As you probably know only too well, in a Fibonacci Sequence of numbers, each successive term is the sum of the two preceding terms, so all you need to do to start it off is specify the first two terms. The classic example is the sequence of what are often called the Fibonacci Numbers:
(0), 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, . . . . . .
The wonderful properties of this sequence – for instance its appearance as slanting sums in Pascal’s Triangle, how the ratio between successive terms closes in on the Golden Ratio alternately from above and below, how successive terms are always co-prime and how sunflowers seed spirals tend to sum to successive terms – are well known. How Fibonacci, also known as Leonardo de Pisa, made his original discovery has long remained obscure but can now be revealed.
Before proceeding to this important part of the post (scroll down to a repeat of the opening image), I’ll justify the blog title by briefly investigating Lattice Labyrinth tessellations based on adjacent Fibonacci numbers. Here are two Fibonacci Lattice Labyrinths, with separation parameters (5,8) being successive Fibonacci Numbers, meaning that each tessellation repeats itself if you count 5 across and 8 up/slantingly-up the lattice.
Supertile area S = Fundamental Domain/Repeat Unit R = (5² + 8²) = 89
(e² + ef + f²) = (5² + 5×8 + 8²) = 129 Supertile area S =2(5² + 5×8 + 8²)/3 = 86
Fundamental Domain/Repeat Unit R = 2(5² + 5×8 + 8²) = 258
Whether such Fibonacci Tessellations partake of general family charateristics I’ve yet to discover, but the numbers associated with the areas of their supertiles certainly do.
Here is a table of the particular families into which low-order examples fall, where Ch. stands for the Chinese Family, Serp. for the Serpentine Family, both being on the square lattice, on which I denote the Fibonacci separation parameters by (a,b), Tre. for the Trefoil Family and Dia. for the Diamond Family, both the latter being on the Triangular Lattice, on which I denote the Fibonacci separation parameters by (e,f).
Fibonacci N’S: 1 1 2 3 5 8 13 21 34 55 89 144 Sep.Parameters (1,1) (1,2) (2,3) (3,5) (5,8) (8,13) (13,21) (21,34) (34,55) (55,89) (89,144) Squ. Lattice Fam. Serp. Ch. Ch. Serp. Ch. Ch. Serp. Ch. Ch. Serp. Ch. (a² + b²) 2 5 13 34 89 233 610 1597 4181 10946 28657 Tri. Lattice Fam. Dia. Tre. Tre. Tre. Dia. Tre. Tre. Dia. Dia. Tre. Tre. (e² + ef + f²) 3 7 19 49 129 337 883 2311 6051 15841 41473
Well, there’s a beautiful property! The values of (a² + b²), the sums of the squares of successive terms of the Fibonacci Sequence are themselves alternate terms of the sequence. (Can we find a function of successive Fibonacci numbers that generates the missing alternate members of the sequence: 1, 3, 8, 21, 55, 144 . . .?)
No doubt this is another well-known property of the Fibonacci Sequence, maybe constituting another of those “Are you a real mathematician?” tests. If you are or can become a REAL mathematician then the underlying reason for that result is immediately obvious to you. If it isn’t immediately obvious then join me among the also-wrangled (to coin a phrase).
The series of values of (e² + ef + f²), called Loeschian Numbers, start off as alternate members of another Fibonacci Sequence: 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, . . . . but thereafter to get the successive Fibonacci Loeschians we have to modify the Fibonacci procedure of adding successive pairs to get the next number by the quaint expedient of alternately adding and substracting unity, 1, which, despite a net zero effect that one might naively expect, causes the two sequences to diverge quite rapidly.
1, 3 ,4 ,7, (4+7+1)=12,19, (12+19-1)=30, 49, (30+49+1)=80, 129, (80+129-1)=208, 337, (208+337+1)=546, 883, . . . there surely has to be another way to generate this sequence?
Enough of serious recreational mathematics, on to the REAL subject of this blog post, some less serious (though still worthwhile) recreational mathematics.
It is not widely appreciated that Leonardo de Pisa, better known as Fibonacci, came upon his celebrated eponymous sequence during a visit to the urinal in the gents’ toilet (American; men’s room) to perform a Number 1, which, as he tells us in his Liber Abaci, happened on a Spring day (“die vernali”) during the year 1198. The term“urinal” is derived of course from the mediaeval Latin verb urinare, to urinate, water your horse, point Percy at the porcelain, shake hands with the Vicar or otherwise euphemise.
This was not Leonardo’s first visit to the official Pisa peeing place, and he was familiar with the reluctance of patrons to stand next to each other whilst performing. Whenever possible, new arrivals would make for a stall well separated from other practioners, the aim being to leave at least one empty stall between yourself and your nearest neighbour. The behaviour he witnessed persists to this day and can best be observed and quantified in the gents’ at a busy motorway service station, such as those operated by Welcome Relief.
It ocurred to Leonardo to wonder how many different ways there might be of accommodating patrons at the urinal such that no gemtleman would have his privacy invaded by another standing next to him. Being a mathematician he contemplated not merely the particular porcelain of Pisa, but the general case, searching for a formula for a urinal with n stalls or, failing that, an algorithm (a term then newly borrowed from the Arabic) for arriving at the solution for any positive value of n.
Suppose we designate an empty stall by a zero (another concept but lately imported via Arabia) and an occupied stall by the universally recognised symbol. An example of the notation heads this post.
So Fibonacci had failed to come up with a formula in terms of n, the number of stalls, but had found an elegantly simple generating algorithm, obtaining obtain each answer 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 adding 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, the array of stalls being now joined end-to-end, finite but boundless. I reproduce his analysis below:
(Assonance and alliteration courtesy of Rev. John Drackley)