The first instance of pre-computer fractals was noted by the French mathematician Gaston Julia.
He wondered what a complex polynomial function would look like such as the ones named after
him (in the form of z2 + c where c is a complex constant with real and imaginary parts). The
idea behind this formula is that one takes the x and y coordinates of a point z and plug them
into z in the form of x + i*y where i is the square root of -1 square this number and then
add c a constant. Then plug the resulting pair of real and imaginary numbers back into z run
the operation again and keep doing that until the result is greater than some number. The
number of times you have to run the equations to get out of an 'orbit' not specified here can
be assigned a colour and then the pixel (x y) gets turned that colour unless those coordinates
can't get out of their orbit in which case they are made black. Later it was Benoit Mandelbrot
who used computers to produce fractals. A basic property of fractals is that they contain a
large degree of self similarity i.e. they usually contain little copies within the original
and these copies also have infinite detail. That means the more you zoom in on a fractal the
more detail you get and this keeps going on forever and ever. The well-written book 'Getting
acquainted with fractals' by Gilbert Helmberg provides a mathematically oriented introduction
to fractals with a focus upon three types of fractals: fractals of curves attractors for
iterative function systems in the plane and Julia sets. The presentation is on an
undergraduate level with an ample presentation of the corresponding mathematical background
e.g. linear algebra calculus algebra geometry topology measure theory and complex
analysis. The book contains over 170 color illustrations.
