Friday, April 09, 2010

The Mathematics of Honeycomb: Part I

The fact that honeycombs are hexagonal is rather well-known. However, this is only a two-dimensional description of the honeycomb. The three-dimensional description of the honeycomb structure is more interesting, but most people is unaware of it.

Technically, the basic unit of a honeycomb is special type of hexagonal prism bound by six trapeziums, the familiar hexagonal top and a base formed by three rhombuses. When many of these basic units are glued together side-by-side, a slab is formed. And when two of these slabs are glued back-to-back, a honeycomb is formed.

The first person who took the trouble to measure the angle of the honeycomb structure is a French astronomer named Giacomo Filippo Maraldi. In 1712, Maraldi measured many samples of honeycomb cells and concluded that the angles of the trapezoidal sides and rhombic bases are always consistent: the smaller angle of the rhombus/trapezium is always about 70o. By postulating that the rhomboidal angle and trapezoidal angle are exactly equal, Maraldi was able to compute this angle exactly, that is, 70o32'.

Several years later, a French biologist named René Antoine Ferchault de Réaumur took up the same problem and postulated that the angle of the rhombic base is related to the minimization of the construction material of honeycomb. This make sense from an evolutionary point of view for nature will select and favor bee species that is able to economize its resources. Réaumur asked this question to his Swiss friend Johann Samuel König. Being a mathematician, König was able to utilize his calculus skill to solve the problem. König's result was 70o34', which disagrees with Maraldi's result by 2'.

In 1743, the Scot mathematician Colin Maclaurin gave the problem a fresh shot and solved the problem by geometric method, and concluded that the rhomboidal angle is 70o32'. This result is similar to that of Maraldi's. Maclaurin also pointed out that there was a mistake in König's earlier computation. König's results was off by 2' because there was a misprint in the logarithm table he used.

In Part II of the article, I will give a calculus-based demonstration on why the rhombic angle of the honeycomb is 70o32' or cos-1(1/3).

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