Quaternion Coils

A point on a circle carries one number: an angle. A point on S^3 can carry a quaternion — four numbers w + xi + yj + zk — which is enough structure to describe rotations in three dimensions. This chapter introduces just the checked algebra and the one sign surprise that the spin lessons later depend on.

Lesson Goal

Learn the two rules that make quaternions more than points on a circle — non-commutative multiplication and the q / −q sign ambiguity — and see each one as a concrete, checked fact.

The Idea

Quaternions multiply with three imaginary units i, j, k obeying:

i² = j² = k² = −1
i·j = k        j·i = −k

The second line is the headline: order matters. Swapping the factors flips the sign. On an ordinary circle, adding angles never cared about order; here it does. That non-commutativity is exactly what lets quaternions encode 3D rotation, where “turn then tilt” differs from “tilt then turn.”

A unit quaternion has w² + x² + y² + z² = 1 — a point on S^3. The unit quaternions are closed under multiplication, and for a unit q the conjugate q* = w − xi − yj − zk is its inverse. Those are the checked algebra seeds below.

Worked Example

Multiply i and j both ways:

i · j = k
j · i = −k

Same two inputs, opposite results. Now check a unit quaternion: q = (0, 1, 0, 0) = i. Its norm is 0² + 1² + 0² + 0² = 1, so it sits on S^3, and q* = −i, with q · q* = i · (−i) = −i² = 1. The conjugate really is the inverse.

The sign surprise: q and −q are different quaternions, but they produce the same 3D rotation when you use them to rotate a vector (rotation uses q twice, as q v q*, so the two minus signs cancel). That is why later spin lessons say S^3 is a double cover of rotation space — two quaternion records, one physical turn.

The Double Cover Is Real Now

The reason q and −q give the same rotation has a name: the double cover S³ → SO(3). On the actual real quaternions, its algebraic core is now Lean-proved:

conjugation v ↦ q v q⋆ by a unit quaternion preserves the norm — it is a genuine isometry, the property that makes it a rotation;
it preserves the real part, so it acts on the pure-imaginary copy of ℝ³;
it is a monoid homomorphism: rotating by a·b is rotating by a then b;
the real unit quaternions are exactly +1 and −1, and they act trivially.

Put together: conjugation is a norm- and vector-preserving homomorphism that glues q to −q — the heart of the two-to-one cover. And the kernel is now exactly computed: a unit quaternion fixes every vector if and only if it is +1 or −1 (the hard direction: acting trivially forces it to commute with everything, hence to be real, hence ±1). So the two representatives glued together are precisely q and −q, and no others. (What is not yet claimed, tracked as roadmap: that the action hits every rotation — surjectivity onto SO(3).)

Common Mistake

The proved core above is the algebraic heart of the double cover, not the whole theorem: surjectivity onto SO(3) and quotient-topology formalization are still roadmap, not claimed. And a checked algebra fact is never a claim about physical robotics. See What “Proved” Means Here.

Checkpoint

Compute j · k and k · j. Do they agree? What does that tell you about the order of quaternion multiplication? (Hint: j·k = i, k·j = −i.)

Source Trail

Paper source: S3 Quaternion Coils

Spin source: S3 Spin Double Cover Roadmap