same as Vector Mandelbrot.

- Newtonian Moon
Inverted variation of Fractal Explorer formula, a variation on Nova fractal formula. 
Having power 2 it do looks like inverted Nova. But unlike Nova, larger powers produces more 
but flatter moons. Power 5 fractal with modified factor create 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 FE moon: 
http://edo555.deviantart.com/gallery/34751689#/d4lto7v 

- Star
Star made of fractal. Star sides are equation power -1. 
Coefficient allows to modify star geometry, imaginary factor curves the star, 
real streches the fractal. 
Example: 
http://edo555.deviantart.com/gallery/34751689#/d4n48di


- Vector Mandelbrot (c)
Unit vector is z/|z| but mathematicaly meaningless functions alsou produces interesting things. 
Positive factor produces unkempt mandelbrot with grass like stalks. Negative factor produces 
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 larger even power like 8 insted of square produces 'pillow' squares with
round edges.  
Small positive factor about 0.15 creates grass like mandelbrot, negative factor more than -0.3 creates
mandelbrot sets with rings or squares, but factor -1 generates julia sets made of rings. Imaginary 
values rotates things. 
Some ideas are from  Kerry Mitchel and Toby Marshall implementation of this tweak.


- 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.


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3D Fractal formulas:
(Malinovsky3D)

- Gilgamesh Head Fractal. (c)
Fractal (in cube power) resembles some mayan or sumerian relict (both had round headed statues),
so the "Gilgamesh". Alsou an interesting julias and square power fractal.
This one was result of another version of true 3D mandelbrot. Equation rises z in some power, +C, then
switches number parts, again raises in power, switches number parts till all 4 quaternion 
number parts had been both real, imaginary i, j and k. Hence super rotated interference quaternion.
Having enermous power in single iteration it needs just 2 iterations to generate the final shape, so
low orbit values will produce smooth colouring. Zoomed it repeats circle interference pattern.
In 2D this produce nothing, becouse in 2D Chaos Pro optimises off unused j and k parts, so just zero.

Having powers 3,3,3,3 and K4=-1 generates robotic Bender head, K2=-0.5 & k3=0.5 Buddha's head.

Description:
http://www.fractalforums.com/3d-fractal-generation/gilgamesh-head-the-3d-mandelbulb-like-fractal/
Animation:
http://www.youtube.com/watch?v=0F2_V_5zmZc 


- Mandelbulb DE variations.
The basic mandelbulb equation with z based bailout and few additions. 1st is c=recip(pixel); sphere 
inversion of mandelbulb. 2nd is unit vector formula for mandelbulb working just like unit vector 
tweak for a mandelbrot. Like for mandelbrot z=z+ factor*z/|z| negative factor will enlarge mandelbulb
and positive will shrink the bulb. Both + and - 1i creates seacucumber shaped bulb. 
Non-mathematical function like z=z+z/round(z) having factor 1 generates right angled bulb. 
Z=ZxZ/|Z| had no factor ir must be 1. 3rd is z=z*(golden ratio) generating  irregular surfaces. It affects
only last z and bailout, but will increase colour speed. Strenght depends on bailout value, the lower is
the bailout, the more irregular will be the surface. 
Added checkbox to use 8th power modulus in triplex calculations instead of normal square modulus. 
This approximates the bulb to cube shape, hence the "Boxed bulb".


- Quaternion Mandelbrot. (c)
This generates ordinary 2D mandelbrot set, version of 3D mandelbrot set, and ordinary quaternionic julia.
Equation is based on notation, that in quaternionic numbers y^2(=i^2)  = z^2(=j^2) so quaternion
number mandelbrot is just boring rotation surface. 
This formula in 3D do have features of 2D mandelbrot, so zooming in reveals mandelbrot spirals, 
but only in bulbous side regions and as 3D are insides of 2D, they tend to disapear.
http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/ 


- zAlter 3D mandelbrot. (c)
Mandelbrot formula in alternative 3 part complex numbers. Defined that i^2=-1, j^2=-i, i*j=1, j beeing
sort of pseudo imaginary unit. This is something in between quaternion numbers and tricomplex
numbers, so it generates interesting shaped mandelbrots (but so so julias). Alsou zooming into 
and checking "inside" reveals elaborate inner structure of the shape. 
Had been able write just 2, 3, 4 powers, as it requires manual algebraic multiplication according 
to number definition and formula lenght grows with each additional power.  (Ar, Bi, Cj) x (Ar, Bi, Cj)
In 2D formula it creates just ordinary escape time mandelbrot fractal.


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Alsou 2D formulas exept those in 2D colection are written in quaternion friendly manner, so will 
produce at least something 3D. Menorah and some julias are pretty good.


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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 a plain text: this is result of hours of my intellectual work, it is released to be used, do with code whatever you like, just when modifying mention the name of the original author (me).

p.s.
And sorry for my bad English;)