Box-Counting Fractal Dimension Estimator

Point-Cloud Pattern

Theoretical D ≈ 2.00 (log 4 / log 2)

Point count 2500

Grid scale level (k) k = 4

Box-Counting Readouts

Box size s

0.500

Boxes hit N(s)

Estimated dimension D (regression over all 6 scales)

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How it works

Box counting estimates the fractal dimension of a point set by covering it with a grid of cubes of side s and counting how many cubes N(s) contain at least one point. For a self-similar set this follows a power law:

N(s) ∝ s^(−D)
D = lim(s→0)  log N(s) / log(1/s)

In practice you sweep several scales s₁ > s₂ > … , plot log N(s) against log(1/s), and the slope of the best-fit line through those points is the estimate of D — the same measurement used to size up coastlines, cell-membrane boundaries and rough surfaces in real pattern-recognition pipelines. This simulator grids the domain at k = 1, 2, 4, 8, 16, 32 divisions per side (s = 2/k), counts occupied cells at every level, and fits:

D̂ = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)²
   where x = log k,  y = log N(k)

Pattern buttons — switch the underlying point cloud; each has a known theoretical dimension shown above the buttons for comparison against the measured D.
Point count — more points resolve the fine scales more accurately (sparse clouds under-count small boxes).
Grid scale level — chooses which of the 6 box grids is drawn on the 3D scene; the regression itself always uses all 6 levels at once.

Pattern points Occupied grid boxes

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