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 nx 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 nz has an extremely large magnitude then often a small value of ε can be found such that  n-1(z+ε) -(n-1z)  forcing n(z+ε) to a very small value and giving  n+1(z+ε)1 and  n+2(z+ε)z+ε.

Tetration - Periods Mandelbrot Tetration
Tetration by period: nz → 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.