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:
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:





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binomial variation images
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