Saturday, 28 March 2009

Fibonacci and the Baker's Dozen

cover for 'The Abyss of Time: an architect's history of the Golden Section', by Martin Hutchinson
Last year I did some typesetting for a 1970's book by Martin Hutchinson, on archaeology and the Fibonacci Series.

A couple of things stuck in my head that I hadn't come across before. One was the idea that the ribbed stonework of North-European Gothic cathedrals might have been inspired by the ribbing in Northern European longboats. The other was that the Fibonacci Series may once have been used as the basis of an international prehistoric system of weights and measures (which kinda overlaps with Alexander Thom's work on the existence of a possible standardised "megalithic yard").

At first sight, this second idea looks a bit anachronistic ... surely the Fibonacci Series is a comparatively recent invention, with perhaps a few obscure older precedents in ancient texts, and could only have been of interest to a very limited number of people in ancient times?
Well, if you think of the Fibonacci Series as a mathematical thing, sure ... but if you've ever worked on a delicatessen stall, it should strike you that actually, the qualities of the Fibonacci Series make it an ideal system for quickly measuring out and bagging standardised quantities of food and other measurables, if your customers (or staff!) aren't especially numerate.

If you've ever used an old-style kitchen counterweight balance with weights that go up in powers of two, then you'll already be used to the idea of using binary for a weights and measures system. The binary system lets you measure out any integer quantity of something, but it's a bit fiddly. If you run a busy market stall, you don't want to be carefully measuring out whatever weights a customer might ask for. You want a simple set of pre-packaged sizes.

The binary series is the first member of a family of additive systems that form an Extended Fibionacci Series. But as a trader, we don't want to be only supplying our product in quantities that are powers of two - that's not customer friendly. We want a system that does 1, 2, 3 ... and then has units where each step is somewhere in the vicinity on one-and-a-half times the previous size. And that's where the next member of our Extended Fibonacci Series comes in. This second member of the family is the usual Fibonacci Series. It's the basis of an ideal weights-and-measures system for people who can't multiply or divide, and maybe don't even have a strong grasp of number. You can present them with a set of pre-set sizes that you can name, that can be created by stacking rods or blocks together, and the simplicity of the system means that they can easily check for themselves that you aren't cheating them. All they have to do is learn and recognise how the units stack together.

Traditional pre-metric weights and measures (such as the old Imperial system) tended to be based on multiples of threes and fours and sixes and twelves, with a few fives and tens thrown in for good measure. There seems to be a strong influence here from ancient Sumerian mathematics, with its emphasis on base-60 (which allows a large number of convenient integer divisions with integer results). The Sumerians get credited with the decision to use a factor of 360 for the number of degrees in a circle, and for using sixty divisions for minutes into degrees (measuring angles) or minutes into hours (measuring time).

But one of the odd features of many pre-metric (ie non-decimal) systems was the appearance of "thirteen" in some of the definitions of units.
Thirteen has no right to exist in any multiplicative system of weights and measures. It's a prime number! And it's so close to twelve (which divides so nicely into 2, 3, 4 and 6), that there's no obvious reason why we'd want to use multiples of thirteen in a system instead of multiples of twelve.

Except that twelve doesn't appear in the Fibonacci Series, and thirteen does. So all those thirteens in the old archaic weights and measures systems might be leftovers from a more primitive tradition of weighing and measuring, where people created larger sizes by clumping one each of the two smaller sizes together. They might have been the last echoes of an old pre-Sumerian tradition.

Habits and traditions are sometimes passed down through human societies long after the original meanings have been lost, as a kind of behavioural fossil. If Hutchinson's hypothesis is correct, this may be one of the oldest.

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