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.