The Rule of Turns
A slide rule for SO(3), the group of rotations of space, after Hamilton.
Every rotation of a sphere is a turn about some axis through some angle. Draw it as an arc of a great circle, half as long as the angle, on the circle whose pole is the axis, and let the arc slide freely along its circle. Then two rotations compose like two sides of a spherical triangle: slide the arcs until they meet head to tail, and the third side is the product, read off the globe as a slide rule is read.
Plate I. The globe, two slides, and the cursor that closes the triangle.
Plate II. One step, repeated, cuts its own scale: a ring for the point, and a ring for the turn.
Plate III. The commutator a b a⁻¹ b⁻¹ of two small turns about perpendicular axes.
Plate IV. A globe turned through 720°, and the ribbon that ties it to the frame.
¶Commentary
Rotations and quaternions. A rotation of space about a unit axis n through an angle θ corresponds to the unit quaternion q = cos θ/2 + sin θ/2 (n₁i + n₂j + n₃k), and it acts on a vector v, written as a pure quaternion, by v ↦ q v q⁻¹. Composing rotations is multiplying quaternions: first q₁, then q₂, is q₂q₁. The unit quaternions form the group SU(2), a three-sphere, and q and −q give the same rotation. So SU(2) covers SO(3) twice.
Hamilton's arcs. Hamilton called a unit quaternion a versor, and in his Lectures on Quaternions (1853) drew it as an arc of a great circle on the unit sphere, an arc that may be slid along its own circle without changing the versor. For unit vectors A and B the quotient B A⁻¹ is the versor whose arc runs from A to B. Then B P⁻¹ · P A⁻¹ = B A⁻¹: arcs that meet head to tail at P compose to the arc from the first tail to the last head. That one line is the whole instrument.
Why half the angle. A reflection in the plane perpendicular to A, followed by a reflection in the plane perpendicular to B, is a rotation about A × B through twice the angle between A and B. So an arc of length θ/2 stands for a rotation through θ, and an arc of 180° stands for a full turn. Arcs longer than a half circle are allowed, and they are the other sheet of the cover: the arc of 360° − φ in one direction gives −q for the arc of φ in the other.
Rodrigues. Euler had already written a rotation with four parameters, the components of q, and Olinde Rodrigues in 1840 found how the parameters of two rotations combine to give those of the product. His rule is Hamilton's product, three years before quaternions. The same paper's formula v cos θ + (n × v) sin θ + n (n·v)(1 − cos θ), for turning one vector, is what draws every arc on these plates.
Order matters. The products qbqa and qaqb are conjugate, qaqb = qa(qbqa)qa⁻¹, so they turn through the same angle about different axes. On the rule this is the two triangles on either side of the crossing, mirror images in the plane through the two poles, with cursors of equal length on different circles.
Curvature. On a flat rule, sides add. On the sphere the triangle's third side falls short of the sum, and the shortfall is the curvature at work. For small turns ε about perpendicular axes the commutator a b a⁻¹ b⁻¹ is a rotation through ε² + O(ε³) about the third axis, the bracket [X, Y] = Z of the Lie algebra. The pole it carries round encloses an area that agrees to the same order, as the area of a small loop agrees with the turning of a vector carried round it, by Gauss and Bonnet.
Two sheets. A loop of rotations from the identity back to itself, once round about a fixed axis, cannot be shrunk to a point in SO(3); twice round, it can. The fundamental group π₁(SO(3)) is ℤ/2, and SU(2), being a three-sphere, is simply connected. The belt shows it with a ribbon instead of a loop: one twist is stuck, two can be undone. The demonstration is usually credited to Dirac, and the same sign change is the one an electron's wave function picks up under a single full turn.
Further. W. R. Hamilton, Lectures on Quaternions (1853); S. L. Altmann, Rotations, Quaternions, and Double Groups (1986), which uses the turns throughout; J. Stillwell, Naive Lie Theory (2008), for SU(2), SO(3), and the bracket.