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Part I: Foundations

Part I is the tutorial half of this site. Four chapters build the vocabulary and the measurement tools that the research manuscripts in Parts II–V take for granted: where fractal geometry came from, how dimension is defined and estimated, what fractals look like in one to four dimensions, and a glossary of the terms and equations used throughout.

The Four Chapters

Chapter Page What it covers
1 History of Fractal Mathematics From Cantor to Mandelbrot to modern complex dimensions; self-similarity versus self-affinity; fractals enter biology; an appendix of machine-checked proofs
2 Mathematical Foundations of Fractals Hausdorff dimension, box-counting and differential box-counting, power laws, the Hurst exponent and lacunarity
3 Fractal Dimensionality (1D through 4D) 1D through 4D fractals with examples: 1/f noise, Brownian surfaces, Mandelbrot and Julia sets, the Menger sponge, DLA, branching networks and spacetime
4 Glossary and Equations Quick reference for terms and formulas, including the Barnsley fern IFS maps and the ten key equations
  1. Chapter 1 first if you are new to fractals: it motivates every later definition historically and ends with the self-similar/self-affine distinction that organises the whole project.
  2. Chapter 2 next, for the measurement tools (box-counting, differential box-counting, the Hurst exponent) that the Part II methods apply to leaves, branches and forest canopies.
  3. Chapter 3 can be read selectively by dimension: §3.1 for signals and time series, §3.2 for images and terrain, §3.3 for branching networks and DLA, §3.4 for growth dynamics.
  4. Chapter 4 is a reference rather than a read-through; return to it whenever a symbol or term is unfamiliar.

Readers who come for the biology can start at Chapter 1 §1.5 and continue straight to Part II: Metabolic Scaling Theory.

Companion Notebooks

Where to Go Next