Matpic
mathematics, in pictures
Concept

Uniformity

Convergence at a rate that works everywhere at once, rather than separately at each point. It is the difference a quantifier order makes, and it is what allows a limit to be exchanged with an integral or a sum.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

j is one more than i, counting round — as a grid, with both quantifier readings. A grid of marks for a relation, with the row and column facts the two quantifier orders ask about.

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

logic · Quantifiers Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

analysis · Arc length One curve, 3 parametrisations, 3 lengths. Several maps from an interval with the same image drawn side by side, each with marks at equal parameter steps and its length beneath it — the same point set reported at different lengths.

The length belongs to the journey

Three maps from an interval with exactly the same image, and three different lengths. The picture of a curve is the set of points it passes through, and that set does not determine how far anything travelled along it.

analysis · Arc length

Named alongside it

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

CounterexampleApproximationArc lengthContinuityDerivativeLimitConvergenceGibbs' phenomenonNegationParametrisationPredicate logicQuantifier

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