On Angular Measurement: Part Two

Post two of angular measurement. Note that a ‘slice’ is equal to tau radians. Note that this is the difference between wandering in the wilderness and coming down from the mountain with the Word.  Ah well.

In the last post, we talked degrees and radians. Radians are less arbitrary in some sense, but less intuitive also, at least for most people, and I believe this to be more than just training.

Here’s an example. If you want to know what one radian is in degrees, it’s c. 57.2957795 of them. This does a weird one on our intuition. When we divide a circle into various angles, most of the easy ways give us nice numbers: six ways is 60 degrees, ten ways is 36 degrees, twelve is 30 and so on. Why would the relationship between a single radian and a single degree be irrational?

The reason is Pi, which is itself irrational. A whole circle of arc is defined as 2 * pi radians, because the radian is defined in terms of the relationship between a circle’s radius and its circumference. This is known as circular reasoning.

There are eminently good reasons for this definition, as those who retained their trigonometry know. But when using angles for real work in the world, the value of this mathematical abstraction is not always obvious, and the degree is used by reflex. The real conversion between degrees and radians is 1 degree = 2pi/360 radians; it’s not rational, or pretty, but at least it’s exact.

I propose we simplify our lives, and define a new unit, which I will call the metric slice. A slice is defined as such: 1 slice = 2 pi radians.

Picture a steaming, delicious pi, fresh from the oven. Slice it in half, and each side is half a slice. Slice it into thirds, and each is a third slice. Slice it into twenty-thirds, and everyone will have one-twentythird slice.

Or if you prefer, your .5 slice could be Pi radians. Your seventeenth slice could be pi/8.5 radians, or 2pi/17 for the homies in the know. Conversion is simple, you divide 2pi by your slice. Since you have to remember 2pi to work with radians in the first place, this is free.

Conversion to degrees is also simple: a one-eighth slice is 360*slice, in this case, 45. I predict this will not be hard to remember.

Fun fact: the number of slices cut by a harmonic oscillator in a second is its frequency in Hz! Beat that, radians!

The reason this is so much abundantly simpler is that the natural unit of the angle is the circle, and should be 1. There’s irrationality built into the way the universe is composed; we should tuck it as far out of the way as possible, to ease stress on our poor wet minds.

There’s also a unit of solid angle, which we may as well call the ‘halo’. The concept is so obvious as ato require little elaboration. The steradian, by contrast, is a headache-inducing extension of the idea of the radian into 3 dimensions. To convert, multiply halo by 4 pi, making 1/4 halo equal to pi steradians as expected. If you’re willing to guess that a solid wedge projected through one face of an icosahedron has 1/20th halo of solid angle, you’re well on your way to understanding the utility vs 1/5 pi steradians.

I’m convinced that the use of halo and slice units can significantly simplify and aid understanding of trigonometry and conic sections, among other areas of interest, perhaps even offering a simpler link between these disciplines and symmetry theory. I haven’t had the time to significantly pursue this; in the meantime, I offer the slice and halo as conceptual aids.

The standards, natural dimensions and scales we choose have consequences, in the physical world and in our intellectual world also. Since a standard is a concept we apply, not an object with independent existence, applying different standards to the same measurement can and typically does have important effects on the ways in which the standard is used.

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Published in: on June 29, 2010 at 7:21 pm  Leave a Comment  

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