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Sphere Packing

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In 1611, the great physicist and mathematician Johannes Kepler published a short book called 'De Niva Sexangular' (On the Six-pointed Snowflake). The booklet had originally been written as a New Year's present for Kepler's friend Wacker von Wackenfels. Kepler relates that as he was musing over an appropriate present for his friend, “.... by a happy chance water-vapour was condensed by the cold into snow, and specks of down fell here and there on my coat, all with six corners and feathered radii.

Upon my word, here was something smaller than any drop, yet with a pattern; here was the ideal New Year's gift, .... the very thing for a mathematician to give, .... since it comes down from heaven and looks like a star.”

In the booklet Kepler considered the geometrical structure of a variety of natural objects including bee's honeycombs, pomegranates and crystals, as well as snowflakes. Kepler put forward the prophetic idea that the regular structure of crystals was due to the geometrical arrangement of their constituent atoms. In the process Kepler posed a famous mathematical problem now known as The Sphere Packing Problem or simply Kepler's problem. The problem is to find the densest packing of equal-sized spheres.

The equivalent problem in two dimensions is much simpler to analyse. Kepler showed that the densest packing of equal-sized circles in a plane occurs when the circles are arranged like the hexagons in a honeycomb or regular tessellation. Each circle is then touched by six surrounding circles. We say that the circles have Kissing Number six.

When grocers stack oranges each layer is packed in this hexagonal arrangement. With the oranges in each layer resting on the gaps in the layer below. Each orange within the stack is touched by six oranges within the same layer and also by three oranges in the layer above and three in the layer below, so the Kissing Number of the oranges is twelve. The centres of these twelve oranges are situated at the vertices of a cuboctahedron. Kepler described this packing, but could not prove that it was the densest possible.

For almost 400 years mathematicians were unable to find a proof. In 1998, Thomas Hales presented a proof that relies on a very complicated computer program. Whilst Hales' mathematics appears to be correct, the computer program has not been refereed, so there is not yet a consensus about whether the proof is valid. The packing is shown in the animation on the left.

Chemists know the grocers' packing as the face-centred cubic packing and it is indeed the usual arrangements of equal-sized atoms in crystals. The name derives from an alternative construction of the packing. We fill space with equal sized cubes arranged so that there are no gaps. We now position spheres so that their centres are situated at the centre of each of the faces of the cubes. If the spheres are the right size to just touch each other we have produced the grocers' packing.

It is possible to fill space with a regular arrangement of tetrahedra and octahedra without leaving any gaps. This honeycomb of tetrahedra and octahedra is also closely related to our packing of spheres. The centres of the spheres in the packing are positioned at each of the vertices of the honeycomb.

The animation on the right shows John Robinson's sculpture 'Evolution'. The structure of the sculpture is closely related to the honeycomb of octahedra and tetrahedra. The sculpture is composed by following a sequence of triangular faces taken alternately from tetrahedra and octahedra in the honeycomb, pairs of triangular faces making the diamonds that form the sculpture. The structure is similar to Buckminster Fuller's 'Octet Truss'.

Text by Nick Mee from the CD-ROM 'POLYTOPIA II: Honeycombs and Polytopes'. Animations by Nick Mee and John Robinson.