Fractals

Fractals are complex geometric shapes characterized by repeating patterns at different scales, often exhibiting self-similarity. This means that zooming in on a fractal will reveal the same pattern, albeit smaller, as the larger structure. They are found in nature, art, and can be generated mathematically. 

Fractals draw attention to the scale/shape/size problem in perspective systems, where new structure appears as the projection scale increases. Fractals also exhibit a never-ending amount of new detail or repeating patterns of structure, as the scale of view is increased. Fractals effectively negate the size/distance law of natural perspective, because identical shapes/ patterns are seen on multiple scales or viewing magnifciations. 

Figure 1: Fractal image created by computer.

Fractal Characteristics

  • Self-similarity: The pattern repeats itself at different scales. 
  • Infinite complexity: Fractals can be infinitely complex, with details revealed upon continued magnification. 
  • Fractional dimension: Fractals often have a dimension that is not a whole number, existing “between” dimensions. 
  • Generated by iterative processes: Fractals can be created by repeating a simple process over and over. 

Examples of Fractals

  • Nature (natural fractal): Trees, coastlines, rivers, snowflakes, broccoli, and seashells.
  • Mathematics (artificial fractal): The Mandelbrot set, Sierpinski triangle, and Menger sponge. 

Fractal Creation (by natural and artifcial methods)

  • Recursive algorithms: Repeatedly applying a rule to create a pattern. 
  • Iteration of mathematical equations: Using complex equations and feedback loops to produce fractal shapes. 
  • Stochastic processes: Using randomness and probabilities to generate patterns, like coastlines.

N.B. The processes, and methods by which natural fractals are created are not yet fully understood and may involve natural versions of the mathematical methods mentioned above or other unknown processes.

Fractals in Art

Fractals have been a source of inspiration for artists and can be used to create intricate and beautiful artworks. 

Fractals in Science

  • Computer graphics: Generating realistic landscapes and textures.
  • Chaos theory: Understanding complex and unpredictable systems.
  • Neuroscience: Studying the brain’s structure and connectivity.