Star Vistas

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

I acquired another 4 hours of data on this object and Noel put the whole thing together – now totalling some 9-hours of imaging time at f#2 on an 11″ reflector!  This is NGC1333 a beautiful reflection nebula in Perseus and it is surrounded by the most amazing dust clouds as you can see.  This is one of the “busiest” regions I have ever imaged.  Clearly another page in Star Vistas II

Noel recently processed the Caldwell 10 dataset and managed to pull out the faint red open cluster (IC116) towards the top left of the image.  Note also the dark nebulosity running diagonally across the field of view.  There are a total of 7 (yes-seven) catalogued open clusters in this image – Cassiopeia is a very rich star field region.  Another beautiful Star Vista – courtesy of Cassiopeia – for Star Vistas II.

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.

Back to some deep-sky imaging and two more brand new objects for Star Vistas II.  Recently captured from the New Forest Observatory we have a nice little open cluster (in fact there are three open clusters in the field of view) in Cassiopeia – Caldwell 10.  Next to this we have a globular cluster and Messier object – M15 – in the constellation Pegasus.  The C11/Hyperstar III/SXVF-M25C is a great combination for star clusters and star fields as you can grab a great dataset in just one evening’s (4-hour) imaging.

I have asked some friends to put up the Golden Solid angle on their sites to try and find where this might occur in Nature.  Some people in trying to help with a reply have gone astray with both the mathematics involved (which aren’t that complex) and the concept.  So here I will try to explain a little more about the Golden Solid angle (and solid angles in general as this doesn’t seem to be a generally understood concept).

An ordinary (planar) angle is defined by considering a circle of radius r (make r=1 for simplicity).  Now consider a length of arc on the circumference of the circle of length L, this will subtend a planar angle at the centre of the circle defined as L/r = L radians.  Now the total circumference of a circle is 2 x Pi x r so that if again we have r=1, then the total angle about the central point of a circle is 2Pi radians.  2Pi radians is therefore equivalent to 360 degrees, Pi radians is equivalent to 180 degrees, and Pi/2 radians is a right angle.  So far so good I hope.

Now let’s move onto the slightly more involved concept of a SOLID angle.  This is no more difficult in reality to the planar angle, it’s just that we don’t use it much (if at all) in every day life.

The unit of solid angle is the STERADIAN and it is defined as follows.  Consider a sphere of radius r, and consider some area on the surface of the sphere of area A.  Then the solid angle subtended by the area A at the centre of the sphere is A/r x r steradians.  The total surface area of a sphere of radius r is 4 x Pi x r x r so by using our definition of solid angle we see that the total solid angle about a point is 4 x Pi x r x r / r x r or simply 4Pi steradians (this is precisely why 4Pi turns up in the permeability of free space – but that’s another story).

Solid angle in steradians (or in square degrees) is of importance to astronomers too as it gives an indication of the size of an object in the sky – but as solid angle isn’t generally understood this also means that the apparent size of objects in the sky is also not well-understood.  When astronomers say that the Sun and Moon subtend about half a degree – they are talking PLANAR degrees and that the Sun and Moon are about half a degree in (planar) diameter.  That’s fair enough, but to put things into perspective we should know what looking out into one hemisphere means in terms of steradians (or square degrees) as it is only by looking at the “sphere of space” above us in this way that we can get some measure of how BIG our total field of view is.  A hemisphere is 2Pi steradians and if we convert this to square degrees we can get some idea of how big the celestial sphere is for an observer with a telescope with a typical field of view of 1 square degree.

We can go back to our PLANAR definition of angle to work this one out.

Pi radians = 180 degrees, so

Pi x Pi steradians = 180 x 180 square degrees, so

4Pi steradians = whole celestial sphere = 4 x 32,400/Pi square degrees  = 41,252.96 square degrees, so

2Pi steradians = celestial hemisphere = 20,626.48 square degrees.

So our observer with a 1 square degree field of view would have a roughly 1 in 20,000 chance of randomly hitting a selected object!

As a corollary:  1 steradian = 3,282.81 square degrees or equivalently 1 square degree = 3E-4 steradians.

Returning back to the Golden Solid angle!

We now consider a sphere whose surface area has been divided into two, one of area unity and one of area phi (the golden ratio or 1.618…) and the unity surface area will subtend some solid angle, let’s call it gamma, at the centre of the sphere.  In exactly the same way we define the Golden Ratio on the line, or the Golden angle for the circle, we can come up with an equation for the Golden Solid angle for the sphere:

(4Pi – gamma)/gamma = 4Pi/(4Pi – gamma) which is a quadratic in gamma which can be solved in the usual way to give:

gamma = 1.52786Pi steradians or 15757.2 square degrees.  If you (for whatever reason) wanted to take a slice through the sphere to see what PLANAR angle this solid angle corresponded to you will get a planar (solid) angle of 152.7 degrees – though I’m not sure what use this information is except that it is NOT the same as the Golden planar angle of 137.5 degrees – which IS interesting.

So I return to my original question – anyone seen the Golden Solid angle anywhere in Nature???

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