From JOHN WELLS
In regard to your feature ‘How to succeed in stacking’ (13 July), a
three-dimensional cube viewed along an edge appears as a two-dimensional
square, whereas viewed along a major diagonal it appears as a regular
hexagon. A four-dimensional cube viewed along an edge appears as a three-dimensional
cube, but a few years ago I wondered how it would appear if viewed along
its major diagonal (imagine you live in 4-D space, and each of your eyes
produces a 3-D image of objects). I calculated the answer to be the rhombic
dodecahedron.
Clearly the most efficient way of packing n-dimensional cubes is in
a simple cubic lattice (leaves no spaces, so gives optimal density of 1.0),
thus rhombic dodecahedrons pack because they represent a 3-D view of packed
4-D cubes. I wonder why the most efficient way of packing 4-D (or 3-D) cubes,
when viewed along a major diagonal, produces the same structure as the most
efficient way of packing 3-D (or 2-D respectively) spheres.
John Wells Oxford
