
Atlas of Tetration
The mastery of tetration requires the application of the theoretical to the
explaination of the empirical. The goal of this atlas is to catalog
the major structures of the tetration fractal. The most interesting features are a result of ^{n}x taking very small or large values where x is a real number, typically a negative real number and n is a whole number. The nature of tetration is such that usually if ^{n}z has an extremely large magnitude then often a small value of ε can be found such that ^{n1}(z+ε) ≈ (^{n1}z) forcing ^{n}(z+ε) to a very small value and giving ^{n+1}(z+ε)≈1 and ^{n+2}(z+ε)≈z+ε.


Tetration by period: ^{n}z → z

Tetration by escape: ^{∞}z → ∞



Tetration by period and escape

Fractal on left with a unit grid intersecting at the Gaussian integers

The
main
area of investigation is in the
fractal above. The large red area is the area of convergence of the
tower function; in other words it is period one. The red area could be
considered the "sphere" of influence of the number 1 which is a
superattracting fixed point. The green area is period three.
Sarkovskii's Theorem states that a map containing period three must
contain all periods from one to infinity.
The logo in the upper left conner of the page is a varient of the above
fractal only displaying period one, two and three. The fractal border
immediately below the banner is from the above fractal and contains the
x axis.
The
pseudocircle
is the period
two area around 0; the yellow circle to the immediate left of the red
area and can be seen in the fractal border.
The
recursive structure
at 1 is a structure that reappears as one zooms into 1.
The
star
is the area around 2.5 forming the left "end" of the fractal before the green period 3 area begins.
The
isle
is at 4.12 a can be seen as the dot at the center left as well as on the fractal border under the banner.

