MANDELBROT SET

Harold Brochmann

Review

As discussed elsewhere, a representative recursive function is
Zn+1 <- Zn2 + C
For example, if we let Z0 = 0 and C = 1, we get this sequence:
Z1 <- 02 + 1 = 1
Z2 <- 12 + 1 = 2
Z3 <- 22 + 1 = 5
etc.

Next, we do a quick review of addition and multiplication of complex numbers.
Addition:
(A + Bi) + (C + Di)
(A + C) + (B + D)i
A specific example:
(2 + 3i) + (4 + 5i)
6 + 8i
Multiplication:
(A + Bi) (C + Di)
AC + (AD)i + (BC)i - BD
(AC - BD) + (AD + BC)i
{ i is the square root of -1}
A specific example:
(2 + 3i) (4 + 5i)
8 + 10i + 12i - 15
-7 + 22i

 

Recursion of Complex Numbers

If we let Z0 = 0 + 0i and C = (0.30 + 0.50i), and apply the recursive function
Zn+1 <- Zn2 + C
we get the following sequence:

 

Z1 <- ( 0.00+ 0.00i)2 + (0.30 + 0.50i) = 0.30 + 0.50i

Z2 <- ( 0.30+ 0.50i)2 + (0.30 + 0.50i) = 0.14 + 0.80i

Z3 <- ( 0.14+ 0.80i)2 + (0.30 + 0.50i) = -0.32 + 0.72i

Z4 <- ( 0.32+ 0.72i)2 + (0.30 + 0.50i) = -0.12 + 0.04i

Z5 <- (-0.12+ 0.04i)2 + (0.30 + 0.50i) = 0.31 + 0.49i

Z6 <- ( 0.31+ 0.49i)2 + (0.30 + 0.50i) = 0.16 + 0.81i

Z7 <- ( 0.16+ 0.81i)2 + (0.30 + 0.50i) = -0.33 + 0.75i

Z8 <- (-0.33+ 0.75i)2 + (0.30 + 0.50i) = -0.16 + 0.01i

Z9 <- (-0.16+ 0.01i)2 + (0.30 + 0.50i) = 0.33 + 0.50i

Z10<- ( 0.33+ 0.50i)2 + (0.30 + 0.50i) = 0.16 + 0.82i

Z11<- ( 0.16+ 0.82i)2 + (0.30 + 0.50i) = -0.35 + 0.76i

Z13<- (-0.35+ 0.76i)2 + (0.30 + 0.50i) = -0.15 +-0.04i

Z14<- (-0.15+-0.04i)2 + (0.30 + 0.50i) = 0.32 + 0.51i

Z15<- ( 0.32+ 0.51i)2 + (0.30 + 0.50i) = 0.14 + 0.83i

Z16<- ( 0.14+ 0.83i)2 + (0.30 + 0.50i) = -0.37 + 0.73i

Z17<- (-0.37+ 0.73i)2 + (0.30 + 0.50i) = -0.10 +-0.04i

Z18<- (-0.10+-0.04i)2 + (0.30 + 0.50i) = 0.31 + 0.51i

Z19<- ( 0.31+ 0.51i)2 + (0.30 + 0.50i) = 0.14 + 0.81i

Z10<- ( 0.14+ 0.81i)2 + (0.30 + 0.50i) = -0.34 + 0.72i

Z20<- (-0.34+ 0.72i)2 + (0.30 + 0.50i) = -0.11 + 0.00i

If this sequence is plotted on the complex plane and the points connected with segments, we get this pattern:
Keeping initial Z at 0.00+0.00i, but using for example, 0.27+0.28i, -1.01+0.54i and -0.52+0.24i for C, and again plotting the sequence of numbers on the complex plane, we get these patterns:

The last value for C results in a pattern that 'flies off the plane', while the others remained in the vicinity of the origin.

If you experiment with many different values of C, you will find that two types of patterns occur: those that remain in the vicinity of the origin and those that become infinite after some number of iterations.

If those values of C which produce the former patterns are coloured black while those that 'fly off' are given different colours dependent on the rate at which they leave the origin, the following image emerges:

The black region is known as the Mandelbrot Set.

The Mandelbrot Set may therefore be defined as those complex values for C which when iterated according to the function
Zn+1 <- Zn2 + C
with initial value of Z = 0 + 0i result in a sequence of complex numbers that remain in the vicinity of the origin.

The outline of the Mandelbrot Set is infinitely convoluted and has self-similarity. It is a fractal.

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