Having spent the whole day on the Golden Ratio, the Fibonacci sequence, the Golden (Planar) Angle, and Phyllotaxis – here is a Mathematica produced piccie of the Sunflower seed-head pattern using 2,000 “seeds”.
The well-known Fibonacci sequence formed by summing consecutive terms looks like:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89……….
And a well known property of this sequence is that if you take the ratio of consecutive terms you get closer and closer to the Golden Ratio the higher up the sequence you go. So if we divide 89 by 55 we get 1.61818181…….
We can create a Fibonacci series incorporating the variable x in the following way:
1, x, (1+x), (1+2x), (2+3x), (3+5x), (5+8x), (8+13x)…
Now a remarkable property of this sequence is that once again, if we take the ratio of consective terms in the sequence, we get the Golden Ratio – Phi – provided x=Phi, which I think is a remarkable result.
What this means is that looking at the sequence with Phi replacing x we get:
1, Phi, (1 + Phi), (1 + 2*Phi), (2 + 3*Phi), (3 + 5*Phi)…
And if taking the ratio of consective terms gives us Phi, this means that:
(1 + Phi) = Phi^2
(1 + 2*Phi) = Phi^3
(2 + 3*Phi) = Phi^4
The first result comes directly from the basic definition of Phi itself, namely that (1 + Phi)/Phi = Phi
Taking a deeper look at the above sequence I was able to come up with a general result, namely:
Phi^n + 2*Phi^(n+1) = Phi^(n+3)
It’s been a loooong time since I’ve done one of these – so I’ve got a bit out of practice. But this was tonight’s Space Station pass at 5:37 p.m. I got part of a nice long 6-minute pass and you see it disappears off the top of the screen where there is the glow from a very bright Moon. I will do better next time 
This is what Einstein was referring to of course when he came up with the Einstein-Podolsky-Rosen paradox for Quantum Mechanics. I have written about this subject before and I even researched it for most of a Sabattical without coming to any definite conclusions. It still didn’t make sense to me. Then, maybe about a year ago, someone made a throwaway remark that made the whole thing crystal clear. The initial pair of particles created at time t=0 can be described by A SINGLE WAVEFUNCTION! And there is all you need to know. If the initial state can be described by a single wavefunction then it is absolutely no surprise whatsoever that if you measure a property of one of the particles at a later time t, then you can infer the same property for the other particle at the same time. There is no magic. There is no spooky action at a distance. Instead there is a single wavefunction which completely describes the situation. Now why this isn’t mentioned everytime there’s a discussion on the EPR paradox is completely beyond me.
In my different forms of photography there can be a vast range of exposure times used. For instance, in the solargraph image above the total exposure time is 6 months, or around 16 million seconds (16 x 10^6 seconds). The colliding water drop image, taken using an ultra high speed flashgun, has an exposure time of just 10 microseconds or 1 x 10^-5 seconds. The total range of exposure times from the shortest I do to the longest is therefore an almost unimaginable factor of 1.6 x 10^12!
I loaded up the pinhole camera last December Solstice to get a Solstice-to-Solstice solargraph over the New Forest Observatory domes. Here is the result of a 6-month exposure time.
A new Prime Number has been found using the mega-computing array at the New Forest Observatory. The new number is: 4324748322195*2^1290000-1
110 hours and 38 minutes later out came a Sierpinski pyramid in Rigid.Ink black PLA, 162mm along a base edge and 0.2mm resolution.
Rigid.Ink will be taking a number of my 3D prints to a Birmingham exhibition later this year. As I didn’t want to send them my original Mandelbulb they kindly gave me a spool of Gold/Bronze PLA to print out a backup.
I have removed most of the support material from the Mandelbulb, but now have several thousand fine hairs to trim off – but these images give you the basic idea.
