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 – it gives an indication of how BIG it is up there!
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 would get an 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.
For completeness: The solid angle corresponding to the Golden angle of 137.5 degrees is 1.275Pi steradians.
So I return to my original question – anyone seen the Golden Solid angle anywhere in Nature???
Very interesting. I can’t think of an example of when I have seen a golden square angle in nature, but I will let you know if I do. Have you? May I ask, what is the origin of this question?
I am experimenting with an unusual antenna based on a Geometric Series using the golden section as the base of logarithms. It is documented here, if you are interested:
It was shown to me in a “very vivid dream”, and I felt compelled to make one.
The ordinary planar Golden Angle appears everywhere in Nature and is the basis of Phyllotaxis. The Golden Solid Angle – which I seem to be the first to come up with – I haven’t seen in Nature and that’s what I am asking for.