Rotation Is What Perpendicular Does

Contents

1. The thing the trick is hiding
2. One line of algebra
3. i is the quarter turn
4. Euler’s formula, demystified
5. Where the minus signs come from
6. The board demo, measured
7. Two true things that sound contradictory
8. The same right angle, four more times
9. Where it stops
10. On “math is why”
11. Exercises

01 The thing the trick is hiding

Write the Taylor series for cosine on a board. Plot two terms, then three, then four. Each polynomial hugs the curve near zero and then hurls itself off to infinity. Keep going. Somewhere around fourteen terms a circle appears — and the room reacts, every time, because it looks as though a shape has been conjured out of arithmetic.

It has not. And the reason the demonstration is worth doing is not the surprise, it is what the surprise conceals: the circle was never in the polynomial. The circle was in one relationship, stated at the very beginning, and the polynomial is a laborious way of writing that relationship down.

The relationship is perpendicularity. That is the whole lesson, and the rest of this piece is spent earning it.

02 One line of algebra

Take anything that changes — a position, a velocity, a state — and call it x. Ask what happens to its length. Length is easiest handled squared, and the squared length is just x · x, so differentiate that:

Read it slowly, because it is the entire mechanism. If the change is perpendicular to the thing, the dot product is zero, so the derivative of the squared length is zero, so the length never moves. And conversely: if the length is fixed, the change had to be perpendicular. There was no other option available.

So “why do things go in circles instead of growing or shrinking” has an answer with no physics in it at all. A perpendicular change cannot make you bigger. All it can do is aim you somewhere else.

Figure 01

The only move a perpendicular change is allowed to make

03 i is the quarter turn

So write down the quarter turn explicitly. In the plane, rotating a point ninety degrees counterclockwise sends (a, b) to (−b, a). Call that operation J. Do it twice and you are pointing backwards:

Figure 02

Four quarter turns and you are home

1. J⁰ = I(a, b)start
2. J¹(−b, a)90°
3. J² = −I(−a, −b)180°
4. J³(b, −a)270°

That isomorphism is worth sitting with, because it dissolves the word “imaginary” completely. i is not a strange number that got invented to solve an equation. i is the quarter turn, wearing a letter. Multiplying by it is the only thing it does. And i² = −1 stops being a rule to memorise and becomes an observation about doorways: turn left twice and you are facing the way you came.

Nothing about i is imaginary. It is the most concrete instruction in mathematics: turn left, and do not change size.

04 Euler’s formula, demystified

Now put the two pieces together and watch the famous identity fall out with nothing left over.

Ask for the thing whose rate of change is its own quarter turn. Start it at 1 on the real axis. That is one short differential equation:

Two facts drop out immediately. Because i is a rotation, the velocity is perpendicular to the position — so by section 02 the length never changes, and the point stays on the unit circle. And because i is a rotation and not a stretch, |z′| = |z| = 1 — so the point moves at unit speed. Constant speed one on a circle of radius one means that after time t it has swept exactly t of arc.

Figure 03

Reading the identity off the motion

What you asked for

z′ = i z,   z(0) = 1

Velocity perpendicular to position, same magnitude. Unique solution, written e^it.

↓  unit speed on the unit circle for time t  ↓

Where it therefore is

e^it = cos t + i sin t

The point that is t of arc around the unit circle. Its coordinates are, by definition, the cosine and sine of that arc.

A note on the history, since it is usually told badly. Euler published the formula in the Introductio in analysin infinitorum of 1748, though he had it around 1740. Roger Cotes had reached an equivalent statement in logarithmic form in 1714 — but it was a geometrical result about arc length, it was never exponentiated, it carries a misplaced factor of i as printed, and it takes no account of the complex logarithm being multivalued, which is precisely the thing Euler later got right. De Moivre has a genuine prior claim to a different formula, (cos x + i sin x)ⁿ = cos nx + i sin nx, and never wrote an exponential of an imaginary quantity in his life.

05 Where the minus signs come from

Students accept the cosine series and never ask the obvious question: why do the signs alternate? Nothing else in elementary mathematics alternates for free.

Expand e^it as a series and sort the terms by their power of i. Powers of i cycle with period four — 1, i, −1, −i — which is Figure 02 again. Even powers land on the real axis with a sign that flips every step; odd powers land on the imaginary axis the same way.

That is the answer. The alternation is not a quirk of trigonometry; it is the record of a turn that keeps coming back around. Two quarter turns is a reversal, and a reversal is a minus sign, and the series is simply counting them.

One piece of rigour that should not be skipped, because it teaches a real habit: splitting a series into two sub-series and rearranging the terms is not free. It is legal here because the exponential series converges absolutely. Say so in front of the class; students who learn that term-shuffling always works will be wrong later in a way that is hard to unlearn.

06 The board demo, measured

Now back to the demonstration, and let us put an exact number on it.

First the classic picture: partial sums of the cosine series, each one clinging to the curve near the origin and then diverging catastrophically. Every additional term buys a little more good behaviour and then the polynomial’s true nature reasserts itself.

Figure 04

Partial sums of cosine: each one hugs, then bolts

• cos t, exact
• partial sums, 2 to 5 terms

Now plot the same partial sums the other way. Instead of graphing cosine against t, take the partial sums of e^it and plot them in the plane — real part across, imaginary part up. This is the version worth putting on the board, because you can watch the circle assemble itself.

Figure 05

The circle assembling itself, term by term

• the exact unit circle
• N = 6 and N = 10
• N = 14
• N = 18

And here is the number for the board, measured rather than estimated. Take “on the circle” to mean within one percent of radius one, and ask how far around each partial sum gets before it fails that test:

Figure 06

How much circle each term buys

• N = 6100°
• N = 8148°
• N = 10212°
• N = 12236°
• N = 14272°
• N = 16317°
• N = 18360°

07 Two true things that sound contradictory

A sharp student will now object. If the series converges to cosine everywhere, why does any of this happen? And if the partial sums fly off to infinity, in what sense does it converge?

Both statements are true, and the resolution is entirely about the order of two quantifiers.

Converges everywhere

Fix a value of t — any value, a million if you like — and then let the number of terms grow. The error goes to zero. The radius of convergence is infinite; cosine, sine and the exponential are entire functions.

Needs more terms further out

Fix the number of terms instead, and then let t grow. You are looking at a fixed polynomial, and a non-constant polynomial is unbounded, while cosine never leaves [−1, 1]. It must eventually fail, and it does.

The technical name for the difference

Convergence is pointwise, and uniform on any bounded interval — but not uniform on the whole real line. That distinction is the entire content of the demonstration, and it is worth naming out loud, because it is the first place most students meet a quantifier order that actually matters.

And the bound is exact

For sine and cosine, every derivative is bounded by 1, so Lagrange’s remainder gives |R_N(t)| ≤ |t|^N+1/(N+1)!. Put t = 2π in it and you can predict Figure 06 before you plot it. Do not carry this bound over to the exponential, where it is false — there you pick up a factor of e^|t|.

Which returns us to the point of the whole exercise. The polynomial is not wrong and the circle is not emerging. The circle was fully determined the moment you wrote z′ = iz. The series is a slow, local, term-by-term reconstruction of something that one right angle already specified completely, and the further you walk from where you started building, the more terms the reconstruction costs you.

08 The same right angle, four more times

If the argument in section 02 is really that general, it should turn up wherever a perpendicular relation does. It does, and recognising it is worth more than any individual result.

The harmonic oscillator. Write the state of a spring as the pair (ωx, x′) and the equation x″ = −ω²x becomes u′ = ωJu — literally the same equation as z′ = iz. The state rotates in phase space at constant rate, and its conserved length is the energy. This is why part three‘s ladder alternated: the phase-space state was turning the whole time.

The LC circuit. Charge on the capacitor and current through the inductor do exactly the same dance, with the conserved length again the total energy sloshing between the two. Same equation, different hardware.

Quantum mechanics. In part four we found the i sitting in the Schrödinger equation. Here is what it is doing. From iħ ∂ψ/∂t = Ĥψ and the self-adjointness of Ĥ, the quantity ⟨ψ|∂_tψ⟩ comes out purely imaginary — so its real part is zero. The rate of change of the state is perpendicular to the state, in exactly the sense of section 02, and that is why the norm is conserved and evolution is a rotation rather than a decay. One precision the textbooks skip: the orthogonality is with respect to the real inner product Re⟨·,·⟩. In the complex inner product an energy eigenstate has ⟨ψ|∂_tψ⟩ = −iE/ħ, which is not zero at all.

And the spinning top. Gravity pulls down on a gyroscope and it refuses to fall — it precesses instead. To the extent that the angular momentum points along the spin axis, the gravitational torque is perpendicular to it, so by section 02 it cannot shorten L; it can only swing it round. Two honesty notes, because this example is usually oversold: the torque is exactly perpendicular to the symmetry axis, not exactly to L, so |L| is conserved only in the fast-spin approximation; and a top released from rest generally nods as it goes round, so steady precession is an average, not the full motion.

09 Where it stops

Every lesson in this series carries a ledger, and this one has three genuine limits.

Not every smooth function is its series

Take f(x) = e^−1/x² for x ≠ 0 and f(0) = 0. It is infinitely differentiable everywhere, and every derivative at the origin is zero — so its Taylor series there is identically zero, while the function plainly is not. Smooth is not the same as analytic. Cosine is well behaved; do not let students conclude that everything is.

The quarter turn is a fact about the plane

“Multiply by i to rotate” works because a plane has exactly one rotational direction. In three dimensions rotations do not commute — turn a book twice in one order, then the other, and it ends up facing differently — so no single number can encode them. Hamilton chased a three-dimensional version for years and there is none to find: the only real division algebras have dimension 1, 2, 4 or 8. That is Frobenius in 1878 for the associative case, and Hopf and then Bott, Milnor and Kervaire to close it without associativity. His answer in October 1843 was to go to four.

And the slogan slightly overreaches

“Perpendicular gives you a circle” is not quite right, as Figure 01 already conceded. Perpendicularity alone gives constant length. What produces the circle is the stronger statement z′ = iz, which fixes the magnitude of the change as well as its direction. The defensible version is the one in section 04: the thing whose change is its own quarter turn moves at unit speed on the unit circle, and reading off its coordinates gives Euler’s formula.

10 On “math is why”

Which brings the series back where it started. Part one built a circle out of straight lines. Part six says why it could not have built anything else.

On Max Tegmark’s reading — the one this series runs on — the mathematical structure is not a description laid over the physical world; it is the world, and physical properties are positions inside it. Rotation is the sharpest case yet, because there is nothing left over once you have the relation. A cyclotron, a spring, an LC circuit, a wavefunction and a spinning top are not five phenomena that happen to resemble each other. They are five places where the same structure — change perpendicular to state — is instantiated, and the circle is not a picture we draw of them. The circle is what that structure is.

Put the direction of explanation the right way round and the whole series reads differently. The circle does not explain e^it. e^it — turning that never changes length — explains the circle.

Fourteen terms do not create a circle. They spend fourteen terms catching up with a right angle.

11 Exercises

1. Prove it both ways Show that x′ ⊥ x implies |x| is constant, then show the converse. Then find a curve in three dimensions that satisfies the condition and is not a circle, and say exactly which hypothesis of section 03 it fails.
2. Verify the isomorphism Multiply [[a, −b], [b, a]] by [[c, −d], [d, c]] and confirm you get the matrix for (a+bi)(c+di). What does the determinant turn out to be, and why should you have expected that?
3. Predict Figure 06 Use |R_N(t)| ≤ |t|^N+1/(N+1)! at t = 2π to find the smallest N guaranteeing 1% accuracy all the way round. Compare with the measured value of 18 and explain why the bound is conservative.
4. Find the perpendicular Pick any oscillating system not listed in section 08 — a pendulum, a tuned circuit, a predator-prey cycle, a swaying bridge. Identify the state and its rate of change, and determine whether the relation really is perpendicular or only loosely resembles one. Cases where it fails are the more instructive ones.
5. Argue the other side Take the position that the series is the real object and the differential equation is the summary — that cos simply is its power series and the geometry is an interpretation laid on top. Make the strongest case in one paragraph, then say what section 10 has to give up if you are right.