Pi aside: Degrees are even worse

Make no mistake: When I run the world, π will be set aside in favor of τ. For those who haven’t heard, there’s a movement to use 6.28318530718… (that is, 2π) as the basic variable relating to circles rather than 3.14159265359…. I personally feel that there’s a strong pedagogical motivation at the secondary education level (specifically, τ is the circumference of a unit circle), but beyond that, it doesn’t matter.

However, all the chants of “pi is wrong” aside, I think there’s an even larger elephant in the room: The fact that, before we even mention π, we divide a circle into 360º. π is not an arbitrary number; it’s motivated by mathematical logic. For instance, while τ represents the perimeter*of a unit circle, π represents its area. We can represent the perimeter of a circle with either τr or πd. There’s ample justification for using π. There’s even justification, by the way, for using η, which is 1.57079632679… or π/2. These are not arbitrary or random numbers.

360, on the other hand, embeds a number system that’s thousands of years old and should have been forgotten.

We use base 10, although when I run the world, we’ll use base 8.† But I’m not that radical yet, so let’s keep using base 10. The metric system fully embraces this notion by filling everything with multiples of 10. Once upon a time, there was even something of an attempt to replace degrees with a metric equivalent, using the right angle as the base; these are called gradians. There are 100 gradians in a right angle, and 400 in a circle.

Gradians stumbled, and one of the most obvious reasons is that they’re messy to use with the two of the three most rudimentary triangles. The “special” right triangles are the scalene 1:2:3 triangle (30°-60°-90°) and the isosceles 1:1:2 triangle (45°-45°-90°). Equilateral triangles are 1:1:1 (60°-60°-60°). Converting these three to gradians gives us 33 1/3^g-66 2/3^g-100^g, 50^g-50^g-100^g, and 66 2/3^g-66 2/3^g-66 2/3^g, respectively. Messy, messy.

To see where gradians go wrong, we have to look at the base system that 360 comes from in the first place. The Babylonians had a fascination with base 60. Why base 60? Presumably because the Babylonians were mutants with five arms on each side, thus giving them 60 fingers and toes. (Actually, it probably had to do 60 being the smallest number that is cleanly‡ divided by 2, 3, 4, 5, and 6, but I like the image of those Babylonian Indian Gods.)

But it’s base 60 they were obsessed with, not 360. 360₁₀ = 60₆₀, which isn’t nearly as special in base 60 as it looks using our base 10 numeral system. It’s just 6 · 10. Big whoop.

I believe that’s because, for the Babylonians, the basic angular shape wasn’t the circle or the right triangle: It was the corner of an equilateral triangle. There happen to be six of those that can be packed into a circle. The Babylonians thought that the worthiest size of an angle was the one that allowed for 60 (that is, 10₆₀) to be packed into a corner of the most balanced of triangles.

So: Keep the baby, dump the bathwater. What happens if we divide the corner of an equilateral triangle into 100 wedges, instead of 60? What happens?

Our three basic triangles become 50^w-100^w-150^w, 75^w-75^w-100^w, and 100^w-100^w-100^w. A circle has 600^w. A triangle has 300^w. A square has 600^w. A pentagon, 900^w. A right angle is 150^w.

A related option would be to divide these numbers by 5, or even 25, to make the numbers less daunting, but I don’t think that really provides much of an advantage, and it requires us to rely on minutes and seconds (or, really, their base 10 equivalents) more quickly.

Granted, this system is still arbitrary. Dividing the circle into wedges using something other than a multiple of η (such as π or τ) is going to be arbitrary. But at least it’s arbitrary in base 10. We could tell students clearly: The corner of an equilateral triangle is the basic unit, because a three-sided regular polygon is the simplest polygon there is. It makes sense to chop that up into some multiple of 10, because we use base 10. Our fingers have generally prevailed over the whimsies of the Babylonians: Let’s complete the process.

* When I run the world, we’ll also stop using a specific word for the perimeter of a circle.

† We apparently use decimal (base 10) for the most speciocentric of reasons: That’s how many fingers we have. If cartoon characters ran the world, it would be octal (base 8). But base 10 is ugly, particularly when trying to interface with the binary (base 2) used by computers. 0.1₁₀ is the infinitely repeating 0.0001100110011…₂. Binary, meanwhile, creates unwieldy numbers. Octal and hexadecimal (base 16) allow for clean interface with binary while keeping numbers at a reasonable length. The programmer inside me says that hexadecimal is superior to octal, and logically, it is. However, octal is closer to decimal and requires no tweaking to our existing number system, while hexadecimal cheats by co-opting 6 letters to be numbers, leading to confusing among the uninitiated. So logic be damned, I’m rooting for octal.

‡ When I run the world, we’ll also stop using “even” to refer to both numbers divisible, ahem, evenly by 2 and to numbers that are the multiple of two integers.