How Straight Lines Make a Circle

Every trigonometry course I have ever seen starts the same way: draw a unit circle, put a point on it, and define cosine and sine as its shadow on the two axes. Rotation is assumed, and the functions are read off it. That order works, but it hides something. It makes the circle a prerequisite — a thing you must already believe in before the mathematics starts. Students memorize it. Few of them ever see what causes rotation.

This lesson runs the causality the other way. We will start with nothing but a rule for making straight arrows — each one perpendicular to the last, each one shorter — and we will watch a circle assemble itself out of them. No circle is defined anywhere in the setup. It arrives as a conclusion.

The circle isn’t an ingredient. It’s the result.

Take the exponential series and feed it an imaginary input. Nothing else is needed:

Look at how each term relates to the one before it. To get term n+1 from term n, you multiply by exactly one thing:

Multiplying by i is a pure 90° left turn. Multiplying by θ/(n+1) is a shrink. That is the entire engine: turn left, shrink, repeat. Because every factor of i lands on the cycle +1, +i, −1, −i, every single term is either purely real or purely imaginary — so when you draw the terms as arrows laid tip-to-tail, every arrow is exactly horizontal or exactly vertical, and each is perpendicular to the one before it.

Now the punchline hiding in plain sight: the horizontal steps, added up, are the series for cos θ. The vertical steps, added up, are the series for sin θ. Cosine is not a shadow of a circle here. Cosine is how far east the walk ends up; sine is how far north.

The demo below is the whole argument, live. The left pane is the top-down view: the arrow chain for the current angle θ, one term at a time. The right pane is the side view: the graph of the partial sums of cos x. Add terms with +1 term. Then press ▶ sweep θ together with trace tip and watch the pink tip draw a curve nobody asked for. Only after it closes should you press reveal unit circle.

One number in the readout deserves attention: the tip’s distance from the origin. The construction says nothing about that distance staying at 1. It just does — for every θ, once enough terms are in. Constant distance from a fixed point is the definition of a circle, and here it is an output of turn-left-shrink-repeat, not an axiom.

The worry every student has about an infinite series: if I keep adding things forever, why doesn’t the sum wander off? Why doesn’t term nine ruin what terms one through eight built? The answer is the factorial. The n-th arrow has length θⁿ/n!, and n! grows so much faster than θⁿ that each correction is smaller than the last — brutally so.

There is a second, structural reason the early picture survives, and it matters more than the shrinking: adding a term never edits the terms before it. The five-term polynomial contains the three-term polynomial verbatim — same coefficients, untouched. New terms are appended, not blended in. Near zero, where θⁿ is microscopic, the new term contributes essentially nothing, so every partial sum agrees there; farther out, where the old polynomial was starting to fail, the new term does its work. Press zoom near 0 in the demo and look: every curve lies on top of every other curve near the center. Higher-order terms are polite. They fix the frontier and leave the settled territory alone.

The sequence matters. The whole point is that students see the circle arrive, so do not show it early. This is the order I would run:

1. Start with one arrow

   Set terms to 1. One acid arrow of length 1 pointing east. Ask the room: where is the circle? There isn’t one. There is nothing here but a straight line.

2. Add terms one at a time at a fixed angle

   θ = 1 works well. Each press of +1 term adds one perpendicular correction, visibly smaller than the last. Say the rule out loud each time: turn left, shrink.

3. Sweep θ with the trace on

   The pink tip starts drawing. Do not name the shape. Let someone in the room say the word “circle” first — it will not take long.

4. Only now reveal the unit circle

   The dashed circle lands exactly on the trace. The thing they were told to assume in trigonometry has just been manufactured from straight lines in front of them.

5. Zoom near zero on the graph

   Every partial sum agrees near the center. This is the quiet lesson underneath: each term has a job, the jobs do not overlap, and the low-order terms are load-bearing and never touched again.

Euler wrote this series down in the 1740s. Nearly three centuries later it is still, I think, the shortest honest answer to a question most students never get to ask: what actually causes rotation? Not a circle. A rule. Turn left, shrink, repeat — and the circle takes care of itself.