Fractal encompasses any object that has similarity at different scales and orientations. Classic examples of "real-world" fractals are ferns, broccoli, clouds and even coastlines and mathematical fractals can be created that replicate natural phenomena. Fractal art is generally an art created from mathematical formulas by digital artists using software specifically written for the creation of complex fractals. There are various types of mathematical fractal including escape-time fractals, IFS, flame fractals, L-systems and others. Each type is artistically distinctive in form.
Benoit Mandelbrot (a Polish Mathematician b.1924 but who moved to France when young) coined the term "Fractal" in 1975: A fractal is generally "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole", a property called self-similarity. One of his tutors was Gaston Julia, French Mathematician 1893-1978. "Although he was famous in the 1920s, his work was essentially forgotten until Mandelbrot brought it back to prominence in the 1970s through his fundamental computer experiments." An interesting and remarkable fact is that Gaston Julia investigated fractals without the aid of computers (in the 1920s). Julia and Mandelbrot mainly investigated escape-time fractals.
Mandelbrot Set: z = z^2 + c (z and c are complex numbers)
Mandelbrot's method: z0 = (0,0), z1 = z0^2 + pixel, z2 = z1^2 + pixel, z3 = z2^2 + pixel and so on. As iteration progresses the new z value (zn) will either become progressively larger (tend to infinity) or it won't. The Mandelbrot "Set" consists of points that do not tend to infinity and are not normally coloured in Fractals.
Julia's method: Also using z = z^2 + c
Julia took the start value from the pixel and used a fixed constant as the value to add. z0 = pixel. z1 = z0^2 + constant, z2 = z1^2 + constant, z3 = z2^2 + constant and so on. In fact for every pixel on the Mandelbrot version there is a corresponding Julia version. This is interesting from an artistic viewpoint because the geometric detail in a given Julia matches the geometric detail from near the corresponding position on the Mandelbrot.
True fractals require infinite iterations but obviously that's not feasible, so fractal software has to have ways of stopping iteration and deciding whether a given pixel is "inside" or "outside" the "Set". To test if a given pixel is "outside" is done quite simply - if at any time the new z value has a magnitude greater than a given test value (bailout value) then that pixel is assumed to be "outside" i.e. it's assumed that the iteration values are tending to infinity, this is where the phrase "escape-time" comes from. For "inside" pixels this outside test would never stop the iterations, so for "inside" pixels we simply limit the number of iterations performed for a given pixel. If the calculations for a given pixel reach the maximum iterations count without the pixel "bailing out" (i.e. passing the "outside" test) then we assume that the pixel is "inside". The larger the bailout test value used and the more iterations allowed before deciding a pixel is "inside" then the closer to the true fractal the resulting image will be.
*The* "Mandelbrot Set" is used to mean the fractal generated from z=z^2+c using Mandelbrot's method. *A* "Mandelbrot Set" is used to mean a fractal generated from a general iterated formula using Mandelbrot's method. Similarly *the* original "Julia Sets" are any fractal generated from z=z^2+c using Julia's method and "a Julia Set" is used to refer to a fractal generated from a general iterated formula using Julia's method.