*Originally posted by wormer*

**for the demisional packing part. could you 3explain where you got those numbers such as square(3)*r and what r is? maybe im stupid or blind but i dot follow**

If a plane is covered with circles that do not overlap then the densest arrangement is the hexagonal lattice with each circle touching six others. So any three neighbouring circles will be just touching. We can tile the plane with equilateral triangles with their vertices on the centre points of the circles. This means that the packing density is the ratio of the area overlapped by one of the triangles with the circles it's vertices are on.

First we find the area of overlap. Equilateral triangles have interior angle 60°. So that means each triangle overlaps with one sixth of each circle. Each triangle overlaps three circles so the total overlap is half the area of a circle. If the circles all have radius r then that is an area of πr^2 / 2.

To get the area of the equilateral triangle we note that the sides join the centres of two touching circles of radius r, so the triangle's sides are of length 2r. Cut the triangle in half to form two right angled triangles. Pythagoras' theorem gives us the relation between the lengths of the sides. So we have a^2 + b^2 = c^2, where a is the height, b the base length and c the length of the hypotenuse. The hypotenuse is the side length of the original equilateral triangle, 2r, and the base length is half of an original side length. So we have that a^2 + r^2 = 4r^2, which after a small amount of algebra gives a = r sqrt(3). So the area of the equilateral triangle is sqrt(3) r^2.

Dividing these two areas gives the packing density π/(2 sqrt(3)) ~ 0.907.

In my earlier post I seem to have quoted 0.928, that is a typo and the 17,775 figure is correct.