
Projective Fractals








^{∞}z

^{∞}(z^{1/z})

^{∞}(Log(z^{1/z}))

^{∞}(Log^{2}(z^{1/z}))

The first row displays periods and the second row displays the
complementary escaping fractals. The first column is the standard
tetration fractal. The area of convergence in red is bounded by
Equation 1 where 0<= x <= 1 and displayed in Figure 1. The points
at x = 1/n are on the boundary between period 1 and period n.

Equation 1


Figure 1

The second column is a projection of
the first such that the inner red
area displays the location of the fixed points of the first set of
fractals. The boundary for the area of fixed points for the area of
convergence is given by Equation 2 where 0<= x <= 1 and displayed
in Figure 2

Equation 2


Figure 2

The third column is based on the Lyapunov characteristic number
where the bounding curve is the unit circle. The fourth column is based
on the Lyapunov exponent and forms a line on the axis of pure imaginary
numbers. This is a nice demonstration of the area of convergence of the
tower function. It is also a nice demonstration of the limitation of
visual "proofs" since the point at 1 is convergent but its convergence
is not visually apparent.

