Quaternions from First Principles

A conceptual introduction to quaternions, built from geometric and algebraic motivations, followed by concrete representations and examples, especially their role in rotations.

quaternionsalgebrarepresentation-theorygeometry

1. Why quaternions exist at all

The story of quaternions begins with a very practical question:

How should we represent rotations in three-dimensional space?

Complex numbers already give a beautiful answer in two dimensions. If we identify the plane with the complex numbers, then multiplication by exp(iθ) = cos(θ) + i·sin(θ) is exactly a rotation by angle θ. Rotations become algebra.

Naturally, one asks: Can we do the same thing in three dimensions?

William Rowan Hamilton tried for years to find a three-dimensional analogue of complex numbers. He eventually proved something disappointing but fundamental:

There is no way to build a 3-dimensional division algebra that behaves like the complex numbers.

So Hamilton relaxed a constraint. He allowed one extra dimension.

That extra dimension is not an accident—it is the price of encoding 3D rotations algebraically.

2. Defining quaternions from scratch

A quaternion is an expression of the form

q = a + b·i + c·j + d·k

where a, b, c, d are real numbers, and the symbols i, j, k satisfy the multiplication rules:

i² = j² = k² = ijk = -1

From these rules, all others follow:

  • ij = k, jk = i, ki = j

But crucially:

  • ji = -k, kj = -i, ik = -j

Quaternion multiplication is non-commutative.

This is not a flaw. It is a feature that mirrors the geometry of 3D rotations.

3. Scalar and vector parts

Every quaternion splits naturally into two pieces:

  • Scalar part: the real number a
  • Vector part: the expression b·i + c·j + d·k

This suggests identifying the vector part with a vector in R³. We can write q = (a, v) where v is a 3D vector.

Under this identification, quaternion multiplication becomes:

(a, v)(b, w) = (ab - v·w, aw + bv + v×w)

This formula already hints at something deep: the dot product and cross product of R³ are built into quaternion multiplication.

4. Conjugation, norm, and inverses

Define the conjugate of a quaternion by flipping the sign of the vector part:

q̄ = a - b·i - c·j - d·k

Then define the norm (squared) as:

|q|² = q·q̄ = a² + b² + c² + d²

Key facts:

  • |q|² ≥ 0, and |q| = 0 if and only if q = 0
  • |pq| = |p|·|q| (multiplicative property)
  • Every nonzero quaternion has an inverse: q⁻¹ = q̄ / |q|²

Thus, the quaternions form a division algebra over the real numbers.

5. Unit quaternions and rotations

The real power of quaternions appears when we restrict to unit quaternions: those with |q| = 1.

Write a unit quaternion as:

q = cos(θ/2) + u·sin(θ/2)

where:

  • u is a unit vector in R³
  • θ is a rotation angle

This is not a coincidence. It encodes rotation.

6. Quaternion representation of rotations

Identify a vector x in R³ with the pure quaternion (scalar part zero):

X = 0 + x₁·i + x₂·j + x₃·k

Then define the action:

X ↦ q·X·q⁻¹

This map:

  • preserves length
  • preserves angles
  • produces a rotation of R³

In fact:

Every rotation in R³ arises this way.

Example: rotation about the z-axis

Let q = cos(θ/2) + k·sin(θ/2).

For a vector (x, y, z), identified with X = x·i + y·j + z·k, the transformation X ↦ qXq⁻¹ gives:

(x, y, z) ↦ (x·cos(θ) - y·sin(θ), x·sin(θ) + y·cos(θ), z)

This is exactly the standard rotation matrix about the z-axis.

7. Representation-theoretic viewpoint

From a representation theory perspective, several important facts emerge:

7.1 The group structure

The unit quaternions form a group under multiplication, denoted Sp(1), which is isomorphic to SU(2).

This group acts on R³ by conjugation: X ↦ qXq⁻¹.

The image of this action is SO(3), the group of rotations.

The map from SU(2) to SO(3) is:

  • surjective (every rotation is hit)
  • two-to-one (q and -q give the same rotation)

This explains the famous double cover of rotations: SU(2) → SO(3).

7.2 Spin and half-angles

Because the map is two-to-one:

  • q and -q represent the same rotation
  • a 2π rotation corresponds to -1 in SU(2), not the identity

This is the algebraic origin of spinors and half-integer spin in quantum mechanics.

8. Matrix representations

Quaternions admit faithful matrix representations.

One standard embedding maps a quaternion a + b·i + c·j + d·k to a 2×2 complex matrix in SU(2).

Under this identification:

  • quaternion multiplication ↔ matrix multiplication
  • unit quaternions ↔ special unitary matrices

This connects quaternions directly to Lie groups and representation theory.

9. Why quaternions are still used

Quaternions are not historical curiosities. They remain indispensable because:

  • they represent rotations without gimbal lock
  • they interpolate rotations smoothly (slerp)
  • they encode geometry directly, not via coordinates
  • they expose the deep structure behind SO(3)

In short, quaternions are geometry made algebraic.

10. Summary

  • Quaternions extend complex numbers by adding two extra imaginary directions.
  • Non-commutativity is essential and geometrically meaningful.
  • Unit quaternions encode rotations in R³.
  • Representation theory reveals their identity with SU(2).
  • Spin, double covers, and half-angles emerge naturally.

Quaternions are not just a computational tool—they are a window into how algebra, geometry, and symmetry fit together.

There’s always more to explore, but I am glad I am on the path.

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