Question

The blood vessels in a lung branch into ever smaller capillary vessels to facilitate the exchange of oxygen with the blood carried by the capillary vessels. One hypothesis is that the resulting form of the branching maximizes the surface area covered by the vessels. We can test this hypothesis with a very simple (and admittedly not too realistic) model of a 'lung' in two dimensions. Consider a blood vessel of unit length, which branches into two vessels each of length f, with f < 1. The two smaller vessels are 120 degree apart. Each of them then branches in the same way, with the new branches reduced in length by the same factor f. Repeat this process ad infinitum. We obtain then a fractal tree, which is called the Golden Tree. For too small a reduction factor, say f = 0.5, for example, you will find that large gaps of space remain that are not covered by the branches of the tree. With too large a reduction factor, say f = 0.7, you will find that the branches overlap.
Find the optimal f that yields an arrangement with the branches just about to touch.

Answer 1

SOLUTION:

GIVEN THAT