To David Makin and DarkBeam*"Please show how you generalise to 3 dimensions (and no more)."*

"Okay but computers don't understand those elegant concepts, they need explicit expressions"Hello. You want clarified computer formulas. So I give you

I simply use Mandelbrot formula An+1= An^2 + Ac or An+1= (An*An*An...) +Ac at any power and dimension. Or logistic equation.

The mutiplication * of absoliens A0*A1 is discrete circular convolution on positive coordinates.

Attention: there are 4 positive coordinates for absoliens in 3D because we work on tetrahedron base (u, i, j, k) were u is "real" and i, j, k imaginary half-axes.

In 3D this function multiplication is below. In any dimension convolution stays very symmetric, commutative and associative (that's one of the main interests of my proposal):

// absoliens 4A3 multiplication A0 * A1, discrete circular convolution equivalents to complex multiplication

vector<double> operator* (vector<double>A0, vector<double>A1)

{

vector<double> AC(4,0);

AC[0]=A0[0]*A1[0]+A0[3]*A1[1]+A0[2]*A1[2]+A0[1]*A1[3];

AC[1]=A0[1]*A1[0]+A0[0]*A1[1]+A0[3]*A1[2]+A0[2]*A1[3];

AC[2]=A0[2]*A1[0]+A0[1]*A1[1]+A0[0]*A1[2]+A0[3]*A1[3];

AC[3]=A0[3]*A1[0]+A0[2]*A1[1]+A0[1]*A1[2]+A0[0]*A1[3];

double m=mini(AC,4);

for (int i=0;i<4;i++) AC*=AC**-m; // canonic reduction from symmetric equation in absolien base: u+i+j+k=0*

return AC;

}At the begening before calculation, we need to convert the sample vector from classic euclidian base R3 (screen+Oz) to tetrahedron one (u, i , j, k), that is equivalent to transform usual R3 and vectors space to absoliens space. The function is:

// gives absoliens coordinates in 4A3 base (u, i, j ,k) from V in R3 base (U, V,W)

vector<double> coordR3to4A3(vector<double> V)

{

vector<double> A(4,0);

double coe=-1.0/(2.*RA3);

A[0]=0.; // we choice u coordinate arbitrary null

A[1]=coe*(V[0]+V[1]);

A[2]=coe*(V[0]+V[2]);

A[3]=coe*(V[1]+V[2]);

double m=mini(A,4); // canonic reduction relative to axis symmetry, all resulting coordinates are positive and one null at least

for (int i=0;i<4;i++) A*=A**-m; // if m<0, Ai becomes positive or null*

return A;

}

We obtain 3D big slice of Mandelbrot set: boring but the elementary stratum that we find again in 4D dimension but in several different orientation very mixed.

Not so complicated, computers also like elegance !

I am trying to translate my site for details and to show you the change of paradigm that implies (R cube geometry is not the world of absoliens. Absolien universe is geometry of simplexes: regular triangle, tetrahedron, pentachore etc) .

My purpose is not to find the grail of fractals (sorry) if it exists. My grail is the true/natural generalization of complex numbers and finally to improve the philosophical comprehention of numbers and space geometry associated