This is Mandelbrot/Julia program.

Download. 568KB. Self melting exe file.
Source(C++ Builder 4.0) Download. 231KB. Zip file.

Version 01.03 (01/09/00). Added Study.
Version 01.04 (03/05/00). Added Distance Option.
Windows 98 : OK
Windows 95 : not checked
Windows NT : NG
Please try demo in file menu.

I cannot take responsibility, even if any trouble occurs
in your computer owing to this program.

Interface

Characteristics

1.This program solve Mandelbrot equation.
2.Roots can be calculated, so many images can be made automatically.
3.And classifications of Mandelbrot images are possible.

Algorithm

1.How to solve equation

Mandelbrot iteration
z(0) = p(ixel)
z(i) = z(i-1)^2 + p

Mandelbrot equation of dimension m : z(m) = 0
dimension 0 : p = 0
dimension 1 : p^2 + p = 0
dimension 2 : (p^2 + p)^2 + p = 0
dimension 3 : ((p^2 + p)^2 + p)^2 + p = 0
-----------------------
If p is a root, then z(i) cannot diverge, because z(m+1) = z(0),
and p is included in Minibrot.

Modify a little.
dimension 0 : f(0) = -p = 0
dimension 1 : f(1) = -p �} sqrt(-p) = -p �} sqrt(f(0)) = 0
dimension 2 : f(2) = -p �} sqrt(-p �} sqrt(-p)) = -p �} sqrt(f(1)) = 0
-----------------------
dimension m : f(m) = ... = -p �} sqrt(f(m - 1)) = 0
where,
f(0) = -p
f(i) = -p �} sqrt(f(i - 1)) , (i = 1 to m)----- (1)

So, �} splits Mandelbrot equation into 2^m parts,
and roots can be gotten easily using Newton method.
�} is memorized as Nth Root in this program.

2.About hidden area

From (1)
�} sqrt(f(i - 1)) = f(i) + p -------- (2)
So
f(i - 1) = (f(i) + p)^2 ------------ (3)
Let p is the root of f(m) = 0, then using (3)
f(m) = 0 = z(0) - p
f(m - 1) = p^2 = z(1) - p
f(m - 2) = (p^2 + p)^2 = z(2) - p
-----------------------
f(m - i) = z(i) - p --------------- (4)
From (2)(4)
�} sqrt(f(m - 1 - i)) = z(i) -------- (5)

Let p is a pixel, and calculate z(i), then (5) means that
if (m - 1 - i)th �} is +, and z(i)'s real part is minus, then p cannot be root. --> hidden area
if (m - 1 - i)th �} is -, and z(i)'s real part is plus, then p cannot be root. --> hidden area
because sqrt(any complex number).real() >= 0

Next image shows hidden area.
Coloring : red --> root point (at center but not clear)
--------dark blue --> non hidden area until appointed dimension
--------other --> hidden area (colored by fabs(z(at hidden dimension).real()))

3.Reverse Analysis (in Root menu) : Root (X, Y) -> (Dimension, Nth Root)

Exact Root (X, Y) is not needed. Enclose minibrot with box, then click inside.
(Dimension, Nth Root) will be calculated.

How to solve Dimension
Calculate arg(z(i)) at four points of box, then if root exists in box,
arg around box is 2pi, otherwise 0.

How to solve Nth Root
Calculate z(i).real() at center of box, then use (5).

Mandelbrot Images Link

Spanky Fractal Database
Fractal pictures & animations
The Beauty of Chaos
chaotic n-space network

---