Well this ChatGPT business just got a whole lot weirder!
I first asked ChatGPT about the Golden Solid Angle and at first it didn’t understand what I meant and came back with the conventional 2-dimensional golden angle. I then defined what I meant by the Golden Solid Angle and it then came back with the correct result, but did it by a method I hadn’t even thought of which was by using the solid angles themselves! Extraordinary! So this is what ChatGPT came back with.
We need to divide the total 4Pi steradians of a sphere into two solid angles such that their ratio is the Golden Ratio.
Mathematically this would involve finding two solid angles A1 and A2 such that:
A1/A2 = Phi (where Phi is the Golden Ratio)
Given that the total solid angle around a point is 4Pi, you have:
A1 + A2 = 4Pi
From this you can solve for A1 and A2 in terms of the Golden Ratio Phi.
Let A1 = Phi . A2
Phi . A2 + A2 = 4Pi
A2(Phi + 1) = 4Pi
Phi^2 . A2 = 4Pi
A2 = 4Pi/Phi^2 which I derived earlier by another (equivalent) method.
Therefore, A1 = 4Pi . Phi/Phi^2 = 4Pi/Phi
ChatGPT then goes on to say that the Golden Solid Angle doesn’t correspond to any specific, natural phenomenon like the 2-D golden angle, but it provides a geometrical discussion of space that maintains the same principle. I haven’t been able to find any natural occurrences of the Golden Solid Angle either. How about you?
