Of course there is this famous paper: R. K. Guy and J. L. Selfridge, <a href="/A003018/a003018.pdf">The nesting and roosting habits of the laddered parenthesis</a> (annotated cached copy) which is mentioned in (and visible from) A003018. Incidentally this sequence could use more terms. There is also a link to Math Overflow for "related questions", and a link to the OEIS Index for other sequences involving parenthesizing. Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sun, Apr 26, 2020 at 6:39 PM James Propp <jamespropp@gmail.com> wrote:
It's well known that the expression i^i takes on an infinite set of values if we understand w^z to mean any number of the form exp (z (ln w + 2 pi i n)) where ln is a branch of the natural logarithm function.
Since all values of i^i are real, all values of i^i^i (by which I mean i^(i^i))) are on the unit circle, and in fact they form a countable dense subset of the unit circle.
I can't figure out what's going on with i^i^i^i, though. I've posted a Mathematica notebook at http://jamespropp.org/iiii.pdf containing some intriguing images starting on page 2. Each image shows the points in the set of values of i^i^i^i lying in an annulus whose inner and outer radii differ by a factor of 10.
Can anyone see what's going on?
Also, what happens for taller towers of exponentials? Does the set of values of i^i^...^i become dense once the tower is tall enough?
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun