Rotation
Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.
A multiplication that remembers the order
Four characters — i² = j² = k² = ijk = −1 — define a multiplication in which ab and ba are different numbers. Everything follows from them, including the fact that a rotation of a solid body has two names, and that two full turns are needed to get one of them home.
![The flow of a linear equation, and the matrix that runs it for one unit of time. Paths of points moving so that their velocity is [0.25, −1.2, 1.2, 0.25] applied to their position, with the position after time 1 marked on each; the matrix taking start to finish is e^A.](/f/the-flow-of-a-linear-equation-and-the-matrix-that-runs-it-15ef5202.svg)
The exponential of a square
The series for e makes perfect sense with a matrix in it. What comes out solves a system of equations the way the ordinary exponential solves one, and a skew matrix exponentiates into a rotation with no trigonometry anywhere.
The five solids as three groups
There are five regular solids and only three groups of rotations between them, because a solid and its dual share their symmetries exactly. The largest of the three is the smallest group with no way of coming apart, which is why the general equation of the fifth degree has no formula.
A rotation of four-space takes two of them
One unit quaternion, conjugating, turns three-space about an axis. Two of them, multiplying from the left and the right, turn four-space — and a rotation of four-space has no axis at all, but two independent angles and two planes it spins in.
The sphere that complex numbers live on
Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.
Cars that park, and trees that grow
Three cars arrive at a one-way street with three spaces; each has a favourite space, drives to it, and takes the first free one from there on. Of the 27 lists of favourites, exactly 16 let every car park — the same 16 as the labelled trees on four points. The reason is a circular street with one extra space, on which every list parks and exactly one rotation of it leaves the extra space empty.
The matrix that has no logarithm
Every square matrix has an exponential, and a matrix exponential is always invertible. The converse fails, and it fails in a way that can be read straight off the eigenvalues — a real matrix with eigenvalues −1 and −2 is no exponential at all, while minus the identity is the exponential of a whole continuum of matrices that do not even commute with each other.
The trisectors on a sphere almost agree
Draw Morley's construction on a sphere and the triangle the trisectors cut out is not equilateral — but its sides agree to within a tenth of a percent even when the angles add to 222°. Every step of the rotation proof survives on the sphere except one, and that one is the angle sum.
The point nearest in total to three corners
Three towns want one depot with the least total road to all of them. The answer is not the centroid or any of the classical centres: it is the point from which every side of the triangle is seen at 120°, found by building equilateral triangles on the sides. Three straight lines through it have the same length, and that length is the least total. When an angle of the triangle reaches 120°, the answer jumps to that corner — and the same 120° rule shapes soap films and the shortest networks joining many points.
An instrument that computes with circles
An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.
A rotation chosen at random
Turn a solid body to an orientation chosen at random and it has usually been turned a long way: on average 126.5 degrees, and by less than ten degrees only three times in ten thousand. Unit quaternions make the right meaning of 'at random' a picture, and two recipes that look fair fail it in opposite directions.
Named alongside it
The objects these essays reach for when they reach for this one.
QuaternionSymmetry groupComplex numbersConformalDeterminantDimensionDouble coverEigenvalueGroupMatrix exponentialMobius transformationPermutation