Fractals are mind-bending shapes that repeat their patterns at different scales. They're everywhere in nature, from coastlines to broccoli, and they've revolutionized how we see the world around us.

Mathematicians have been fascinated by fractals for centuries, but it wasn't until the 1970s that Benoit Mandelbrot coined the term. Since then, fractals have found applications in fields ranging from computer graphics to biology.

Introduction to Fractals

Definition of fractals

• Complex geometric shapes exhibit self-similarity across different scales
  • Fine structure remains detailed at arbitrarily small scales
  • Too irregular to be described by traditional Euclidean geometry (lines, circles, polygons)
• Key characteristics of fractals
  • Self-similarity: Similar patterns appear at different scales (zooming in reveals repeated motifs)
  • Infinite detail: More intricate details emerge upon magnification (coastlines, snowflakes)
  • Fractal dimension: Non-integer dimension exceeds topological dimension (between 1D and 2D or 2D and 3D)
  • Iterative generation: Created by repeating a simple process over and over (recursive algorithms)

Self-similarity in fractals

• Property where a part of the fractal resembles the whole fractal structure
  • Exact self-similarity: Fractal appears identical at different scales (mathematical fractals)
  • Statistical self-similarity: Numerical or statistical measures preserved across scales (natural fractals)
• Self-similarity observed at different magnifications or scales (zooming in or out)
• Examples of self-similarity in fractals
  • Koch snowflake: Each side composed of smaller versions of the entire shape
  • Sierpinski triangle: Composed of smaller triangles, which contain even smaller triangles (recursive structure)

Fractals in Nature and Mathematics

Examples of natural fractals

• Fractals in nature
  • Coastlines and geographic features (rugged, jagged shapes)
  • Branching patterns in trees and blood vessels (self-similar branching)
  • Snowflakes and crystal growth patterns (intricate, repeating structures)
  • Romanesco broccoli and other plants with self-similar structures (spiral patterns)
• Fractals in mathematics
  • Mandelbrot set: Complex plane fractal defined by $z_{n+1} = z_n^2 + c$
  • Julia sets: Related to Mandelbrot set, defined by $z_{n+1} = z_n^2 + c$ with fixed $c$ (varied shapes)
  • Cantor set: Created by repeatedly removing the middle third of a line segment (dust-like appearance)
  • Sierpinski carpet: Created by repeatedly removing smaller squares from a larger square (intricate square pattern)

History of fractal geometry

• Early observations of self-similarity in nature by philosophers and mathematicians (ancient Greeks, Leibniz)
• 1872: Karl Weierstrass discovers a function continuous everywhere but differentiable nowhere (early fractal-like behavior)
• 1904: Helge von Koch introduces the Koch snowflake (early mathematical fractal)
• 1915: Wacław Sierpiński describes the Sierpinski triangle (another early fractal)
• 1918: Felix Hausdorff introduces the concept of fractional dimension (foundation for fractal dimension)
• 1975: Benoit Mandelbrot coins "fractal" and develops fractal geometry
  • Mandelbrot's "The Fractal Geometry of Nature" popularizes fractals (seminal work)
• 1980s onwards: Fractals gain widespread recognition and applications
  • Computer graphics, chaos theory, physics, biology, and more (diverse fields)