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Concept

Symmetry group

The set of all transformations leaving an object where it was found, taken together with composition. Its size measures how symmetric the object is, and its structure distinguishes objects that a count of symmetries alone cannot tell apart.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Solids with every corner alike and more than one kind of face. truncated tetrahedron, cuboctahedron, truncated cube, icosidodecahedron, each cut from a Platonic solid and drawn in projection; every edge in each is the same length and every vertex is surrounded by the same faces.

Thirteen more when one word is dropped

The list of regular solids stops at five because the definition asks for two things at once. Ask for only the second — every corner alike — and thirteen more appear, each of them cut off a Platonic solid at a depth found rather than chosen.

geometry · Regular polyhedra 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 the right triangle at an eighth of a turn: 16 directions, and a surface of genus 2. A polygonal billiard table with a long trajectory drawn on it, the finite set of directions that trajectory takes, and the arithmetic of the surface it unfolds into.

A table folded into a surface

Unfolding a square billiard gives a straight line on a torus. Unfolding any table whose angles are whole fractions of half a turn gives a straight line on some surface — and which surface it is decides how hard the dynamics will be.

dynamics · Billiards 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 The twenty-four unit quaternions, at the corners of a four-dimensional solid. The twenty-four units of the Hurwitz quaternions drawn as the vertices of a 24-cell projected into three-space, with the ninety-six edges joining units at distance one.

The integers among the quaternions

The obvious integer quaternions are the ones with whole coordinates, and they are the wrong ones. Sixteen more, with every coordinate a half, have to be let in — and with them comes a division algorithm, twenty-four units instead of eight, and a regular solid that exists only in four dimensions.

algebra · Quaternions The octonion multiplication table, drawn as seven lines. The Fano plane with its seven points labelled by the imaginary octonion units and its seven lines carrying an arrow each, giving the products of every pair.

What is lost at eight

Double the quaternions and division still works, but ab times c and a times bc are no longer the same number. What survives is a weaker law that turns out to be enough for a great deal — and the whole multiplication table fits into a picture of seven lines.

algebra · Quaternions Two hundred and forty roots, built and counted. An eight by eight grid whose upper cells stand for the pairs of coordinates a root can use, beside bars counting how many roots stand at each angle from a given root: 1, 56, 126, 56 and 1.

Two hundred and forty directions

The quaternions have twenty-four units and they are the vertices of the most symmetric object in four dimensions. Eight dimensions has two hundred and forty of them, and the quaternions turn out to be how they are built — twice over, with a hundred and ninety-two left to explain.

algebra · Quaternions

Named alongside it

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

QuaternionDimensionDualityEuler characteristicGroupLatticeNormPlatonic solidsPolyhedronRotationAlternating groupArchimedean solid

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