Gone astray ...

A sphere coloured by surface area in a ratio 1:φ. The red upper section subtends at the centre of the sphere Professor Greg Parker's Golden Solid Angle.

Oh, dear. Further to the previous post, I notice that Professor Greg Parker writes in Some more on the Golden Solid angle (continuing the topic of his request for natural instances of a solid angle of 1.52786*π steradians):

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.

I assume this means me, as I've been the only one to comment so far. No, I haven't gone astray. My analysis produced exactly the same result as Greg's for gamma, his Golden Solid Angle (γ = 1.52786*π = 4.799913751 steradians) so nothing is wrong with that side of things.

What I have done, that he seems to dislike, is to analyse it in terms of a practical working way to identify the magic angle he's looking for. No-one, not even a specialist, can be expected to look at an object and say, "Aha, that's a solid angle of 1.52786*π steradians." His definition is perfectly accurate mathematically, but near-useless practically. We need a visualization for a rough guide, which I've provided (above), and a simple way to ascertain more accurately whether an object has that solid angle.

Here's an example. Do the white end caps of the pool ball (left) subtend a Golden Solid Angle at its centre?



There's no way to reach inside and apply some hypothetical ghostly solid-angle protractor, but what we can do is look at the cross-section (right - not too close, to avoid perspective distortion) find the centre O and the angle θ subtended by the edge of the end cap.

Then the solid angle subtended by the cap = 2*π*(1-cos(θ))



We can see it's about 55 degrees, so the solid angle is

2*π*(1-cos(55*π/180)) = 2.679298269 steradians

So, no, our pool ball end cap is too small for γ which is 4.799913751 steradians. As I said in the previous post, we're looking for an object where θ is about 76.345 degrees. The same kind of analysis would apply if we were looking at something other than a sphere, such as a cone-shaped plant or animal structure. There's no way to measure solid angle directly; you need to measure θ on a cross-section, then calculate the solid angle using 2*π*(1-cos(θ)).

The angle at the blunt end of some species of cone shell, incidentally, looks as if it might be near the required angle.

PS: Sorry about the problem with the comments; I switched them off while using this site to prototype another. Comments are now enabled (and I actually publish mine).

Update: Greg has posted an update More on the Golden Solid angle still disputing my view that's there's not likely to be significance in dividing a sphere's surface area in the ratio 1:φ. He writes:

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

Yeah, I know. But this is an essentially fallacious argument: that because something works in 2D, it will if generalised to a different dimension. It won't necessarily: an example in this very area is the Golden Rectangle, which has the nice property that if you chop a square off the end, you get a smaller Golden Rectangle. This property doesn't generalise to 3D; there is no cuboid where you can chop a cube off the end and get a smaller object geometrically similar to the original. Or another geometrical example: equilateral triangles tessellate on a plane thus; but regular tetrahedra can't tessellate 3D space.

There's no reason to assume a geometric property that depends on the specific properties of 2D packing will work in 3D.