# Fractals

Many
people are familar with the Mandelbrot set of the quadratic map which
provides a sort of visual summary of the different Julia sets of the
quadratic map. Tetration can be studied using the Mandelbrot set of the
expondential map; fractals on this site are of this type unless
otherwise noted. The Mandelbrot set of the expondential map provides a
way of representing ^{∞}z; many of the points "boil off", they escape to infinity as with ^{∞}2. The points that don't escape many settle down to a single value as with both ^{∞}1 = 1 and ^{∞}-1 = -1; otherwise they are periodic as with ^{∞}0 where ^{1}0 = 0 and ^{0}0 = 1 leads to ^{∞}0 being period two. Often complementary sets of escape and periodic fractals will be presented together.

The fractals on this site have been produced using Fractint with Mathematica doing image processing. Many of the fractals took several hours to
generate on an 333MHz PC. Points in the quadratic Mandelbrot set are
known to escape to infiinity once they achive an absolute value greater
than four; this is known as the bail-out value. Many of the tetration
fractals here have used a bail-out value of 10^{36} and are
significantly more detailed than when a bail-out value of 10,000 is
used. More of the fractals here are of the period type which are much
more computationaly intensive that the escape fractals; being allowed
up to 30,000 iterations to discern the period of a given point.

- The Atlas of Tetration - Explores the different areas of the tetration fractal.

- Fractal Parameter Plane of Exponential Map - The variant of tetration fractal studied by the academic community based on the iterations of a*e
^{z}instead of a^{z}. Very important in understanding the features of the tetration fractal on the x axis for x<0.

- Projective Fractals - Another tool for understanding the features of the tetration fractal, particularly the area of convergence of the power tower function</a>.

- Similar Fractals - Interesting example of two different tetration related fractals by
period that are identical for the majority of the fractals.

- Orbits - The above pages show where points eventually go under iteration, this shows an example of how they get there.