Joined: July 2006
|Quote (hereoisreal @ Oct. 19 2006,16:14)|
|Take six pennies and place them around the seventh and see if they “fit”. They "rest on the seventh."|
This is just because the hexagonal lattice arrangement is the optimal regular circle packing on a two-dimensional plane, which was proven by Gauss in 1831 for the case of only lattice packings and by Tóth in 1940 for the general planar case. As far as I can find any sign of, the general idea of the proof rests around the observation that in the hexagonal lattice packing (and no other), if you connect the centers of any three circles in contact, you get an equilateral triangle.