Chapter 9 of 0–21
Angles and waves
The unit circle, sine and cosine, frequency, and the angle-addition identities.
Sources for this chapter
The trained network represents numbers as angles. This chapter builds the trigonometry it uses, from scratch.
Angles in radians
An angle measures an amount of turning. We measure it in radians: a full turn is 2\pi \approx 6.283 radians, half a turn is \pi \approx 3.1416, a quarter turn is \pi/2 \approx 1.571. Angles that differ by whole turns point the same way: \theta and \theta + 2\pi are the same direction.
The unit circle, cosine and sine
The unit circle is the circle of radius 1 around the origin. Start at the point (1, 0) and turn counter-clockwise by an angle θ: you arrive at a point whose horizontal coordinate is the cosine \cos\theta and whose vertical coordinate is the sine \sin\theta:
Values to remember. \theta = 0: point (1, 0), so cos 0 = 1, sin 0 = 0. \theta = \pi/2 (quarter turn): (0, 1). \theta = \pi: (−1, 0). \theta = 3\pi/2: (0, −1). And for any θ, \cos^2\theta + \sin^2\theta = 1 (the point is at distance 1 from the origin).
Because \theta and \theta + 2\pi are the same point, cos and sin are periodic function: they repeat every 2\pi. The frequency of a wave is how many times it repeats over a range. On our 113 numbers we use the waves
which complete exactly k full cycles as x runs from 0 to 113. The angle per unit step is w_k = 2\pi k/113; for k = 14 it is ≈ 0.7785 radians.
A hand turning around the circle draws both waves: its height is the sine and its horizontal position the cosine. Press ▶ to turn it twice around.
Numbers as angles: the clock
Placing the number x at angle w_k x on the unit circle turns the 113 numbers into 113 points on a circle — a clock face with 113 hours, visited k steps at a time. Because w_k \cdot 113 = 2\pi k is a whole number of turns, x and x + 113 land on the same point: wrapping around modulo 113 happens automatically.
The running example at frequency k = 14. The point for a = 100 is at angle w_{14}\cdot 100; since 14\cdot100 = 1400 and 1400 \bmod 113 = 44, this is the same point as 44/113 of a full turn. For b = 50: 14\cdot 50 = 700, 700 \bmod 113 = 22, i.e. 22/113 of a turn. The correct answer c* = 37 sits at 14 \cdot 37 = 518, 518 \bmod 113 = 66 — exactly 44 + 22. Adding the angles of a and b lands on the angle of a + b.
Press ▶ next to a to walk a around the clock and watch a + b follow it.
Rotation, and the angle-addition identities
A rotation turns every point around the origin by the same angle. Rotating by x and then by y is the same as rotating by x + y. Rotating the point (\cos y, \sin y) by an angle x can be done with a matrix–vector product (chapter 3):
The result must be the point at angle x + y, i.e. (\cos(x+y), \sin(x+y)). Comparing coordinates gives the angle-addition identities identities [26]:
Replacing y by −y (and using cos(−y) = cos y, sin(−y) = −sin y) gives the subtraction version
Numerical check with x = 0.5 and y = 0.3: \cos 0.5 \cos 0.3 - \sin 0.5 \sin 0.3 = 0.8776\cdot0.9553 - 0.4794\cdot0.2955 = 0.8384 - 0.1417 = 0.6967, and indeed \cos 0.8 = 0.6967.
These two lines are the whole trick of the network. If you know \cos and \sin of w a and of w b — four numbers — you can compute \cos and \sin of w(a+b) using only multiplications and additions. Chapter 12 shows the network doing exactly that.
Why several frequencies? For a single k, a cosine wave is largest at one point but also nearly largest at others (e.g. \cos(w_{14}\cdot 8) = 0.998 is almost 1, the paper's App. B example [1]). That is why the network will combine several frequencies — see “constructive interference” in chapter 12.