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
Since writing the above I have found that using Phi to represent the Golden Ratio we have:
Golden Angle = 2 Pi / Phi squared
Golden Solid Angle = 4 Pi / Phi squared