Fractal Geometry

William L. Nowell


Fractal geometry, introduced in 1977 by Benoit B. MandelbrotBenoit B. Mandelbrot
(1924 - 2010)Benoit B. Mandelbrot
The Father of Fractal Geometry
, is one of the newest branches of geometry. A fractal is a figure or surface generated by successive subdivisions of a simpler polygon or polyhedron, according to some iterative process. More simply, fractals are shapes made of parts that are reduced-sized copies of the whole. Fractals describe complex shapes of naturally occurring phenomena such as snowflakes, clouds, leaves, lightning bolts, blood vessel systems, and even spiral galaxies.

A classic fractal, known as the Sierpinski triangle, is shown below. Notice that the second triangle contains three scaled down replicas of the first triangle. The same is true at each stage of the development.

Stages of the Sierpinski triangle
Stages of the Sierpinski triangle.

Types of Fractals

Fractals are not all the same. Fractals can be grouped into six major categories. These major categories can be further subdivided into many subgroups. The major categories are as follows.

  • Classic (geometric) fractals. These fractals are constructed by applying iterative processes (doing the same thing repeatedly) to common geometric objects such as lines, triangles, and cubes. Classic fractals are the original fractals, with roots dating back to the 17th and 18th centuries. This is long before the term fractal was coined. These classic fractals are examined in this brief fractal geometry tutorial.

    Julia Set
    Julia Set
  • Complex polynomial fractals. These fractals are generated by recursive complex valued polynomial functions. They are the fractals most commonly seen in fractal art. An example of a complex polynomial fractal is the Julia set pictured at the right.

  • IFS (Iterated Function Systems). IFSs are most often generated by contractive affine transformations. However, Möbius and projective transformations may also be used to generate these fractals. One of the best-known IFSs is the Barnsley fern at the bottom of the page.

  • Strange attractors. These fractals use iterations of a map or solutions of a system of initial-value differential equations that exhibit chaos.

  • L-Systems. L-systems (or Lindenmayer systems) were invented by a biologist and botanist to model plant growth and development. Many plants exhibit fractal patterns and L-systems are useful for generating fractal plant structures.

  • Plasma fractals. Used in CAD programs to produce clouds, fire, etc.

In addition to the major categories listed here, there are many subgroups including fractal terrains and even fractal music! Fractal terrains were used to draw landscapes in the movie Star Trek II: The Wrath of Kahn (one of my favorite movies, but I digress). In fractal music, the results of a fractal algorithm are applied to audio parameters to produce music.

Barnsley fern fractal
Barnsley fern

Next, we take a look at one of the first known fractals, the von Koch snowflake curve.

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