Matpic
mathematics, in pictures
Concept

Rotation

A motion of the plane or of space that fixes a point or a line and preserves every distance and orientation. Composing two gives another, so the rotations leaving an object where it was form a group, whose size measures how symmetric the object is.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

The quaternion multiplication table, from i² = j² = k² = ijk = −1. A four-by-four multiplication table of the quaternion units, with the row giving the left factor, every entry computed from Hamilton's rule, and the pair that differs between the two orders marked.

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.

algebra · quaternions 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.

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.

analysis · the exponential Every turn that leaves a cube where it was. A cube in wireframe beside a table of its rotation axes: 3 of order 4, 4 of order 3, 6 of order 2, totalling 24 turns including the one that does nothing.

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.

geometry · regular polyhedra

Named alongside it

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

Alternating groupAssociativityComplex numbersConjugacy classConjugateDeterminantDifferential equationDimensionDivision algebraDualitye, the numberEigenvalue

All concepts