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.
Entries (RSS)
“Unfortunately this is quite incorrect! Go down one dimension …”
… where we know it works, but there’s no reason to assume that it’ll generalise to 3D. As an example, take the Golden Rectangle: chop off a square, and you get a smaller Golden Rectangle, and so on. But there’s no possible cuboid where you can chop off a cube and get a smaller version of the original in that manner.
“I have written a correcting comment to Ray but he has not posted it on his site yet”
How did you send it? Comments are enabled; there’s no sign of yours having been posted.
I sent the reply using the comment reply box on your site. Never mind. I am not sure what you mean by “where we know it works” – where we know what works? That there are natural world examples of the Golden angle?
I don’t believe that the inability to decompose a rectangular cuboid in a similar manner to a Golden Rectangle has any bearing on the question or the problem.
The Golden Solid angle was arrived at by going from the case of dividing a line into the Golden Ratio, moving up one dimension by curling the line round on itself to give us the Golden angle, and finally rotating the circle about its diameter to give us a sphere and the associated Golden solid angle.
It seems you are happy with the Golden angle, but likewise there is now way of chopping off a part of the circle and getting a smaller version of the original in that manner either, so I don’t really understand what you’re implying.
All I’m saying is that if there’s some geometrical property of interest/significance in x dimensions, there’s no guarantee that its analogue will do the same in y dimensions. Phi, being a solution of a quadratic, is solidly rooted in 2D geometry.
I think that is why I am asking for examples of the Golden Solid angle in the Natural World – this was the question being asked in the first place. Phi is not only the solution of a quadratic of course.