Ooh, while in the shower...
You can write a piecewise continuous function to describe behavior for all positive values:
0 <= k <= sqrt(3) - 1
g(k) =a linear function going through the 2 points (0, pi/6) and f(sqrt(3) - 1)
g = pi/6 + k * [ f(sqrt(3)-1) - pi/6 ] / [sqrt(3) - 1]
sqrt(3) - 1 <= k <= 1/[sqrt(3)-1]
f(k)
1/[sqrt(3)-1] <= k < inf
1/g
pi/6 + [sqrt(3) - 1] / (k * [ f(sqrt(3)-1) - pi/6 ])
This function has the property of f(1/k) = f(k) like you want and properly describes it all without any nonphysical nonsense.
(I finally understood why pi/6 is the desired answer for f(0)... Just like you would expect from that picture)
edit: 1 final session of pondering ...
1/g approaches 0 at infinity and we want to construct it such that we approach pi/6. The concept remains that it's an inverse function to g, but I need to be more careful about dealing with the constant.
edit2: @ below
Ah yes indeed, see I was thinking in terms of efficiency decreasing linearly with volume, and skipping over the fact that k^3 is proportional to volume.
You can write a piecewise continuous function to describe behavior for all positive values:
0 <= k <= sqrt(3) - 1
g(k) =
sqrt(3) - 1 <= k <= 1/[sqrt(3)-1]
f(k)
1/[sqrt(3)-1] <= k < inf
This function has the property of f(1/k) = f(k) like you want and properly describes it all without any nonphysical nonsense.
(I finally understood why pi/6 is the desired answer for f(0)... Just like you would expect from that picture)
edit: 1 final session of pondering ...
1/g approaches 0 at infinity and we want to construct it such that we approach pi/6. The concept remains that it's an inverse function to g, but I need to be more careful about dealing with the constant.
edit2: @ below
Ah yes indeed, see I was thinking in terms of efficiency decreasing linearly with volume, and skipping over the fact that k^3 is proportional to volume.