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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 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 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

Named alongside it

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

GroupCardinalityCommutativityComplex numbersConjugateCountingCounting argumentDimensionDivision algebraExhaustive searchGroup actionIdentity element

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