Category Archives: Special Projects

Prepare to be assimilated

Had a few minutes of clear sky last night and as Leo is in a good position I put the mini-WASP onto Regulus.  Had a bit of a shock when I processed the data this morning to find we had some unwelcome visitors on our doorstep.

Actually the Borg ship is  a Fractal rendering courtesy of Incendia – a Spanish guy who has created the most incredible piece of Donationware I have come across.  I will post a couple of renderings from Incendia shortly.

 

Towers of Power – the unit imaginary Tower of Power, and the face of God

Many years ago I picked up the superb book “Mathematics: The Science of Patterns” by Keith Devlin and came across a most fascinating result that I had never seen before.  If we take the unit imaginary I to the power of itself, i.e. we form I^I we end up with a real number.  Now this looked like real mystical stuff to me so it completely grabbed my attention for quite a few months.  I satisfied myself that you do indeed get a real number, something like 0.207879576350762….. when you form I to the power I and then wondered what would happen if you took the result to the power of I and then the result of that to the power of I and so on.  This I later found was called creating “Towers of Power”.  The graph in the accompanying figure (see the single point on the real line at a little over 0.2) shows the result of taking I to higher and higher powers of I.  The only real number formed is that created by I^I as I first saw in Devlin’s book.  The rest of the all the other I to the power I iterations are imaginary numbers.  Now before I plotted out this graph I really thought I’d stumbled upon something really significant here, in fact I wondered if when plotted whether I would see the face of God 🙂  I was a bit disheartened to see the curve I did get, and when I told my mathematical fantasy to a University colleague he said to me “How do you actually know that this isn’t the face of God?” – a rather irreligious fellow, and I think I’d better leave things there.  Anyhow, what is extremely interesting as you can see is that the iterations do not “explode”, on the contrary they converge to a finite limit!!  The limit to the infinite unit imaginary Tower of Power is 0.438 282 936 727 032… + i 0.360 592 471 871 385…

This in itself was equally fascinating to me and by looking into all this complex number mathematics a bit more deeply I came up with two papers which were subsequently published in the Mathematical Gazette.  In the second of these articles I show how to pull out some very interesting fractals hiding in the mathematics of imaginary Towers of Power 🙂

1)  The Mathematical Gazette, volume 81, number 492, November 1997, page 431, “Complex Power Iterations”, Greg Parker & Steve Abbott.

2)  The Mathematical Gazette, volume 82, number 493, March 1998, page 19, A Beautiful New Playground”, Greg Parker & Steve Roberts.  This issue also had one of the fractals I discovered as the cover picture 🙂

If this stuff interests you at all then I strongly suggest you play in this mathematical playground yourself for a little while, I am quite sure there are plenty more fascinating results to be discovered hiding within complex power iterations.

 

Asteroid 2012 DA14

Although I was completely clouded out last night I still managed to grab a 1-minute exposure of asteroid 2012 DA14 through thin high cloud.  Not the best image ever taken with the mini-WASP, but at least I got a record of the thing.  THe asteroid was moving through Draco at this time and the planetarium screen grab in the lower half of the image shows the region in Draco.

 

The three Curtas at the Scientific Artist

A very nice family from Kent brought me the missing Curta for my collection of mint conidition calculators.  The grey Type II on the right is the latest acquisition for the Scientific Artist and sits proudly next to the early Type I in the centre and the early (black) Type II on the far left.  A Difference Engine for the pocket!  No batteries required and 15-digits of precision, it could still outperform the early electronic calculators in accuracy – how cool is that?

Where did I first come across the Curta calculator?

I am pretty sure the first time I saw a picture of the Curta calculator, and therefore became aware of its existence, was when I saw the advert for one in the Scientific American magazine.  I would have been around 16 or so, that means around 1970.  Curiously, 1970 also heralds the demise of the Curta calculator as this is around the time when the first four function electronic calculators made their appearance.  The Curtas looked very interesting to me even back in 1970, mainly because at school we were working with slide rules and therefore had far fewer significant digits at our disposal when performing numerical calculations.  Also, although interesting, they were way outside my budget range as a 16 year old school kid with no job or income, so the Curta was just a very interesting curiosity.

I enrolled for Maths, Physics and Chemistry at “A”-level and we continued to use the slide-rule for numerical calculations, along with log tables (remember them?)  The super-trendy amongst us with rich parents had cylindrical slide rules, where the standard linear slide rule is “rolled around” a cylinder giving a longer effective length to the rule, and therefore the possibility of calculating to more significant digits.  But none of us had a Curta – no other pupil that is, but the teacher for the Pure Maths part of the “A”-level course DID have a Curta calculator, and very proud of it he was too.  He pulled his Curta (I’m pretty sure it was a Type I) from a soft fabric pouch and showed it to us from a distance of several feet, cradling it in both hads for safety.  I remember the feeling of awe being in the presence of this incredible machine.  Clearly this brief moment in time stuck with me for the intervening 42 years and I now find myself in the extremely pleasant position of being able to afford to buy the toys that I could never afford as a kid – the Curta being one such beautiful toy.  I now have superb examples of both the Type I and the Type II Curta calculator, both in pristine condition.

Although I am a William Gibson fan, and read Pattern Recognition some while back, I do not (consciously) recall the Curta calculator even making an appearance in the book (which it most certainly does).  However, the unconscious is a strange and powerful beast, so although I cannot consciously remember reading about the Curta in Pattern Recognition, I am pretty certain my subconscious didn’t miss it, and this has driven my recent resurgence of interest into this most amazing precision engineering achievement 🙂  Thank you Curt Herzstark for creating such happiness with your incredible invention.