Concept

Piecewise-linear

Built from finitely many straight pieces joined end to end, as a function whose graph is a broken line or a surface made of flat faces. Such objects are simple enough to compute with exactly and flexible enough to approximate any continuous one.

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

Every ear carried to a slice of a disc. The corridor's triangulation on the left and the same triangulation of a regular 20-gon on the right, with four points and their images marked; the map is affine on each triangle and agrees on every shared edge.

Every loop is a circle in disguise

Separating the plane is the weak half of what the eye believes about a closed curve. The strong half is that the inside is a disc — that the whole plane can be bent until the curve is a round circle — and for a polygon that is a construction rather than an argument.

topology · Jordan curve

Six formulas on the interval of truth values. Graphs on [0, 1] of six one-variable formulas of Łukasiewicz logic, each a zigzag of whole-number slopes with values 0 or 1 at the ends.

The zigzags a formula can draw

Let truth be any number from 0 to 1, and Łukasiewicz's connectives turn every formula in one variable into a graph. Every graph that appears is a zigzag of straight pieces with whole-number slopes, ending at 0 or 1 — and McNaughton proved in 1951 that every such zigzag appears. A logic of degrees of truth turns out to be a theory of piecewise-linear functions with integer coefficients.

logic · Non classical logic

Named alongside it

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

Algebra Barycentric coordinates Characterisation Closed curve Completeness Continuity Counterexample Disc Exhaustive search Expressive power Homeomorphism Jordan curve

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