Looks like amazon.com in the U.S.A. wants to get rid of its stock of Star Vistas, just over $7 a pop – even with postage that will be a LOT cheaper than amazon.co.uk  So if you would like a copy of Star Vistas, but don’t like the price, order from the States before they run out

Managed to get today’s Astronomy Picture of the Day [APOD] with the wide field (sparkly colour) image of Kemble’s Cascade.  I like this image so much it is one of the permanent “wallpapers” on my home computer.  The little open cluster sitting on the left hand edge of the cascade makes this image perfect IMO

Got today’s EPOD with a shadow self-portrait taken over the New Forest near the winter solstice at mid-day.  Thank you Jim for continuing to show my work

Mr. Ray Girvan has kindly replicated my Golden Solid angle work from 2007 for others to see (with diagrams!) on his site.  Ray states that he can see why the Mathematical Gazette might have been unimpressed by the Golden Solid angle paper I wrote back then, as basically it is no different from splitting up the surface area of a sphere into any other ratio, such as 3:1 for example.  Unfortunately this is quite incorrect!  Go down one dimension to the Golden (planar) angle of roughly 137.5 degrees and you will find this angle appearing time and time again in the subject area of Phyllotaxis – the ordering and spacing of leaves on plants and trees.  It is also the underlying rotation angle in the spiral patterns of the sunflower seed head, the pinecone and the pineapple (and possibly DNA if there are 10.5 base-pairs per turn!).

An unexpected by-product of applying the most irrational, irrational number (phi) to the packing of sunflower seeds is that it leads to a geometric structure with an infinite rotational symmetry which has important applications in modern optics and was Patented by me back in 2002   So the planar Golden Angle appears extensively in the Natural World and this is a direct result of applying the Golden Ratio to dividing up the circumference of a circle into the Golden section, not a ratio of 1:3 or any other ratio – the Golden Ratio.

It is for this reason that I am expecting to see the Golden Solid angle making an appearance in the 3-D packing of objects (seeds, cells, ?) in the natural world, but to-date I don’t have any unambiguous examples of Golden Solid angle packing in Nature.

So the initial question still remains unanswered - can anybody give me an example of 3-D packing of objects in the Natural World according to the Golden Solid Angle?  If anyone can answer this question it will bring something new to the discussion.

I see that Ray believes that my more detailed piece on the Golden Solid angle (below) was just for his benefit.  It was in fact written for people on astronomy forums (where I also posted my question) who didn’t know about solid angles.  I have written a correcting comment to Ray but he has not posted it on his site yet.

The Golden Ratio (and the closely associated Fibonacci series) makes many appearances in the “living world” – here’s my question – not including Mathematics and man-made objects, does the Golden Ratio appear naturally in any inorganic systems?  There is a link between quasicrystals and the Golden Ratio, but I’m looking for a more direct link than these.  Once again, does anyone out there know of a clear example of the Golden Ratio making an appearance in a non-organic system?  If you do – please let me know ASAP

It will not have been mentioned before in this blog, but I like certain aspects of pure mathematics as much as I like deep-sky imaging.  I think most people will have heard of the Golden-Section, or the Golden-Ratio, and how it can be obtained by dividing a straight line up into two sections one of length unity, and the other of length tau or 1.61803398…  What is less well-known is that if you wrap the line round into a circle, so the circle perimeter is divided into lengths of unity and 1.618, then then angle subtended by the unity length of the perimeter at the centre of the circle is 137.507 degrees – or the Golden Angle.

That’s where the story seems to have been left, for a very long time, but I have to wonder, why?  We started with a line (one-dimension), then moved to a circle (two-dimensions), where’s the spherical case (3-dimensions)?  I did a long search a couple of years back and couldn’t find anything on this.  So I wrote a paper on “The Golden Solid Angle” for the Mathematical Gazette, which was in fact turned down as “although the result was new, just having a new result is not necessarily having something worthy of publication” – well that’s a new one for me!  So wishing to stake my claim as the discoverer of the Golden Solid Angle (sent to the Mathematical Gazette on Thursday 14th June 2007) here’s the thing explained for the first time below.

Divide the surface of a sphere into two regions, one of surface area unity, and the other of surface area 1.618…  The surface area of unity will subtend a solid angle gamma at the centre of the sphere.  By noting the total solid angle about a point is 4Pi Steradians, we can derive the following equation for gamma:

(4Pi - gamma)/gamma = 4Pi/(4Pi - gamma)

Giving a quadratic in gamma which can be solved in the usual way to give:

gamma = 1.52786Pi Steradians or 15757.2 square degrees.

Question is, does anyone out there know where the Golden Solid Angle, gamma, makes an appearance in the Natural world (or basically, anywhere)?  If you do then please let me know ASAP

Amazon U.K. are offering Star Vistas at a nice reduction right now – only £21.69, a 40% saving on the RRP!  Get the ideal bloke’s Christmas present now while it’s being offered at such a good price  In fact buy 5 and give some as pressies to your Astronomically-inclined friends

The Double Cluster in Perseus (image a couple of entries below) was the Meridian News weather picture for the evening of 10th October 2009.  I shall prepare a short video of this a little later and post up on the New Forest Observatory website.

With the recent run of clear dry weather we have had recently I have managed to complete a few more deep-sky images listed on the white board.  We now have well over 50% of the deep-sky images required for Star Vistas II completed.  I expect 100% completion by summer 2010 meaning a possible publication date for Star Vistas II of summer 2011

Noel has just processed a recently taken image – this time it is NGC281, the Pacman nebula in Cassiopeia.  Twelve 15-minute subs went into this one, and we have yet another image ready for Star Vistas II