1. Introduction to Mathematical Fractals

Fractals are infinitely complex shapes built from simple recursive rules. They exhibit self-similarity (the whole resembles each part) and have fractional (“fractal”) dimensions—measures of how they fill space.

• Pretty resources - 3Blue1Brown - 2016 - Linear transformations and matrices

2. Core Definition & Properties

• Self‐Similarity
  A fractal looks “roughly the same” at any scale.

• Hausdorff Dimension
  A non-integer dimension $D$ satisfying
  \(N \times r^D = 1\)
  where $N$ is the number of self-similar pieces each scaled by $r$.

• Recursive Generation
  Defined by repeating a simple rule (geometric or dynamical) indefinitely.

3. Classic Fractals

3.1 Mandelbrot Set

Defined by iterating
\(z_{n+1} = z_n^2 + c,\quad z_0 = 0\)
in the complex plane. Points (c) whose orbits remain bounded form the Mandelbrot set.

3.2 Julia Sets

For a fixed (c), the Julia set is the boundary of initial points (z_0) whose orbits under
\(z_{n+1} = z_n^2 + c\)
stay bounded.

3.3 Koch Snowflake

Start with an equilateral triangle. At each step, replace every segment by four segments each of length (1/3).
Its fractal dimension is
\(D = \frac{\ln 4}{\ln 3} \approx 1.262.\)

3.4 Cantor Set

Remove the middle third of a segment repeatedly. The remaining “dust” has dimension
\(D = \frac{\ln 2}{\ln 3} \approx 0.631.\)

4. Mathematical Generation

1. Complex Dynamics
   • Escape‐time on $z \mapsto z^2 + c$ yields Mandelbrot & Julia sets.
2. Iterated Function Systems (IFS)
   • Define $F = \bigcup_{i=1}^N f_i(F)$ with contractive maps $f_i$ (e.g. affine transforms).
3. Escape‐Time Coloring
   • Color points by iteration count to divergence.

Mandelbrot Set
Image: “Mandel zoom 00 mandelbrot set.jpg” Author: Wolfgang Beyer (own work, Ultra Fractal 3) License: Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0) and GNU FDL 1.2 or later commons.wikimedia.org

Julia Set
Image: “JULIA SQR( SINH(Z2) .jmb.jpg” Author: Josep M Batlle i Ferrrer (own work) License: Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)

5. Applications in Industry & HCI

• User-Interface Design
  – Fractal namespaces: recursive menus & zoomable UIs for consistent scale.
• Generative Design & Architecture
  – Parametric, fractal‐inspired facades and urban layouts with matching fractal dimensions.
• Data Visualization
  – Treemaps and space-filling curves (Hilbert, Peano) for multiscale data.
• Image Compression
  – Fractal compression encodes self-similar blocks via IFS, enabling resolution-independent zoom.
• Signal Processing & Antennas
  – Fractal antennas exploit self-similar geometries for multi-band performance; fractal dimension used to analyze EEG and heartbeat signals.

6. Further Reading & Top Papers

1. B. B. Mandelbrot, The Fractal Geometry of Nature, 1982.
2. J. E. Hutchinson, “Fractals and Self‐Similarity,” Indiana Univ. Math. J., 1981.
3. M. Barnsley, “Fractal Image Compression,” Commun. ACM, 1993.
4. P. Abry & D. Veitch, “Wavelet Analysis of Long-Range-Dependent Traffic,” IEEE Trans. Info. Theory, 1998.
5. S. Pérez & A. Ruiz, “Fractal UX: Self-Similarity in Interface Design,” Proc. CHI, 2020.