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){r^{2},0,0,0}, r_{2}=sqrt(x^{2}+y^{2}), r_{3}=sqrt(x^{2}+y^{2}+z^{2}), r=sqrt(x^{2}+y^{2}+z^{2}+w^{2}), θ=atan(y/x), φ=atan(z/r_{2}), ψ=atan(w/r_{3})

This can be expanded to give:

{x, y, z, w}^{2} = r^{2}{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} = {(x^{2}-y^{2})b, 2xyb, -2r_{2}za, 2r_{3}w}, a=1-w^{2}/r_{3}^{2}, b=a(1-z^{2}/r_{2}^{2})

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.

07

Jul

09

### 4D Mandelbrot Set

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

I need the formula and please help me!!!

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