A space-filling arrangement of polyhedra is called a honeycomb. The name, of course, derives from the cells constructed by bees to store honey, as shown in the illustration above. By considering the three regular tessellations, we can see that the cube, the hexagonal prism and the triangular prism will fill space without leaving any gaps. These three polyhedra can be constructed by extruding the square, hexagon and triangle into the third dimension respectively. Similarly these three honeycombs can be derived from the three regular tessellations.

The above animation zooms through a honeycomb composed of hexagonal prisms.

There are other polyhedra that will fill space by themselves. One is the rhombic dodecahedron. Another is the truncated octahedron. The above animation shows the honeycomb of truncated octahedra. There are several semi-regular honeycombs that are composed of more than one type of polyhedron. For instance, octahedra and tetrahedra form a honeycomb that is equivalent to the face centred cubic lattice used by chemists to model the atomic structure of many metals.

The above animation shows the honeycomb composed of cuboctahedra and octahedra. The vertices of this honeycombs are the midpoints of the edges of the cubic honeycomb.

In 1887, Lord Kelvin conjectured that a honeycomb of truncated tetrahedra would partition space into cells of equal volume with the least surface area. The faces of the truncated octahedra are curved slightly in order to meet at the proper soap film angles. The conjecture has now been overthrown. Denis Weaire and Robert Phelan have found a froth whose surface area is 0.3% less than Kelvin's. This structure, shown above, uses two kinds of cells, a dodecahedron and a tetrakaidecahedron, which has two hexagonal and twelve pentagonal faces.

The above animation shows the honeycomb of truncated octahedra.

Text and animations by Nick Mee from the CD-ROMS POLYTOPIA I: Tessellations and Polyhedra and POLYTOPIA II: Honeycombs and Polytopes - Published by Virtual Image