Dimension theory

Fractal dimension

Classical geometry calls a curve 1-dimensional, a surface 2-dimensional, a solid 3-dimensional. But the Koch curve is more crinkled than any smooth curve; the Cantor set is sparser than any interval. Fractal dimension is the bridge: a non-integer number that measures how much space a set really occupies as you zoom in.

Similarity dimension (Moran’s formula)

For a self-similar set made of N copies of itself at scale ratio r< 1, the similarity dimension is

dim = log N / log(1/r)

derived from Moran’s equation N · r^d = 1. Examples:

• Cantor set: 2 copies at r = 1/3, dim = log 2 / log 3 ≈ 0.6309.
• Koch curve: 4 copies at r = 1/3, dim = log 4 / log 3 ≈ 1.2619.
• Sierpinski triangle: 3 copies at r = 1/2, dim = log 3 / log 2 ≈ 1.5850.
• Menger sponge: 20 copies at r = 1/3, dim = log 20 / log 3 ≈ 2.7268.

Box-counting dimension

Works for any compact set, not just exactly self-similar ones. Cover the set with axis-aligned boxes of side ε. Count how many boxes meet the set: N(ε). The box-counting (or Minkowski-Bouligand) dimension is

dim_B = lim ε→0   log N(ε) / log(1/ε)

For a smooth curve N(ε) ~ 1/ε, giving dimension 1. For a surface N(ε) ~ 1/ε², giving dimension 2. For the Koch curve you find N(ε) ~ 1/ε^{1.262}.

Hausdorff dimension

The mathematically canonical definition, due to Felix Hausdorff (1918). For each s > 0, define the s-dimensional Hausdorff measure of a set by infimal sums of diameters raised to the s-th power. There is a unique critical value of swhere this measure jumps from infinity to zero. That value is the Hausdorff dimension. For sets satisfying mild regularity (the open set condition), Hausdorff and similarity dimensions agree.

Why dimension can be non-integer

The Koch curve has length growing without bound at every scale, so it’s “bigger than 1D.” But it has zero area, so it’s “smaller than 2D.” The dimension log 4 / log 3 measures precisely how it fills the plane: cover it with boxes of side εand you need ≈ (1/ε)^1.262 of them, more than a curve, less than a surface.

Coastlines and the Richardson effect

Lewis Fry Richardson noticed that coastline length depends on the ruler used: the shorter the ruler, the longer the apparent coastline. Mandelbrot reframed this in 1967: total length scales as L(ε) ~ ε^1−D, where Dis the fractal dimension of the coastline. Britain’s west coast comes out near 1.25, Norway’s much higher, Australia’s smoother.

Reference values

References

• Hausdorff, F., “Dimension und äußeres Maß,” Math. Ann., 1918.
• Mandelbrot, B., “How long is the coast of Britain? Statistical self-similarity and fractional dimension,” Science, 1967.
• Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, Wiley, 2014.
• Edgar, G., Measure, Topology, and Fractal Geometry, Springer, 2008.
• Fractal catalog