Fibonacci Fibbing

© Donald Richardson, 2006

The Fibonacci Series of numbers:

0 1 1 2 3 5 8 13 21 34 55 89 144 .................

was formulated by the mediaeval mathematician, Leonardo of Pisa (?1175-?1230), who was known as 'Fibonacci'. Starting from zero and progressing to infinity, it is formed by the neat trick of adding the two preceding numbers to form the next in the sequence. It has long been maintained by some theorists that these numbers are the aesthetic key to an ideal system of proportions. And it can be related to another system of good proportions - theGolden Section- a geometric ratio going back to ancient Greece, which is expressed in a simplified and rounded arithmetical form as 1:1.618. However, the practical application of either system in art, architecture and design has been limited and - where it is claimed to have been applied - difficult to discern.

Both the Fibonacci Series and the Golden Section (also know as the Golden Mean or 'dynamic symmetry') are currently in peoples' minds due to their having been featured in theThe da Vinci Code. Here they were given religious connotations and, reportedly, sponsored interest in some school art classes - so it is appropriate that we check out their veracity.

The Fibonacci Series is sometimes validated as a mathematical metaphor of deep natural processes by reference to how a pair of rabbits may quickly develop into a plague. (However, to accept this, we have to ignore the first term in the series because, of course, nothing can come from nothing!)

But, one can understand people being intrigued - even fascinated - by the concept. Each of these integers seems to have a magical or mythical vibrance, and the sequence seems to surpass mere logic and exist in some rarefied spiritual space.

Zero - Nothingness, The Void - is followed by two expressions of Unity, side by side, like twin planets revolving about a white-hot Sun; but, also, they are the digits of computer technology.

Then comes Duality: the Eternal Couple, male and female, day and night, left and right. The Eternal Opposites. The four-square basis of all arithmetic. (Yet - 2+2=.NOT 3!)

But 3 is the Triad, the Holy Trinity, the trident, the tripod, the stool of both the milk-maid and the Delphic Sibyl, the prophetic and mystical pyramid (IM Pei's new entrance to the Louvre is also inThe da Vinci Code), the geometry of Euclid and Pythagoras.

Next, the five fingers of the hand, the days of the (working) week, the cosmic symmetry of the pentacle (also featured inThe da Vinci Code), the Stigmata, Dürer'sMelencolia I.Eight seems stolid and dull in this company. But 13 is the Devil's number!

And the series has a cosmic embrace: it runs from zero to infinity in a kind of aesthetic perpetual motion.

Yes - intriguing, even mystical. It is no wonder that these arcane quantities and relationships have often been seen as a metaphor for the magic and mystery of proportion in art and design, let alone a clue to the Holy Grail. Dimensions and relationships having the properties of Fibonacci numbers must be automatically ideal and 'beautiful' and - therefore - also 'true' and 'good'. Obviously this is a most potent set of numbers.

But you don't need to be a mathematical genius to see that this is nothing less than specious humbug. If we examine the numbers themselves -

Zero is a totally useless dimension in the real world and any ratio containing it will only be fantasy.

The proportion 1:1 can in no way be construed as 'dynamic symmetry'. It is simple equal, bilateral division, which has rarely been considered a felicitous proportional principle in art or design. And 1:1 is arithmetically the same as 2:2, or 57:57, or 300:300, or 1,730,569: 1,730,569 .................so of what significance is it?

Similarly, 1:2 has never been considered 'dynamic symmetry'.

Thus, the aesthetic applicability of the first four terms of the Series is easily demolished.

Next, the Series' utility is reduced markedly if - as some authorities maintain - its use is confined to the ratios of adjacent numbers (2:3, but not 2:8, for example). And we must eliminate all relationships above the first half dozen in the series because they are too recondite for most humans to intuit (34:55, for example). It may be that they have a deep, unconscious effect, but this is a mater of faith, not reason.

All this means that, for any practical application, the series reduces itself to (possibly)

1 2 3 5 8 and 13. If it is allowed that all of these numbers can relate universally to each other, a reasonably large, yet comprehensible, range of possibilities results:

1:2 1:3 1:5 1:8 1:13

2:3 2:5 2:8 2:13 and so on

However, the system breaks down after 5 because the next number (8) is divisible by 2, thus making 2:8 the ratio of 1:4. The same calculation can be applied to 2:34 (=1:17), 2:144 (=1:72), 3:21 (=1:7), 3:144 (=1:48) and 5:55 (=1:11) - but the numbers 4, 7, 11, 17, 48 and 72 arenotpart of the Series! This appears to be a major logical deficiency in the theory - if not its total demolition!

So, in practical terms, the Series is actually limited to the numbers and ratios 1 2 3 and 5 (all the numbers after 5 having been eliminated by the invalidation of 8), or - if any of these numbers may be related to any other,

1:2 1:3 1:5 2:3 2:5 and 3:5.

Any cursory examination of some of the great works of art and design will reveal that some of these ratios are sometimes in evidence, although this does little to explain their rich complexity. In addition, because most paintings and sculptures contain few straight lines, one has to choose the segment/s of the organic shapes of a portrait or nude from which it is relevant to measure the ratios, exercising a prior aesthetic judgment which will probably be self-fulfilling. And, of course, mathematical approximations are of no help to architects and other designers.Thus, the theory of the use of Fibonacci numbers in art and design is not only theoretically flawed, it is impossible to apply.

The mathematical relationship between the Fibonacci Series to the Golden Section is also a questionable one. Between the numbers in the range of 5 to 34 in the Series, the ratio works out as between 1:1.60 to 1:1.619 - never quite the Golden Section ratio of 1:1.618, but approximating it. However - asThe Oxford Companion to Artpoints out - we can takeanytwo numbers and, adding them to form a series, will result in ratios that approximate the Golden Section: for example, a series

2 2 4 6 10 16 26 42 68 .....................

yields, for most of the adjoining numbers, ratios approximating the Golden Section, the best being the last three (1.615 and 1:1.619). So - what's special about the Fibonacci Series?

All this indicates that we must be extremely cautious in accepting the Fibonacci Series as the aesthetic 'philosopher's stone' - or, indeed, a clue to the Holy grail!