Concept

Space-filling curve

A continuous curve that passes through every point of a square, or of a cube. Peano and Hilbert built the first ones as uniform limits of polygons; they cannot be one-to-one or smooth, and they show that continuous maps alone do not preserve dimension.

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

A square's worth of points, on a line. A unit square with a point marked, the decimal places of its two coordinates woven into one number, and that number marked on a line beneath.

A line with as many points as a square

Interleave the decimal places of two numbers and one number comes out; take every other place back and the two return. The square has no more points than the segment, and dimension turns out to be invisible to counting.

logic · Cardinality

Hilbert's polygons, stage by stage. Hilbert's polygons at stages 1, 2, 3 and 6, each visiting the centres of 4, 16, 64 and 4,096 squares in order, from the bottom-left corner to the bottom-right.

A path that fills a square

A pen moving continuously for one unit of time can pass through every point of a square. Hilbert's polygons, each threading four times as many squares as the last, move no point more than 0.4507 × 2⁻ⁿ from where the previous one put it — and that one estimate, through the theorem that uniform limits stay continuous, is the whole proof. What the limit pays is exactness of a kind: it repeats some points four times, and its points can never be closer than the square root of their parameters allows.

analysis · Uniform convergence

Named alongside it

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

Bijection Continuity Dimension Cardinality Counterexample Decimal expansion Homeomorphism Interleaving Invariance of domain Measure Self-similarity Uniform convergence

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