La Mathematica dell’Arte: let’s think about angles


What if you had the power to change the position of everything in the city while retaining the relationships between each locations. Do you think you’d still be able to get to where you want to go?

Prime: Okay, so that was rational length.

Naught: Yes, now I know how to measure things in your geometry.

Prime: You now know how to measure length in this geometry. What is another magnitude that is vital in geometry in general?

Naught: Lines and angles!

Prime: That’s right. But measuring angles is going to be a little trickier.

Naught: I’m excited, I used to love measuring angles with my protractor when I was younger.

Prime: Haha. It’s a little more abstract as it has been in the past. We’re going to have to recall some topics from before as well.

So as I have said before, I choose to only work with line segments with rational length of 1, which means that the numerator and the denominator of the slope are “coprime”, in other words, the only integer that divides both numbers is 1.

Naught: So like [0, 1] and [2, 9], right?

Prime: Yes, exactly. So say we have two line segments with rational length of 1, denoted L_1 and L_2, which intersect at a point on a normal x-y (Cartesian) plane. Then we can conveniently move those lines so that the intersection is on the origin and L_1 is the line that joins the origin and the point (0, 1).

Naught: You can just do that?

Prime: Yes, with the ASL_2Z transformation we talked about in Post #2, we lose no information about the two lines and their relationship. As mathematicians like to say, without loss of generality.

Naught: Okay, so where does L_2 go?

Prime: We don’t have the freedom to choose where L_2 goes but rational length is preserved to 1. This means that L_2 will be the line segment that connects the origin and a point (a, b) where a and b are coprime in order for the rational length to be 1.

Naught: Hm…let me sort that information. So you’re saying that any two line segments with rational length 1 which meet at a point can be moved to have one of the line become the line from (0, 0) to (0, 1) and the other line from (0, 0) to (a, b)?

Prime: where a and b are coprime. It’s important that they’re coprime because that preserves the rational length of 1 and no matter how many times you subtract b from a, you will never get 0.

Naught: Okay…. [9, 2] are coprime and 9 – 2*4 = 1. [15, 4] are coprime and 15 – 4*3 = 3. Sure, I believe you that if two integers are coprime then you can never get 0 by subtract one from another multiple times. But how is that important?

Prime: That is how the angle is measured. You subtract b from a as many times as you can before going negative. Let’s denote that number as a’. Then the angle is measured to be the fraction a’/b.

Naught: Ohh, that is complicated. Can we do some examples?

Prime: Sure. So using the pair of numbers you mentioned before, [9, 2], then subtract 2 from 9 as many times as you can before going negative. As you had already calculated, 9 – 2*4 = 1. So the rational angle is 1/2 (2 comes from the second entry). With the other pair you mentioned, [15, 4], we have 15 – 4*3 = 3 so the rational angle is 3/4. If you notice, the angle is always going to be a fraction between 0 and 1.

Naught: Okay, that was a lot of math.

Prime: But now you know how to measure length and angle in this geometry!

Naught: Yeah but it was more fun with rulers and protractors.

Prime: But the numbers come out to be so clean here unlike when you measure by hand. Plus the units in degrees, radians and now with the tau manifesto, it’s not as pretty.

Naught: The tau manifesto?

Prime: Yeah, there are a group of people trying to replace pi with tau, which equals 2*pi, to be mainstream. There’s no room for such quarrels in this geometry!


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