Z = f(Z ) n+1 nMost of the time, these functions are polynomials of the second degree:
Z = C 0
2 Z = Z + C n+1 n
Z = C 0
2 Z = Z + A n+1 n
Z = C 0
2 2 Z = Z + A = C + A 1 0
2 2 2 Z = Z + A = (C + A) + A 2 1
2 2 2 2 Z = Z + A = ((C + A) + A) + A 3 2
2 2 2 2 2 Z = Z + A = (((C + A) + A) + A) + A 4 3
etc...
z +z = {x ,y }+{x ,y } = {x +x ,y +y } 1 2 1 1 2 2 1 2 1 2The multiplication ('*') is a little more complicated:
z *z = {x ,y }*{x ,y } = {x *x - y *y ,x *y + x *y } 1 2 1 1 2 2 1 2 1 2 1 2 2 1(even if it is a mathematical non sense, more complicated definitions can be experimented)
{R ,T }*{R ,T } = {R *R ,T +T } 1 1 2 2 1 2 1 2with:
x = R *cos(T ) i i iIn particular, the square of a complex number is then:
y = R *sin(T ) i i i
2 2 (R,T) = (R ,2*T)
{x ,y ,z }+{x ,y ,z } = {x +x ,y +y ,z +z } 1 1 1 2 2 2 1 2 1 2 1 2
{R ,T ,P }*{R ,T ,P } = {R *R ,T +T ,P +P } 1 1 1 2 2 2 1 2 1 2 1 2with:
x = R *cos(P )*sin(T ) i i i i('R', 'T' and 'P' being the three spherical coordinates 'Rho', 'Thêta' and 'Phi' respectively).
y = R *sin(P )*sin(T ) i i i i
z = R *cos(T ) i i i
n n (R,T,P) = (R ,n*T,n*P)this formula being the basis of most of the pictures then obtained.
{R ,T ,P ,A }*{R ,T ,P ,A } = {fR(R ,R ),fT(T ,T ),fP(P ,P ),fA(A ,A )} 1 1 1 1 2 2 2 2 1 2 1 2 1 2 1 2'R', 'T', 'P' and 'A' being the four hyperspherical coordinates 'Rho', 'Thêta', 'Phi' and 'Alpha' respectively:
x = R *cos(P )*sin(T )*sin(A ) i i i i iand the four arbitrary functions (as it will be exhibited later) 'fR', 'fT', 'fP' and 'fA' being defined by default with:
y = R *sin(P )*sin(T )*sin(A ) i i i i i
z = R *cos(T )*sin(A ) i i i i
t = R *cos(A ) i i i
fR(R ,R ) = R * R 1 2 1 2
fT(T ,T ) = T + T 1 2 1 2
fP(P ,P ) = P + P 1 2 1 2
fA(A ,A ) = A + A 1 2 1 2in order to be compatible with the complex multiplication as well as with the Daniel White's results.
P(q) = 1*q + q C
2 fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 2*(T +T ) 1 2 1 2
fP(P ,P ) = 2*(P +P ) 1 2 1 2
fA(A ,A ) = 2*(A +A ) 1 2 1 2
2 P(q) = q + q C
fR(R ,R ) = R *R 1 2 1 2
fT(T ,T ) = 2*(T +T ) + pi 1 2 1 2
fP(P ,P ) = 2*(P +P ) + pi 1 2 1 2
fA(A ,A ) = A +A 1 2 1 2Its bird's-eye view looks like a snow flake with 5 spikes. It was computed using a polynomial 'P' of the second degree. For polynomials of the third, fourth, fifth and sixth degree, the numbers of spikes are 13 (=5*2+3), 29 (=13*2+3), 61 (=29*2+3) and 125 (=61*2+3) respectively.
8 P(q) = q + q C
fR(R ,R ) = R *R 1 2 1 2
fT(T ,T ) = 1*(T +T ) + pi 1 2 1 2
fP(P ,P ) = 2*(P +P ) + pi 1 2 1 2
fA(A ,A ) = A +A 1 2 1 2
2 P(q) = q + q C
fR(R ,R ) = R *R 1 2 1 2
fT(T ,T ) = 1*(T +T ) + pi 1 2 1 2
fP(P ,P ) = 2*(P +P ) + pi 1 2 1 2
fA(A ,A ) = A +A 1 2 1 2
P(q) = 1*q + q C
8 fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 8*(T +T ) 1 2 1 2
fP(P ,P ) = 8*(P +P ) 1 2 1 2
fA(A ,A ) = 8*(A +A ) 1 2 1 2
P(q) = 1*q + q C
8 fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 8*(T +T ) 1 2 1 2
fP(P ,P ) = 8*(P +P ) 1 2 1 2
fA(A ,A ) = 8*(A +A ) 1 2 1 2
2 P(q) = q + {-0.5815147625160462,+0.6358885017421603,0,0}
fR(R ,R ) = R *R 1 2 1 2
fT(T ,T ) = T +T 1 2 1 2
fP(P ,P ) = P +P 1 2 1 2
fA(A ,A ) = A +A 1 2 1 2
P(q) = 1*q + {-0.5815147625160462,+0.6358885017421603,0,0}
3 fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 3*(T +T ) 1 2 1 2
fP(P ,P ) = 3*(P +P ) 1 2 1 2
fA(A ,A ) = A +A 1 2 1 2
P(q) = 1*q + {0,1,0,0}
Fractal(x,y,z,t) fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 5*(T +T ) 1 2 1 2
fP(P ,P ) = 8*(P +P ) 1 2 1 2
fA(A ,A ) = 11*(A +A ) 1 2 1 2It is an original mixing between deterministic and non deterministic fractals.
P(q) = 1*q + {0,1,0,0}
Fractal(x,y,z,t) fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = Fractal(x,y,z,t)*(T +T ) 1 2 1 2
fP(P ,P ) = Fractal(x,y,z,t)*(P +P ) 1 2 1 2
fA(A ,A ) = Fractal(x,y,z,t)*(A +A ) 1 2 1 2It is an original mixing between deterministic and non deterministic fractals.
P(q) = 1*q + {-0.5815147625160462,+0.6358885017421603,0,0}
2 fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 2*(T +T ) 1 2 1 2
fP(P ,P ) = 2*(P +P ) 1 2 1 2
fA(A ,A ) = 2*(A +A ) 1 2 1 2
2 3 P(q) = 1*q - q + q + {-0.5815147625160462,+0.6358885017421603,0,0}
2 fR(R ,R ) = (R *R ) 1 2 1 2
fT(T ,T ) = 2*(T +T ) 1 2 1 2
fP(P ,P ) = 2*(P +P ) 1 2 1 2
fA(A ,A ) = 2*(A +A ) 1 2 1 2