It took a few hours for my last two neurons to process the gem of a solution that ChatGPT gave me to find the Golden Solid Angle. So I woke up at 2:00 a.m. this morning on a bit of a high when I realised that the ChatGPT solution of course immediately applies to the “ordinary” planar Golden Angle. I have never seen the Golden Angle derived in this way before. This is remarkable when you think about it because if the circumference of the circle is divided into lengths of unity and Phi, then of course the subtended angles must be in a 1 to Phi ratio as well since S = r . theta, where S is the arc length, r is the circle radius, and theta is the subtended angle.
The same applies to the sphere of course. With surface areas of unity and Phi for the sphere the subtended solid angles must be in a 1 to Phi ratio since A = r^2 . theta, where A is the surface area, r is the sphere radius, and theta is the solid angle subtended by the area A.
So the procedure for finding the Golden Angle is EXACTLY the same as for the Solid Golden Angle.
We need to divide the total 2Pi radians of a circle into two planar angles such that their ratio is the Golden Ratio.
Mathematically this involves finding two planar angles A1 and A2 such that:
A1/A2 = Phi (where Phi is the Golden Ratio)
Given that the total planar angle around a point is 2Pi you have:
A1 + A2 = 2Pi
From this you can solve for A1 and A2 in terms of the Golden Ratio.
A1 = Phi . A2
Phi . A2 + A2 = 2Pi
A2 . (Phi + 1) = 2Pi
Phi^2 . A2 = 2Pi
A2 = 2Pi/Phi^2 radians or 137.5 degrees
A1 = 2Pi/Phi, A1/A2 = Phi
And unlike the Solid Golden Ratio, the planar Golden Ratio is found extensively in the Natural world.
