tommy1729 Wrote:what is the area of that region?
That's not a stupid question, its actually a good question, and I don't know the answer. But let's see if we can answer that using what is known about this region. Galidakis (1) and I (2) both call it the
Shell-Thron region since these two authors have both investigated this region in great detail. Shell and Thron note that

converges where

and
| \le 1)
(a result which they both attribute to Barrow). So if we want the outer path, then we change the less-than sign to an equals sign, and this should give us the answer. If
| = 1)
, then we can parameterize this as
 = e^{it})
where

. This means that

, and putting this back in the relationship with
b gives the parameterization
where

.
There are some interesting points that can be expressed with this function
 = & 1.4446678610097661337 & = e^{1/e} \\<br />
f(1.4488307492834293737) = <br />
& 2.0477905274644031305 <br />
& + i\ 0.842045503530840715 \\<br />
f(1.927907601568660839<img src=)
=
& 0.8152027425068848021
& + i\ 2.0166402199122300356 \\
f(2.316910654383280043

=
& -0.380979728647791984
& + i\ 0.8997001955459000918 \\
f(\pi) = & 0.0659880358453125371 & = e^{-e}
\end{tabular}
" align="middle" />
While we could use these points to integrate each section of the region, we could also use the parametric integration formula
where
 = \text{Re}(f(t)))
and
 = \text{Im}(f(t)))
. Using numerical integration we find that
which is only the top half of the region, and since
A = 4.02546664046975481171259768713, then
2
A = 8.05093328093950962342519537425 should be the area of the whole region.
Andrew Robbins
(1) I.N.Galidakis
The Birth of the Infinite Tetration Fractal.
(2) A.Robbins and H.Trappmann
Tetration Reference, page 37.