Which Tetrahedra Fill Space?

Abstract

Filling space by fitting congruent polyhedra together without any gaps is one of the oldest and most difficult of geometric problems, and has a fascinating history. It arose first in ancient times in relation to Plato's theory of matter; during the subsequent 2300 years of its development, it has continued to receive its principal stimulus from physicists and others interested in the structure of the solid state. In its most intuitive form, the problem is that of determining the shapes of building blocks- the building blocks of architecture, of inorganic and organic matter, of space itself. Its origin can be traced to Plato's atomic theory: the hypothesis that all matter is the result of combinations and permutations of a few basic polyhedral units. The mathematical question is: what shape must such a unit have if it is possible to fill space without gaps by figures congruent to that single unit? This simply stated geometric problem is still unsolved, despite considerable efforts devoted to it over the ages. That rectangular solids or, more generally, parallelopipeds can be fitted together to fill space was known to the earliest bricklayers, but that any other polyhedra have this property is less obvious. Plato, as we shall see, assumed the existence of such polyhedra, but Aristotle was the first to get down to details. In the process he made a mistake that generated a controversy lasting nearly 2,000 years.