The face of God?

Many years ago I read a most amazing mathematical fact in a book called “Mathematics: The New Golden Age” by Keith Devlin.  What completely blew me away at the time (and actually it still does) was the discovery that i to the power i, where i is the square root of minus 1, gives you a real number!  In fact i^i = 0.20787957… which seemed completely nonsensical to me as a result of taking the square root of minus 1 to the power of the square root of minus 1.  I then sat down with a maths program called Mathcad and started looking at what you got as you took the square root of minus 1 to higher and higher powers of the square root of minus one.  It looked very odd an didn’t make much sense to me at the time, but the numbers I was getting looked like this:

i^i = 0.207879…

i^i^i = 0.947159 + 0.320764i

i^i^i^i = 0.0500922 + 0.602117i

As you can see, there is nothing at all obvious coming out from the sequence so far.  However, when I looked at terms well into the sequence it became clear that this iteration was coming to a limit – how exciting!!  I had never played with this sort of maths before and this was (as far as I was concerned) all brand new and unknown territory.  Certainly a couple of guys in the Maths Department I spoke too had never heard of it – so it must be new 🙂  As I am rather prone to fantasy I wondered if I had stumbled  upon something really deep here, and if I plotted out all these data points, what would I get?  I thought it might reveal the Face of God!  So I used the same mathematics package to plot all the points out and what I got is what you see here, a three-barred spiral with the points all iterating towards a limit somewhere around 0.438 + 0.36i and unfortunately no God-like face (although one irreverent Lecturer did ask me how I knew that wasn’t indeed the Face of God!).  Moving swiftly on – this small excursion into a subject that is not my speciality led to a paper called “Complex Power Iterations” by Greg Parker and Steve Abbott (The Mathematical Gazette, Volume 81, Number 492, November 1997, pp.431-434), my first ever mathematical paper. 

I put this stuff aside for a while but it kept niggling at me and I worked on it again a few months later.  This time I managed (with my PhD student at the time Steve Roberts) to actually extricate a fractal from these mathematical meanderings.  This work formed the paper “A Beautiful New Playground” by Greg Parker and Steve Roberts (The Mathematical Gazette, Volume 82, Number 493, March 1998, pp.19-25) where a couple of very interesting fractals were found.

I subsequently found out that these “Towers of Power” were quite well known long before I re-discovered them, in fact someone had written a paper on the subject as “people seem to keep re-discovering these objects on a fairly regular basis” 🙂  However, I did fortunately bring a few new bits of data to a wider audience, including the values of the interation limits and the full form of the more complex fractal beyond the central cardioid (you need to read the second Mathemtical Gazette paper to see what I’m on about).

All in all it was a very exciting experience to suddenly uncover (what I thought) was an entirely new world hidden within mathematics and at least I can now understand how those mathematicians that actually do make new amazing discoveries feel.


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