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Concept

Conjugate

The function recording, for each slope, how far the line of that slope must be pushed to touch a given curve. It is convex whatever it came from, and applying the construction twice returns the original exactly when the original was convex.

Named by 2 essays across 2 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 x² and its conjugate. Two panels: the curve x² with tangent lines of several slopes, and the conjugate function plotted against slope, whose value at each slope is the intercept of the corresponding tangent.

The function seen from its tangents

A convex function is the upper envelope of its own tangent lines, so it can be described by giving, for each slope, how far the line of that slope has to be pushed down. That description is a second function, and applying the construction twice returns the original.

analysis · Convexity

Named alongside it

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

AssociativityComplex numbersConvexityDimensionDivision algebraDualityEnvelopeGroupInvolutionLegendre transformMultiplicationOptimisation

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