A list of the latest changes in Mandelbrot set -- mandelbox, mandelbulb, burning ship, nebulabrot

A two-dimensional Mandelbrot Set ——Burning Ship

Use the following iterative formula

(x⁴-6*x²*y²+y⁴, 4*|x|³*|y|-4*|y|³*|x|)

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Mandelbulb

This 3D Of Mandelbrot The set uses the following formula , This should be a kind of super complex number , said “triplex”, Ternary complex

 

Mathematically N Power

among ：

In general ,n take 8.

In a nutshell , The computer expression of the square of a ternary complex number is

xx = (x*x+y*y) * (1-z*z/(x*x+y*y));

yy = 2*x*y * (1-z*z/(x*x+y*y));

zz = -2*z*sqrt(x*x+y*y);

For details, see http://en.wikipedia.org/wiki/Mandelbulb

The following is excerpted from http://www.skytopia.com/project/fractal/2mandelbulb.html

{x,y,z}^n = r^n { sin(theta*n) * cos(phi*n) , sin(theta*n) * sin(phi*n) , cos(theta*n) }
...where:
r = sqrt(x^2 + y^2 + z^2)
theta = atan2( sqrt(x^2+y^2), z )
phi = atan2(y,x)

And the addition term in z -> z^n + c is similar to standard complex addition, and is simply defined by:

{x,y,z}+{a,b,c} = {x+a, y+b, z+c}

The rest of the algorithm is similar to the 2D Mandelbrot!

Here is some pseudo code of the above:

r = sqrt(x*x + y*y + z*z )
theta = atan2(sqrt(x*x + y*y) , z)
phi = atan2(y,x)

newx = r^n * sin(theta*n) * cos(phi*n)
newy = r^n * sin(theta*n) * sin(phi*n)
newz = r^n * cos(theta*n)

...where n is the order of the 3D Mandelbulb. Use n=8 to find the exact object in this article.

 

Above, 16 Step MandelbrotBulb

 

Zoom in on a part

 

 

take n=8 Graphics at that time

 

 

The number of iterations is taken as 100 The situation at that time

“ Golden Grand Canyon ”

 

“ Mysterious cave ”

“ Magic ball cabbage ”

“Mandelbrot The garden ”

Eighth order MandelBulb

Christmas coral ball

Ice cream ~~

shell ~~

“ Frozen hell ”

 

 

 

 

 

 

 

 

 

 

 

3D Of Mandelbrot Set ——Mandelbox

A wonderful video demonstration .“ Fly over the Mandelbrot box ”

http://wimp.com/mandelboxflythrough/

Specific algorithm core code ：

for (each axis)
  if (v[axis]>1) v[axis] = 2-v[axis];
  else if (v[axis]<-1) v[axis] = -2-v[axis];
if (v.magnitude() < 0.5) v *= 4;
else if (v.magnitude() < 1) v /= square(v.magnitude());
v = scale*v + c;

 

 For details, see “ Hypercomplex fractal ”

http://www.bugman123.com/Hypercomplex/index.html

n = 100; norm[x_] := x.x; 

TriplexPow[{x_, y_, z_}, n_] := If[x == y == 0.0, 0.0, Module[{r = Sqrt[x^2 + y^2 + z^2], theta = n ArcTan[x, y], phi}, phi = n ArcSin[z/r];

r^n{Cos[theta]Cos[phi], Sin[theta]Cos[phi], -Sin[phi]}]];
Mandelbulb[c_] := Module[{p = {0, 0, 0}, i = 0}, While[i < 24 && norm[p] < 4, p = TriplexPow[p, 8] + c; i++]; i];
image = Table[z = 1.1; While[z >= -0.1 && Mandelbulb[{x, y, z}] < 24, z -= 2.2/n]; 

z, {y, -1.1, 1.1, 2.2/n}, {x, -1.1, 1.1, 2.2/n}];
ListDensityPlot[image, Mesh -> False, Frame -> False, PlotRange -> {-0.1, 1.1}]

 Above, Lambdabulb

 The iterative formula used in the figure above ：{x,y,z}2 = {x2-y2-z2, 2xy, -2xz}

 The figure above is quaternion Mandelbrot Set , The iterative formula used ：{x,y,z,w}2 = {x2-y2-z2-w2, 2xy, 2xz, 2xw}

 

 

The picture above is called Glynn Julia set

 

 

 

The two pictures above are 4D Bicomplex Mandelbrot Set ("Tetrabrot")

The iterative formula used ：{x,y,z,w}² = {x²-y²-z²+w², 2(xy-zw), 2(xz-yw), 2(xw+yz)}

 

  

The above three pictures are called NebulaBrot, yes BuddhaBrot Three dimensional form of

These figures are made up of 20 100 million points are calculated and rendered

 

 

The picture above is called 3D Christmas Tree Mandelbrot Set, The iterative formula used ：

{x,y,z}ⁿ = rⁿ{cos(θ)cos(φ), sin(θ)cos(φ), sin(φ)}
r=sqrt(x²+y²+z²), θ=n atan2(y,x), φ=n atan2(z,x)

 

 

 

{x,y,z}ⁿ = rⁿ{cos(θ)cos(φ), sin(θ)cos(φ), cos(θ)sin(φ)}
r=sqrt(x²+y²+z²), θ=n atan2(y,x), φ=n sin^-1(z/r)

 

 The picture above is called "Roundy" Mandelbrot Set . Iterative formula ：

{x,y,z,w}2 = {x2-y2-z2-w2, 2(xy+zw), 2(xz+yw), 2(xw+yz)}

 

 

Above, Bristorbrot Set

The iterative formula used ：{x,y,z}² = {x²-y²-z²,y(2x-z),z(2x+y)}

The iterative formula used in the figure above ：{x,y,z}² = {x²-y²-z², 2xy, 2(x-y)z}

author David Makin