Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and $$^{n+1}a=a^{^na}$$ for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the principal branch of the logarithm.)

It appears that $^9(-\sqrt2)$ is very close to $1$, but not exactly $1$ -- so that the sequence $\big(^n(-\sqrt2)\big)$ is almost(?) periodic: $$^9(-\sqrt2)-1\approx(4.99+1.51\, i)\times10^{-45}.$$

Is this a mere coincidence or is there an explanation for this?

Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and $$^{n+1}a=a^{^na}$$ for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the principal branch of the logarithm.)

It appears that $^9(-\sqrt2)$ is very close to $1$, but not exactly $1$ — so that the sequence $\big(^n(-\sqrt2)\big)$ is almost(?) periodic: $$^9(-\sqrt2)-1\approx(4.99+1.51\, i)\times10^{-45}.$$

Is this a mere coincidence or is there an explanation for this?