Concept

Ternary expansion

The writing of a number between 0 and 1 in base three, as a sequence of digits 0, 1 and 2. The Cantor set is exactly the numbers with an expansion using no 1, which is why base three is the natural language for its arithmetic.

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

A dust plus itself is a line. C level 6: 64 intervals, length 0.0878; C+C: 1 interval(s) [0, 2].

Two dusts that add up to a line

The middle-thirds Cantor set has no length at all, and yet adding it to itself — every sum x + y of two of its points — fills the whole interval from 0 to 2 without a gap. The reason is a single number, the thickness, and it is exactly 1 for this set. Make the dust a little thinner and its sum with itself collapses to a dust again; make two different dusts and whether they add to a line depends on their dimensions, unless arithmetic gets in the way.

dynamics · Fractal dimension

Named alongside it

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

Cantor set Fractal dimension Measure zero Self-similarity Bijection Box counting Cardinality Geometric series Iterated function system Measure Uncountability

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