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mathematics, in pictures
Concept

Group — where it appears

A set with a way of combining any two of its elements that is associative, has an identity, and gives every element an inverse. It is the standard object for describing symmetry, because the motions that leave something unchanged always form one.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

The permutation (1 3 4 2) drawn as 4 strings, crossing 3 times. A permutation drawn as strings running from a row of numbered pegs to another, with every place two strings cross marked, and the crossing count checked against the number of pairs that are out of order.

The crossings that will not come out even

Draw a rearrangement as strings from one row of pegs to another and count where they cross. The count depends on how the strings are drawn; whether it is odd or even does not, and that single bit is what makes determinants exist and a sliding puzzle unsolvable.

algebra · Permutation parity 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 576 squares of order 4, sorted by whether they associate. Every Latin square of order 4, counted by whether it associates and by which group it is when it does.

Sixteen of five hundred and seventy-six

A Latin square is a multiplication table in which every equation has exactly one solution. Ask it to be associative as well and almost every square drops out — sixteen of the five hundred and seventy-six of order four survive, and they are the two groups.

computation · Latin squares A 4-bit register that visits all 15 nonzero states. The first 15 states of a 4-bit linear feedback shift register with taps at 4 and 1, with the bit that leaves the register at each step; the output shows every nonzero window of 4 bits exactly once.

A memory of four bits

A register holding four bits, shifting them along and adding two of them back, runs through all fifteen nonzero states before it repeats. Which two are added back is a question about a polynomial, and getting it wrong costs fourteen of the fifteen.

computation · De bruijn Two inversions, and the number four points agree on. Four points, their images after one inversion and after a second in a different circle, with the cross-ratio computed at each stage; it is conjugated once and restored twice.

The number four points agree on

One inversion is a reflection and reverses orientation. Two of them compose to a motion, and what that motion leaves alone is a single number computed from any four points.

geometry · Inversion 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 same loop, seen from two places. An annulus with two marked points and a loop based at the first. Two paths join the points, differing by a full turn round the hole, and each carries the loop to a loop based at the second point.

The group a space has at a point

The loops of a space form a group once a starting point is fixed, and the fixing looks like an arbitrary choice that ought to be removable. It is removable, but only up to conjugation, and the residue is exactly what makes a non-commutative fundamental group harder to state than to compute.

topology · Homotopy 3 sheets, and the subgroup they name. A circle with its 3-sheeted cover drawn as a spiral above it, beside a table of the winding classes and whether each lifts to a closed loop. The ones that do are exactly the multiples of 3.

Every cover is a subgroup

A space can be unrolled, and the ways of unrolling it are not arbitrary. They correspond exactly to the subgroups of its fundamental group — index equals sheets, normality equals symmetry — so a question about a group becomes a question about a picture and back again.

topology · Homotopy Sliding one square past another. 4 stages of a slide in which two labelled squares inside a larger one exchange positions without ever overlapping. The larger square's boundary is the base point throughout, and the exchange is what makes the composition commutative.

Why the second group commutes

Replace loops by spheres and the same construction gives a second homotopy group. It is always commutative, and the reason is not a fact about spheres or about any space — it is a two-line argument about any set carrying two compatible operations.

topology · Homotopy Which arrangements sliding tokens reach, on nine graphs, against Wilson's theorem. A table of 9 small graphs drawn as icons, each with the searched count of reachable token arrangements and the fraction of all arrangements it is: a six-cycle 5/120; K₂,₃ 12/24; the 2×3 tray 60/120; a house 24/24; a six-cycle with one chord 120/120; a wheel of six 120/120; θ, inner paths 2, 2, 2 2520/5040; θ₀, inner paths 1, 2, 2 120/720; θ, inner paths 1, 3, 3 20160/40320.

Which graphs let the tokens go anywhere

A sliding puzzle is a graph with a token on every vertex but one. Richard Wilson found in 1974 what every such puzzle can reach, and the answer has a surprise in it — the half the tray is stuck with is not a fact about permutations at all, but about the board being two-coloured — and one exception, a graph of seven vertices that reaches exactly 120 of 720.

algebra · Permutation parity The 9 mirrors of the octahedral group, cutting the sphere into 48 triangles. A sphere crossed by the 9 great circles of the octahedral group's mirror planes, dividing it into 48 triangles with one shaded.

Three mirrors make every solid

Every symmetry of a regular solid, reflections included, is produced by just three mirrors meeting at its centre, reflected in one another over and over — a kaleidoscope. Put a single point between the three mirrors and its reflections are the corners of a solid: the regular solid itself if the point sits in a corner, and every one of its truncated and expanded relatives if it sits anywhere else.

geometry · Regular polyhedra

Named alongside it

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

Fundamental groupHomotopyQuaternionSphereSymmetryAssociativityBase pointComplex numbersCounting argumentDimensionExhaustive searchInvariant

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