unit_circle

Picture, if you would, a circle with a radius of one unit (the 'Unit Circle', as we call it in the biz), and its center at the point (0, 0).

If you placed your finger at the rightmost point on that circle and traced around its circumference counterclockwise, your finger's position would change both vertically and horizontally. At the start, your finger would be one unit right from the center; that's the point (1, 0). As you moved around, you'd be travelling up and left to the top of the circle, at the point (0, 1), then over to the left side at (-1, 0), down to the bottom at (0, -1), and finally back to your starting point at (1, 0) again.

If you tracked just the height of your finger (the y-coordinate of each point you passed through), you would get a graph like on the top right of this pattern. That's where the Sine function comes from!

If you tracked just your x-coordinate (how far left and right you were), you'd get the graph of the Cosine function, as seen on the lower left part of the pattern.

Finally, you might choose to log your position at particular important points on your rotation journey-- where you were after going 30° up, or 45°, 60°, 90°, et cetera. That would give you a chart like this:

That's a lot of numbers! But look-- the coordinates all repeat. The four cardinal directions are just combinations of 0 and 1 (the 1 is sometimes negative), and the others are all fractions with a denominator of 2. Their numerators repeat, too-- there's √3, √2, and 1 (which, heck, is the same as √1, if we want to get real matchy-matchy).

Sometimes those fractions are negative, too, but when they are it's just following the rules of the four quadrants-- up and right are positive, down and left are negative.

If you wanted to make a very compact version of that chart, here's what you could do: forget the negative signs (the four quadrants can keep track of that), and ditch the square root symbol (again, they basically all have one). Mark each space with a series of dots: one dot for one, two dots for two, et cetera, and an 'O' for zero. For fractions, do two layers of dots. A symbol with three dots on top and two on the bottom would mean '√(3)/2'.

And so, that's the design on the lower right! Each angle drawn in gold, with its unit circle coordinate represented by French knots.

Fun fact: there are exactly 100 French knots in this pattern, and it is one of my favorites ever. 🪡