Tierazon v29

Discussion on creating fractal images with the Binomial Formula (BF)

A binomial formula has the following form:

BF = (a+b)^n

In the formula, the following parameters are used to create the fractal image:

a=z
b=1
n=3

To use this formula to create a fractal image the Newton method for finding roots is used for the iteration sequence and the Mandelbrot formula is also used by adding "c" to the formula.

The Newton formula for finding roots is:

z = z-(BF/BF'')

BF' is the derivitive of binomial formula "BF". By adding "C" to the formula, the entire formula becomes:

z = z - ( BF/BF' ) + c

"z" and "c" are the complex variables that represints the initial point on the Cartesian plane. These variables are commonly associated with Mandelbrot images.

The following is a reduction of the Binomial Formula "BF":

BF = (z + 1)^3
BF = (z + 1)(z + 1)(z + 1)
BF = (Z^2 + z + z + 1)(z + 1)
BF = (Z^2 + 2*z + 1)(z + 1)
BF = (Z^3 + z^2 + 2*z^2 + 2*z + z + 1)
BF = z^3 + 3*z^2 + 3*z + 1

The following is the derivative of "BF".

BF' = 3*z^2 + 6*z + 3

Finally the entire formula can be written:

z = z-(BF/BF')+C.
z = z - (z^3 + 3*z^2 + 3*z)/(3*z^2 + 6*z + 3) + c;

The following is an image created with this formula:

An interesting variation of the Binomial Formula (BF)

A variation of the formula is as follows:

The original formula:
z = z - (z^3 + 3*z^2 + 3*z)/(3*z^2 + 6*z + 3) + c;

The variation:
z = z - (z3 + 3*z^2 + 3*z)/(3*z + 6*z + 3) + c;

Note that the only change in the formula is that the "3^2" term is replaced with "3*z" in the denominator.

The following are images created with the variation:

More binomial variation images