Wednesday, August 08, 2007

Naturally Binary

I recently bought a T-shirt with the Eye of Horus on it. I really just liked the way it looked, but since I knew people would ask what it meant, I poked around on the internet to find out.

I was surprised and pleased to discover that, in addition to its significance as a symbol of protection, it also had significance in commerce. Specifically, it was a fractional counting system. The eye consists of six pieces, each corresponding to a power of one-half: 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64. All the pieces together add up to 63/64, which the Egyptians rounded up to 1. By drawing different combinations of eye-pieces, the Egyptians could represent various fractions (e.g., use the 1/2 and 1/8 pieces to get 5/8). These fractions were used to measure quantities of land, grain, medicine, etc.

What fascinates me about this is that it was a naturally occurring use of binary, as opposed to decimal, counting. And the Egyptians are not alone in having used such a binary system. The phrase “two bits” for one-quarter of a dollar, for instance, originated in the division of gold dollars into eight pieces; two “pieces of eight” (another old phrase indicative of the binary counting system) equal one quarter.

There’s also a great practical reason for dividing things in a binary fashion: it requires less reference to standardized measures. Try dividing a piece of paper into five equal pieces without using a ruler. Pretty hard, right? But dividing it into four pieces – or eight, or sixteen – is simple. Just keep folding it in half. Halves are easy to compare. For weights, all you need to do is make sure the scales are even on a simple balance. And you can use the balance repeatedly to get any fraction that’s a power of one-half.

So all this was going through my head this afternoon, as I walked to the Subway to get a sandwich, when I suddenly realized: this probably explains why traditional English weights and measures seem so strange. They are, in many cases, binary. In volume, for example, 1 gallon = 4 quarts = 8 pints = 16 cups = 32 gills. (Odd that there’s no specific name for 2 quarts.) [UPDATE: LP notes in the comments that 2 quarts = 1 "pottle," which makes the binary progression complete.] Since the English system evolved in a time when precise standardized measures were hard to come by, it made sense for people to rely on the comparability of halves. The metric system would not have worked well in ancient times. Another interesting parallel: according to this page, before 1824 a hogshead was equal to 63 wine gallons – probably a result of the same rounding that led the Egyptians to say 63/64 = 1.

Was this obvious to everyone but me? Somehow it had just never occurred to me that there was an underlying binary logic to English weights and measures, but in retrospect it makes perfect sense. Of course, this is all speculative, as I know nothing about the actual history of English weights and measures. And there are some things my theory doesn’t explain, such as the presence of twelves (e.g., 12 inches in a foot) and sevens (e.g., 7 pounds in a clove, 14 pounds in a stone). The twelves might have resulted from a willingness to divide things into thirds as well as halves; note that three feet = 1 yard and three hands = 1 foot. But I’m at a loss to explain the sevens. Maybe a real historian could shed some light here.


Steven Horwitz said...

I suspect you're right about the 12s Glen. Factorable into 2s and 3s (and 4s I guess too) makes it a very useful number.

Anonymous said...

It wasn't obvious to me initially but I've been aware of it for a long time. I had gotten into a discussion about metric vs. US measurements. Me, being an ugly american just had to insist upon the superiority of the US system.

Of course, I got the usual arguments about how metric was so much easier because everything was divisible by ten. My point was that the US System was better because it was more anthropomorphic. 1 cup is a much more human use measurement then anything else. And really is the math that hard?

Someone older and wiser than me (of which there are many such people) pointed out that the real beauty of the US system was its easy divisibility and multiplicity, and that in commerce, and barter it is far easier to use. I've carried that with me every since.

As to 12, it has the same principle as 60, easy divisibility and as such the ancient civilizations regarded 60 highly, ancient Babylon had a base 60 system for example.

LP said...

Wikipedia claims that a half-gallon or 2 quarts was, at one time, called a 'pottle.' I figured someone was making this up, but the dictionary confirms.

I think most people are aware of the power-of-two/doubling progression of volume measurements -- you'll sometimes hear people under their breath saying "2 cups in a pint, 2 pints in a quart," except that it usually breaks down at this point (since 'pottle' is no longer in use) and resumes by jumping to "4 quarts in a gallon" which is easy to remember since a quart is either 1/4 of a gallon or 4 cups.

On 'stone' as a unit of weight: what counts as a 'stone' used to vary depending on what you're measuring. For food items, one stone = 8 pounds (binary again). When the unit was standardized, it was defined so that eight stone = a 'hundredweight' -- an interesting effort to combine intuitive binary measurement with standardized decimal amounts. I looked up 'clove' and learned two things: first, it stands for 'cloven stone' meaning half of a stone (doubling again). Second, in some sources it seems to be 8 pounds, not seven. It looks like the binary theme is even more pervasive than you thought!

Steve_Roberts said...

12s: A friend of mine saw a guy counting on his fingers in the Souk in Amman, except it wasn't the fingers, it was the finger-bones. There are three per finger and four fingers. Using you thumb as the pointer you can easily and naturally count to twelve. Five twelves (60) involves using the whole fingers on the other hand, a gross (144) uses both hands to get to 12 times 12. I'm sure this is the reason for d12 in a dozen, 60 minutes in an hour, etc

Jody said...

I always thought 12's came from moon cycles.

Not that useful for most measurements, but quite useful for calendrical measurements.

Anonymous said...

Since traditional weight-and-measure systems like the English is an example of spontaneous order, and the metric system is one of top-down design and imposition, courtesy of the French government, this is an interesting discussion. Which system is better?

The binary nature of the traditional volume system is both beautiful and functional. Recipes are often given for four people. In a base four system, it's trivial to extrapolate from four to three, five or six. In a base ten system, you quickly run into fractions.

It's not an accident that a foot is twelve inches. Twelve has a wonderful quantity of divisors. Carpenters often have to split lengths into halves, thirds or quarters. With a base twelve system, again, the calculations are trivial.

I know that sevens were used in classic architecture. They didn't have decimals or irrational numbers, but they did know that pi approximates 22/7. Knowing this, they could do some trigonometry, and using a base-seven system simplified the calculations a lot. That doesn't explain the stone measure, but thought I would throw it in.

Is the metric system easier to learn? I grew up with it, so it's natural to me and I don't know how hard it is to learn. I ask because it's one of the advantages often cited for the metric system.

Having the same base across different categories makes for easier conversions -- for instance, lengths to volume. However, it would have been far better to pick a less arbitrary and useless number than ten.

Anonymous said...

12 comes from the zodiac. It has been scientifically proven that the Babylons used the 12 zodiac for the base 12 and the 60.