Fibonacci Goes to the Loo

Fibonacci has a Pee in Pisa

Not many people know thatLeonardo de Pisa, better known as Fibonacci, came upon the celebrated sequence of numbers named after him (usually called his series or just theFibonacci Numbers) during a visit to the loo which he made, as he tells us in his Liber Abaci, on a Spring day (“dius vernalis”) during the year 1198. The term urinal is derived of course from the Latin verb urinare, to urinate, water your horse, point Percy at the porcelain or otherwise euphemise.

This was not Leonardo’s first visit to the official Pisa peeing place, and he had had many opportunities to observe the reluctance of patrons to stand next to each other while performing. Whenever possible, new arrivals would make for a stall well separated from other customers, the aim being to leave at least one empty stall between you and your nearest neighbour.

It ocurred to Leonardo to wonder how many different ways there might be of accommodating clients at the urinal such that no-one would have his privacy invaded by another standing right next to him. Being a mathematician, his irresistible instinct was to consider not merely particular cases, but to consider the general case of a urinal with n stalls and to search for an algorithm (a term newly borrowed from the Arabic) with which to arrive at the solution to this general case. Suppose we designate an empty stall by a zero (a concept but lately imported via Arabia from far-sighted Hindu sages) and an occupied stall by the number 1 (unity). For instance, a seven-berth urinal with three occupants might be represented by (1,0,0,1,0,1,0), this being a case obeying the no-near-neighbours rule. Then, beginning with the zeroth case, a urinal of size zero, we can list all possibilities:

Number of stalls zero: only one case, there being no possible patrons: ()
Number of stalls 1: two cases; one patron, no patrons; (0), (1)
Number of stalls 2: three cases; (0,0), (0,1), (1,0)
Number of stalls 3: five cases; (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,0,1)
Number of stalls 4: eight cases; (0,0,0,0), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1), (1,0,1,0), (0,1,0,1), (1,0,0,1)

A curious sequence, thought Fibonacci, I cannot easily see how it can be expressed in terms of n, the number of stalls, but the generating algorithm is clear enough, I can obtain each term simply by adding the previous two, yielding the sequence
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … and so on ad infinitum (or ad urinatum?).

Leonardo has cheated a little by inserting a first term rather difficult to interpret. The modern reader may like to check a few higher order cases. Having an Italian love of symmetry and the spheres, Fibonacci goes on to consider the case of a circular urinal – presumably reached axially via a spiral staircase( to the top of a tower? Could the lean of Leonardo’s local tower have been caused by water weakening the soil beneath the foundations) so that the array of stalls is joined round behind, finite but boundless. The result of this investigation is not so celebrated, but has an interesting feature. The reader might well like to derive the sequence and algorithm and contemplate the male self-repugnance that is implied!

Calculate while you Urinate! (to our male readers; ladies just calculate while you imagine)

translation and commentary DJM January 2008

Incidentally as this article had climbed only to page fifteen when I Googled worldwide in November, still not many people know of the above; so welcome to the select set of those who are not members of the set of those who don’t know of the above.