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.
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