Angles and waves

The angle between two vectors

The dot product lesson defined cosine similarity as a·b divided by both lengths. That number is exactly the cosine of the angle between the two vectors, so you can turn it back into an angle with the inverse cosine, math.acos:

import math
a = [3, 1]
b = [1, 2]
dot = sum(x * y for x, y in zip(a, b))
cos_sim = dot / (math.hypot(*a) * math.hypot(*b))
print(round(cos_sim, 3), round(math.degrees(math.acos(cos_sim)), 1))

A cosine similarity of 1 is an angle of 0°; 0 is 90°; −1 is 180°. Reading similarities as angles can make them easier to feel: two embeddings at a cosine of 0.5 sit 60° apart, which in a space of 1,536 dimensions is a strong sign of related meaning.

import math
print(round(math.degrees(math.acos(1 / math.sqrt(2))), 1))

Waves: amplitude, period and frequency

As the angle keeps turning, sine repeats forever: up, down, up again, every 2π. That makes it the natural shape for anything that cycles. Three numbers describe a wave:

• the amplitude: how far it swings above and below the middle;
• the period: how long one full cycle takes;
• the frequency: how many cycles fit in one unit of time, 1 ÷ period.

A pattern that repeats every 7 days is sin(2π × day / 7): after 7 days the angle has gone exactly once round. Add a middle value and an amplitude, and you have a model of weekly sales:

import math
for day in range(8):
    print(day, round(100 + 20 * math.sin(2 * math.pi * day / 7), 1))

Day 7 matches day 0, because the wave has completed exactly one period.

Notice what each number in the formula does. The 100 sets the middle line the wave swings around. The 20 is the amplitude: sales rise to 120 at the peak and fall to 80 at the trough. The 7 under the day is the period. Change it to 30 and the cycle becomes monthly; change it to 365 and it becomes yearly. Real data rarely follows a single clean wave, but a forecasting model that knows "there's a weekly rhythm here" can separate that rhythm from the trend underneath it, and that is often the difference between a forecast that's useful and one that chases noise.

import math

sales = []
peak = 0

print(sales, peak)

Positions as waves

A transformer reads all its words at once, so it needs a way to know where each word sits. The original transformer gave every position a list of sines and cosines at many different frequencies. Fast waves change a lot from one position to the next, so they tell neighbors apart. Slow waves barely change between neighbors, but differ across long distances, so they tell far-apart positions apart.

It works like a clock: the second hand distinguishes moments a second apart, and the hour hand moments hours apart. Read all the hands together and every time of day gets a unique pattern. Many recent models use rotary encodings instead, which rotate each word's vectors by an angle that grows with its position: angles again.

import math
for pos in range(4):
    fast = round(math.sin(pos * 1.0), 2)
    slow = round(math.sin(pos * 0.01), 2)
    print(pos, fast, slow)

Angles and waves in AI

Beyond similarity and positions, waves are the language of seasonality in forecasting, where a weekly or yearly cycle rides on top of a trend. Audio is a sum of waves, which is how speech models see sound. Fourier features feed a network sines and cosines of its inputs so it can learn fine detail. And whenever a library function takes an angle, check whether it wants degrees or radians before you trust the answer.

import math
print(round(math.sin(math.pi / 2), 4))