Concept

Double cover

A space that maps onto another so that every point below has exactly two points above it, arranged consistently nearby. The circle wrapping twice round itself is the simplest; rotations of space are covered twice by unit quaternions.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A rotation of four-space, and the two angles it turns through. The four by four matrix of the map sending x to p x conjugate q, beside two dials showing the angle it turns through in each of its two invariant planes.

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.

algebra · Quaternions

Three ways to pick a rotation at random. Mean rotation angle: uniform 126.70° (exact 126.48°), uniform Euler angles 127.91°, uniform axis and angle 90.18°. Densities by 5° bin, uniform: 0.00, 0.00, 0.01, 0.01, 0.03, 0.04, 0.04, 0.06, 0.09, 0.09, 0.13, 0.15, 0.16, 0.20, 0.23, 0.25, 0.28, 0.29, 0.34, 0.36, 0.38, 0.41, 0.44, 0.46, 0.47, 0.52, 0.54, 0.55, 0.57, 0.61, 0.60, 0.64, 0.62, 0.61, 0.64, 0.64.

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.

algebra · Quaternions

Named alongside it

The objects these essays reach for when they reach for this one.

Quaternion Rotation Symmetry group Dimension Group Interpolation Invariant subspace Orthogonal matrix Random walk Semicircle law Uniform distribution

All concepts