On Angular Measurement: Part One

Note: this blog post is about nine months old. I have a way of wandering off from a promising subject and getting involved in other things. In any case, as this whole discussion is now obsolete thanks to tau day, I’m publishing this in its raw form. For a proper treatment of the subject, click through the above link.

Our last discussion of standards dealt with the pure numbers of dimension, namely, base and scale. Lets turn our attention to angles.

When we start to measure the dimension of length, angles will quickly leap to the foreground. Any two points in space have a length between them, and that’s all we can really say. Any three points in space have three lengths defined by their distance. There are also three angles, which are also defined by their distance.

There are a lot of different ways of referring to those angles. The most common here in North America is the ‘degree’. The degree is an example of preferred numbers being baked into the system. In this case, the bakers were the Sumerians, and the degrees were literally baked into tablets of clay.

The Sumerian culture, and their Babylonian successors, used a counting system based on the number 60, called sexagesimal. There’s a reason for this, namely that 60 is the smallest number evenly divisible by the numbers 1 through 6.

This makes fractions easy to work with. We know all about this, actually, when we tell people we’ll see them in ten, fifteen or twenty minutes, a sixth, quarter and third of an hour respectively. Our time system is inherited from the Sumerians, ultimately, and reflects their counting base.

It was the Sumerians also who divided a circle into 360 degrees. 360 is also close to the ratio days:years, which I am not convinced is a coincidence; 360 is a very special number and Mercury, for example, has a precise 3:2 ratio of days to years.

I think the convention was adopted for a simple, practical reason: 360 and 60 are both highly composite numbers. This means they have more divisors than any number smaller than them. This makes them better than any smaller choice for dividing into even fractions, which is one of our favorite things to do with circles.

So 360 makes sense. The metric system, early on, made various abortive attempts to divide a circle by 400 degrees rather than 360, but this never caught on. 400 allows for easy right-angle math, but sucks for anything involving 3, hence 6 and 12 are also awkward.

The metric system, and math in general, has moved on to a more rational system for representing angles. It is the radian, and it gets to the heart of what an angle is: a relationship between a circle and its radius.

We all know that c = 2 pi r, that the circumference of a circle is its diameter times pi. This is the basis of the radian.

Lets think about the face of a clock, and the minute hand. The minute hand is a radius of a circle, and sweeps out one circle every hour. In that hour, the tip of the minute hand will have travelled the distance of the radius times 2pi. It will also have swept through 360 degrees of arc.

In the time it takes for the tip of the minute hand to travel the exact distance of the radius, it will have swept out one radian of arc.

Or, in one hour (360 degrees, one circle) the minute hand will have swept out 2pi radians of arc. 180 degrees is pi radians, 90 degrees is pi / 2 radians, 45 is pi / 4 radians and so on.

This is a better standard, for some purposes. Doing actual math involving angles is utterly dependent on it, for example. It is able to express any rational subdivision of a circle as a whole number divisor, for example, pi/56 radians is 23 even divisions of a circle, which is 15.652…. degrees.

The radian, however, is confusing and counterintuitive, and a lot of smart people don’t really get it. It makes people uncomfortable. There’s a reason for that; my next post will cover that, and propose another way of talking about angles.

Published in: on June 29, 2010 at 7:17 pm  Leave a Comment  

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