04/06/2011, 04:20 PM

till now we considered tetration for 2D complex numbers.

what about 3D numbers ?

are there methods for 3D numbers that are distinct from the 2D ?

the problem might be that the Riemann Mapping Theorem does not apply in 3D. (only conformal mapping are moebius in 3D)

i think there is no 3D solution and we will need to use the 2D solutions and apply them to get a 3D solution with is correct upto its " complex absolute value ".

it should be noted that there are 2 types of 3D numbers.

a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.

( group ring is the correct term here )

and the classical " 3D complex "

a + b w + c w^2 where a , b , c are real and w^3 = 1

( it is trivial to compute the " absolute complex value " , just replace P with i or w with the upper cube root of unity )

the advantage in 3D might be less fixpoints for exp^[r](z) = exp(z) and cycle detection / branch point understanding of the ordinary 2D sexp / slog.

( this relates to some threads like tid616 and tid499 amongst others )

so let me know what you think.

regards

tommy1729

what about 3D numbers ?

are there methods for 3D numbers that are distinct from the 2D ?

the problem might be that the Riemann Mapping Theorem does not apply in 3D. (only conformal mapping are moebius in 3D)

i think there is no 3D solution and we will need to use the 2D solutions and apply them to get a 3D solution with is correct upto its " complex absolute value ".

it should be noted that there are 2 types of 3D numbers.

a + b P + c P^2 + d P^3 where 1 + P + P^2 + P^3 = 0 and P^4 = 1 and a , b , c , d are positive.

( group ring is the correct term here )

and the classical " 3D complex "

a + b w + c w^2 where a , b , c are real and w^3 = 1

( it is trivial to compute the " absolute complex value " , just replace P with i or w with the upper cube root of unity )

the advantage in 3D might be less fixpoints for exp^[r](z) = exp(z) and cycle detection / branch point understanding of the ordinary 2D sexp / slog.

( this relates to some threads like tid616 and tid499 amongst others )

so let me know what you think.

regards

tommy1729