Concept

Cantor set

The set left in the unit interval after the middle third of every remaining piece is discarded, for ever. It has no interval inside it and total length zero, and yet as many points as the interval it came from, which makes it the standard example of a set that measure and cardinality describe quite differently.

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

Middle thirds removed 6 times over. The interval with its middle third removed, then the middle third of each survivor, and so on. The lengths removed are a geometric series adding to the whole interval.

Almost none of it left, and still uncountably many

Remove the middle third of an interval, then the middle third of each piece left, and keep going. The lengths removed add to exactly the whole interval, so nothing measurable survives — and what survives can be paired off one for one with every point of the interval that was started with.

analysis · measure

the Koch curve, after 5 steps. the Koch curve drawn from its own rule: replace the middle third of every segment with two sides of a triangle.

A dimension that is not a whole number

Cover a set with boxes of side ε and count how many are needed. For a line the count grows like 1/ε, for a region like 1/ε². For the Koch curve it grows like 1/ε to the power 1.26, and that exponent is as good a definition of dimension as the other two.

dynamics · fractal dimension

Named alongside it

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

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