Hausdorff dimension
Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.
Infinite on one side and nought on the other
Box counting returns a growth rate. Hausdorff's definition returns a measure — a quantity that is infinite for every exponent below the dimension and zero for every exponent above it, and the dimension is the one place where it is neither.
A carpet with two dimensions
For every self-similar set met so far the two definitions of dimension agree, and the agreement is a theorem about sets built from copies of themselves scaled equally. Stretch one direction more than the other and the two numbers come apart, by an amount that can be computed exactly.
A dimension for every rate of crowding
Spread a unit of mass over an interval by splitting it unevenly, again and again, and the result covers the whole interval while crowding almost all of its weight onto a set of smaller dimension. Every rate of crowding picks out its own set of points with its own dimension, and the whole family is read off one curve.
The room a jagged graph takes up
The graph of a continuous function is a curve, and a curve ought to have dimension one. Build the function by raising midpoints — by an amount that shrinks more slowly than the intervals do — and its graph needs more boxes than any line, at a rate fixed by one ratio. A random path built the same way needs exactly as many.
No set with a line in every direction is small
In the plane over the integers modulo 7 there are 49 points and lines in 8 directions. A set holding a whole line in every direction needs 31 of the points — more than half — and in any dimension such a set fills a fixed share of the space. In the real plane the same sets can have area zero. Over a finite field one polynomial of low degree shows they cannot be small.
The exponent a staircase shares with its set
The Cantor function rises from nought to one on a set of length nought, and it is Hölder continuous with exponent log 2/log 3 — the same number as the dimension of that set. It is not a coincidence: both numbers say that an interval of width r carries mass r to the power 0.6309, and that one inequality proves the dimension and the smoothness at once. Tilt the weights of the construction and the two numbers separate, which shows exactly what the coincidence was measuring.
Where a random walk leaves the snowflake
Start a random walk at the centre of the Koch snowflake and record where it first touches the edge. The edge has dimension 1.26, but the places the walks arrive form a set of dimension one: 60,000 walks spread their entropy by ln 3 a level, not ln 4, and the busiest fifth of the edge takes nine tenths of them. Makarov proved in 1985 that it must be so for every simply connected region in the plane.
A small quotient for every denominator
Pick a denominator, any whole number. Among the fractions with that denominator, is there always one whose continued fraction has no large entries — whose path down the Stern–Brocot tree never turns the same way more than five times running? Stanisław Zaremba guessed so in 1972, for a reason to do with sampling a square evenly. Every denominator up to a hundred thousand obeys, with only 6, 54 and 150 needing a five. The guess is proved for almost every denominator and not for all, and the reason it ought to be true is a dimension: the reals whose quotients stay at most five fill a set of dimension 0.837, enough to give each denominator thousands of good fractions on average, and the open question is whether averages are kept everywhere.
Named alongside it
The objects these essays reach for when they reach for this one.
Self-similarityMeasureBox dimensionScalingCantor setCoveringIterated function systemSelf affinityBinomial coefficientCantor functionConformal mapContinued fractions