Consider the set of all functions of one variable $x\in[0,1]$ that can be constructed from any number of instances of that variable using parentheses and exponentiation only: $$x,\;x^x,\,x^{x^x},\;\left(x^x\right)^x,\;x^{x^{x^x}},\;x^{\left(x^x\right)^x},\;\left(x^x\right)^{x^x},\;\left(\left(x^x\right)^x\right)^x,\; x^{x^{x^{x^x}}},\;x^{x^{\left(x^x\right)^x}},\;...$$ Here are graphs of some functions from this set: Graphs

Looking at these graphs made me think — what is the supremum of arc lengths of these graphs on $[0,1]$? Is there a closed form expression for it? Is there an efficient algorithm that can compute it to an arbitrary precision?

  • $\begingroup$ A wild guess would be $1+\sqrt 2$. $\endgroup$ – Szeto Mar 5 at 22:41

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