07
Jul
09

4D Mandelbrot Set

based on 4D analog to Daniel White’s formula for squaring a 3D hypercomplex number
Here is a 4D version of Daniel White’s formula for squaring a 3D hypercomplex number. In 4D, rotation is about a plane and there are 6 possible rotational matrices to choose from: Rxy, Ryz, Rxz, Rxw, Ryw, Rzw. Following this train of thought, I came up with the following 4D analog to Daniel’s formula by applying three consecutive 4D rotations:
{x,y,z,w}2 = Rxy(2θ)Rxz(2φ)Rxw(2ψ){r2,0,0,0}, r2=sqrt(x2+y2), r3=sqrt(x2+y2+z2), r=sqrt(x2+y2+z2+w2), θ=atan(y/x), φ=atan(z/r2), ψ=atan(w/r3)
This can be expanded to give:
{x, y, z, w}2 = r2{cos(2ψ)cos(2φ)cos(2θ), cos(2ψ)cos(2φ)sin(2 θ), -cos(2ψ)sin(2φ), sin(2ψ)}
This formula reduces to Daniel’s squaring formula when w = 0 and reduces to the regular complex squaring formula when w = z = 0. For faster calculations, this formula can be simplified to:
{x, y, z, w}2 = {(x2-y2)b, 2xyb, -2r2za, 2r3w}, a=1-w2/r32, b=a(1-z2/r22)
Using this formula, I rendered 3D slices of this 4D Mandelbrot fractal at w = 0, z = 0, y = 0, x = 0. Note that the case when z = 0 happens to be the same as Karl’s variation of Daniel White’s formula.

1 Response to “4D Mandelbrot Set”

1. 1 Timothy
September 27, 2009 at 3:25 am

The Mandelbrot Set is a very complicated pile of numbers, which I know, but I can’t get the formula of this set.

I swear that I really need help!!! PLEASE????

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