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

(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

, then we can parameterize this as

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

=

& 0.8152027425068848021

& + i\ 2.0166402199122300356 \\

f(2.316910654383280043

=

& -0.380979728647791984

& + i\ 0.8997001955459000918 \\

f(\pi) = & 0.0659880358453125371 & = e^{-e}

\end{tabular}

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While we could use these points to integrate each section of the region, we could also use the parametric integration formula

where

and

. 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.