Hausdorff dimension
Named by 6 essays across 3 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Self-similarityBox dimensionMeasureScalingCantor setCoveringIterated function systemSelf affinityBinomial coefficientCantor functionCounting argumentDegree