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Concept

Associativity

The condition that bracketing does not matter: combining three things as (a·b)·c gives the same result as a·(b·c). It is what separates a group from a bare table of products, and among all the multiplication tables of a given small size it holds for a vanishing fraction of them.

Named by 7 essays across 4 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 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 Ordinal sums and products, in normal form. A table of ordinal expressions with their Cantor normal forms and whether the two sides of each pair are equal, above two tick lines drawing one such pair.

One step in front of infinitely many

Put one step before an infinite run of them and nothing has changed; put it after and something has. Ordinal addition records that difference, which is why it is not commutative — and why it keeps information that counting throws away.

logic · Ordinals 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 The 14 triangulations, joined by single flips. The flip graph of a 6-gon: 14 triangulations drawn as small polygons and joined by 21 edges, one for each pair differing in a single diagonal.

The solid whose corners are triangulations

Take the triangulations of a hexagon as points and join two of them when a single diagonal can be swapped for another. The result is not merely a graph — it is the edge skeleton of a genuine convex polyhedron, with fourteen corners, three square faces and six pentagonal ones.

discrete · Catalan numbers The one line from which the nearfield plane looks Desarguesian. A grid of the 91 lines of the nearfield plane of order nine shaded by how many of 40 Desargues configurations with that line as axis failed; only the line at infinity has none, and every other line at least 20.

A plane no field built

Every finite field builds a projective plane, and for a long time every known plane was built that way. The plane over Dickson's nearfield of order nine has ninety-one points, ninety-one lines and every incidence right — and Desargues' theorem fails in it on most configurations tried, except for one line, from which it never fails at all.

computation · Finite geometry A subword can be checked with one quantifier block; a factor cannot. Subword aba: monoid 7, J-trivial, degree 3; factor aba: monoid 12, aperiodic, not J-trivial; witness abab / abba.

One block of quantifiers, read off an algebra

Which properties of words can be stated with existential quantifiers alone — 'there are positions, in this order, carrying these letters' — and Boolean combinations of such statements? Imre Simon answered in 1975: exactly those whose syntactic monoid is J-trivial. Of the 57,068 languages that automata of up to four states recognise, 848 pass the test, and every one of them is decided by which words of length three or less it contains as scattered subwords.

logic · Ehrenfeucht–Fraïssé games

Named alongside it

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

Division algebraExhaustive searchCommutativityDimensionGroupProjective planeQuaternionSymmetryCardinalityCatalan numbersClassificationComplex numbers

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