# tetrahedron

Close packing of jumbled tetrahedra. Image: A. Jaoshvili, P. Chaikin, and A. Esakia.

A tetrahedron is a four-sided polyhedron. A regular tetrahedron, one of the Platonic solids, is a regular three-sided pyramid in which the base-edges and side-edges are of equal length. The projection of a regular tetrahedron can be an equilateral triangle or a square. The centers of the faces of a tetrahedron form another tetrahedron.

Research by Alexander Jaoshvili of New York University and colleagues, published
in *Physical Reviews Letters* in 2010,^{1} showed that the
tetrahedron is the most efficient shape for filling a container randomly
with identically-shaped objects. They found that the tetrahedra were packed
tightly enough to occupy 76% of their containers. In comparison,
randomly packed spheres fill up to 64%
of space, while the figure for ellipsoids can be as high as 74%.

### Reference

1. Jaoshvili, A., Esakia, A., Porrati, M., and Chaikin,
P. M. "Experiments on the random packing of tetrahedral dice," *Physical
Review Letters*, 104, 185501 (2010).

## De Malves' theorem

De Malves' theorem, or de Gua's theorem, is the three-dimensional equivalent of Pythagoras' theorem. It states that given a tetrahedron in which the edges
meeting at one vertex, *X*, form three
right angles (i.e. the tetrahedron is the
result of chopping off the corner of a cuboid), the square of the face opposite *X* is equal to the sum of the squares of the other three faces. The theorem is named after the French mathematician Jean Paul de Gua de Malves (1713–1785).