Here is another Solstice-to-Solstice image of the Sun’s path across the sky taken with a tea-caddy tin pinhole camera and FLAT film.  The beer can pinhole camera uses film folded around the inside of the can and leads to a more distorted final image.  Camera opened on 18th June 2011 (3 days before Summer Solstice) and closed on 21st December 2011 (one day before Winter Solstice).

The straight line towards the left of the image is the edge of the roof of the house across the road.  You can see plenty of trees along my southern horizon.  The two light dome-shaped regions are the two New Forest observatories in my garden.

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.

Managed to get a three-dimensional packing of the Sunflower seed-head pattern onto the surface of a sphere (with a lot of false starts) using Mathematica.  Not sure there’s any practical application of this but it was an interesting exercise 🙂

Managed to get today’s EPOD with the image you can see a bit further down the page of a Sunflower Seed-Head and the mathematically simulated seed-head.

Many thanks Jim for continuing to publish my work – both astronomical and like this one, non-astronomical 🙂

I have been going on about intersecting Fibonacci spirals and the like when discussing Sunflower seed-heads, but I haven’t shown a mathematical graph showing just how close the Sunflower seed-head pattern is to a pure maths function.  On the left of this image we have the same (real) Sunflower seed-head image as shown in the article below.  On the right is a mathematically generated graph of points created using the Mathcad 2000 software.  Each X-Y co-ordinate of the points is directly related to the Golden Ratio (phi).  Phi also happens to be the most irrational, irrational number there is.  So it might not be too suprising to find out that the Sunflower seed-head pattern has an infinite rotational symmetry.  An infinite rotational symmetry has applications in optical devices called photonic quasicrystals and so the Sunflower seed-head pattern became the basis of my Patent for an entirely new class of photonic crystal with extremely interesting optical properties.  Lots of mathematics and heavy science – all from a Sunflower seed-head 🙂

Lots of cloud and a bright Moon blazing away last night, so although conditions were no good for deep-sky imaging, there were a few stars to be seen, so time for a different sort of night time photography.  I got out the AstroTrac so that I could take exposures of a few seconds without star trailing, and loaded up the Canon 5D MkII with the Canon fisheye lens.  Using ISO 3200 f#4 and a 3-second exposure time I took the above images which shows (just) the summer triangle directly overhead.  This was actually an experiment to see if it is worth trying to video the space station passing over, again using the Canon 5D MkII and the fisheye lens – but no AstroTrac.  There are two passes tonight (17th August 2011) and if I can see any clear sky at all I will give it a try.

The mini-WASP array was connected up to the two computers for a dry run before transporting everything down to the new observatory.  Cameras, filter wheels, autoguider and Paramount all fired up and ran as expected.  I am hoping for great new deep-sky images to come out of this revolutionary imaging system.  To read more about the mini-WASP array and its development please go to the New Forest Observatory web site.