Concept

Envelope

The curve that touches every member of a family of lines or curves, each at one point. It is the shape a family draws without anyone drawing it, as the chords of a times table on a circle trace out a cardioid.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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

Every residue joined to 2 times itself, on a dial of 199. A circle with 199 equally spaced points and a chord from each point k to the point 2k mod 199, with the 1-cusped curve the chords envelope drawn dashed.

Multiplying every number on the dial at once

Join every residue on a dial to twice itself and the chords draw a heart-shaped curve with one cusp; join each to three times itself and the curve has two. The picture is the whole multiplication map at once, and it holds three facts: the map splits the dial into cycles whose lengths are orders, those cycles on a dial of 2ⁿ − 1 are the binary necklaces of length n, and the curve is the caustic light draws inside a cup.

discrete · Modular arithmetic

Named alongside it

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

Caustic Conjugate Convexity Doubling map Duality Epicycloid Involution Legendre transform Modular arithmetic Necklace Optimisation Orbit

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