Sunday, June 21, 2009

The Mesoamerican Calendar & Vigesimal

Many people know that the Maya calendar "predicted" a "catastrophe" in 2012. But the Maya calendar was strange and interesting. Rather than solar years, the Maya Long Count calendar counted the number of days from the beginning of their mythological epoch thousands of years earlier.

Counting the number of days in 4,000 years is somewhat unwieldy in our numeral system—it nearly a million and a half days. Rather than our decimal (base-10) system, the Mayans used a vigesimal (base-20) system of numbers. In vigesimal, each position in the numeral is worth more: rather than 1s, 10s, 100s, and 1000s, each position denotes 1s, 20s, 400s, and 8000s. Rather than count a million and a half days in the seven digits it would take in decimal numbers, they counted it in just five digits. The Long Count calendar was not a pure vigesimal count; the Mayans modified it so the first two digits would count a period of 360 days rather than 400 days, which was close enough to a solar year to be convenient for their purposes.

Imagine we used a system of counting days, and we only used five digits to do so. The day 13,765 would be the 250th day of the 37th Julian year. The day 99,999 would be the 285th day of the 273rd year. On the next day, the count would reset to zero, because we aren't using a sixth digit for the hundred thousandth day. In a decimal count, this would happen about every 273 years. In the Maya's modified vigesimal count, it would have happened every 2,880,039 days, or roughly every 7,885 years.

If, using our decimal system, we began counting the number of days since April 27, 1821, then December 20, 2012, would be the 69,999th day. The next day (December 21) would be the 70,000th day. Something analogous happens on that day in the Mayan calendar. Obviously, it's a great chance for someone to get rich selling stuff to New Agers.

A couple of months ago, I was working on a fictional puzzle that would use a vigesimal day-count calendar, which is an unfamiliar and confusing concept for most people, and was monkeying around with base-20 math and how to represent it graphically. One of the problems with thinking about numbers in other bases is that they're very difficult to conceptualize for a number of reasons. One is that we don't have many words to describe these numbers. Another is that we don't have adequate glyphs to represent them: there are only ten Arabic numerals.

Most of us have been trained for years in base-10 math, and find it very intuitive, easy to talk about, and easy to think in. The English language has remnants of alternative systems: base-12 or dozenal (dozens, grosses, and great grosses), and base-20 or vigesimal ("Four score and seven years ago..."). These systems are pretty common in the world's cutures, because they are easier to divide. While ten is evenly divisible only by two and five, twenty is divisible by 2, 4, 5, and 10, and twelve is divisible by 2, 3, 4, and 6. The base-60 or sexagesimal system of the Sumerians and Babylonians consisted of six base-10 units, and survives in measurements of time and geometry.

Computer scientists have been using octal and hexadecimal, base-8 and base-16, for decades because they are convenient ways to handle binary, base-2, numbers. But computer scientists don't have numerals to represent them, instead borrowing the letters of the alphabet to stand in for the missing digits. It can be a somewhat confusing system.

In 1968, Bruce A. Martin proposed a new series of numerals that drew out the 15 numerals according to their bit position. I tried some similar ideas, drawing out the numbers in a way somewhat similar to the Maya numerals. There is a design problem with this, though: the numerals are difficult to distinguish and sometimes rely on counting out every digit. In Martin's system, for example, the glyph for three is hard to quickly and clearly distinguish from the glyph for five. Good font design, or using an angle bracket instead of a straight line for the back, could help, but it's an inherent problem.

Working with invented alphabets and scripts, it's clear that a quick and easy way to generate glyphs for sounds is to flip or rotate existing characters. In the Latin alphabet, M, W, q, p, b, and d, are all flipped or rotated. So, I tried something similar for the Arabic numerals, and borrowed the character for 10 from kanji. It's certainly not a perfect system: 1 and 8 rotated look like other symbols, and need to be somewhat modified. Ten looks very like a plus symbol, and they would need to be clearly distinguished in a font. OTOH, for those of us accustomed to the decimal system, it's easy to remember which glyph represents seventeen.

Another reason that it's easy to think in decimals and hard in other bases is that the decimal positions all have names. Ten tens is a hundred, ten hundreds is a thousand, a thousand thousands is a million, a thousand million is a billion, a million million is a trillion. English has some words in dozenal: a dozen dozen is a gross, a dozen gross is a great gross. But what is a score score? I borrowed and twisted around some old Scottish and English words to have some vocabulary to play with.

A score scores is a dubbock; a score dubbocks is a skelling, and a score skellings is a skell. In decimal, a score is 20, a dubbock is 400, a skelling is 8,000, and a skell is 160,000. Thus, two skelling, twelve dubbock, nine score and sixteen is equivalent to (in decimal) twenty thousand, nine hundred, and ninety-six. Somewhere after a skell is a villion, the vigesimal million.

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