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 Question about infinite tetrate andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 04/06/2009, 08:28 AM 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 ${}^{\infty}b$ converges where $b=h^{1/h}$ and $|\ln(h)| \le 1$ (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 $|\ln(h)| = 1$, then we can parameterize this as $\ln(h) = e^{it}$ where $i=\sqrt{-1}$. This means that $h = e^{e^{it}}$, and putting this back in the relationship with b gives the parameterization $ f(t) = {\left(e \uparrow e^{it}\right)} \uparrow {\left(e \uparrow {-e^{it}}\right)}$ where $a \uparrow b = a^b$. There are some interesting points that can be expressed with this function $ \begin{tabular}{rrl} f(0) = & 1.4446678610097661337 & = e^{1/e} \\ f(1.4488307492834293737) = & 2.0477905274644031305 & + i\ 0.842045503530840715 \\ f(1.927907601568660839 = & 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 $ A = \int_{x=\alpha}^{x=\beta} y dx = \int_{t=a}^{t=b} y(t) x'(t) dt$ where $x(t) = \text{Re}(f(t))$ and $y(t) = \text{Im}(f(t))$. Using numerical integration we find that $ \begin{tabular}{rl} A & = \int_{t=\pi}^{t=0} \text{Im}(f(t)) \left(\text{Re}(f(t))\right)^{'} dt \\ & = \frac{i}{2} \int_{h=1/e}^{h=e} h^{(1/h - 2)} \left(h^{1/h} - e^{1/(e^{1/\ln(h)} \ln(h))}\right) (\ln(h)-1) dh \\ & \approx 4.02546664 \end{tabular}$ which is only the top half of the region, and since A = 4.02546664046975481171259768713, then 2A = 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. « Next Oldest | Next Newest »

 Messages In This Thread Question about infinite tetrate - by tommy1729 - 04/01/2009, 09:53 PM RE: Question about infinite tetrate - by andydude - 04/06/2009, 08:28 AM RE: Question about infinite tetrate - by andydude - 04/06/2009, 08:51 PM RE: Question about infinite tetrate - by tommy1729 - 04/15/2009, 04:19 PM RE: Question about infinite tetrate - by andydude - 04/17/2009, 03:33 AM

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