
Packing Pennies in the Plane 
But it can also be demonstrated by a purely
geometric argument.
The figure on the left above shows
the special case. In the figure on the right
the animation generates the other cases.
As the central angle decreases we also generate the image
of the left disc
under the unique
linear transformation which acts by scaling vertically
and horizontally, transforming
the original
(
In summary, the
density will achieve the maximum possible only when there are no
regions of the first or third type and when the central
angle of each of the regions of type II is exactly
The analogous argument fails in 3D, since the smallest cell surrounding a sphere is known to be a regular dodecahedron, a shape which cannot partition all of space. In particular, as Kepler himself well knew and well described, this is not the same as the cell in the densest layout of spheres throughout all of space. It is this disparity between local and global behaviour that offers a serious obstacle to a simple proof of Kepler's conjecture in 3D.