Math Type

Monday, January 20, 2014

History of Fractals

A fractal is an infinitely extending mathematical set that perpetuates the the same image over and over.  Contrary to what many people believe, a fractal does not always have a regular pattern.  Many fractals have a very irregular or even fractured appearance to them.  Fractals are not necessarily a first degree shape either.  Circles, squares, and other 2D shapes if increased, then the area is increased by a factor of 2 because that is the size of its dimensions; however, a fractal does not expand like 2D or even 3D objects.  Fractals grow by a value that is not always an integer.

Allegedly, fractals have been known about since about the 17th century by the mathematician and philosopher Gottfried Leibniz.  It was known by the late 19th century that fractals were recursive and that they were continuous but not always differentiable discovered by Georg Cantor, Felix Klein, and Henri Poincare.  In 1904 Helge von Koch started making figures that represented the ideas of fractals (which can be viewed at  Later that century, around 1918, at about the same time, two French mathematicians Pierre Fatou and Gaston Julia, came up with how fractals map the complex numbers.  By 1975, the name "fractal" was finally coined for these figures.  The name is derived from the Latin fractus meaning "broken" or "fractured."  The name fractal was coined by Benoit Mandelbrot.  Although naming these objects fractals, debate is still had over these objects.  Many people do not know how to formally define a fractal and what its name should encompass, but Mandelbrot was known for saying that fractals are "beautiful, damn hard, and increasingly useful. That's fractals!"  Now the general consensus on what a fractal is, is infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions.

Fractals have been defined as many things, not just geometric objects.  Among other things, fractals can also describe processes of time.  Fractals have been studied in images, structures and sounds, and also found in nature, technology, art, and law.  My area of interest comes from fractals in nature.  Fractals in nature show extended,  but finite, scale ranges.  These fractals are being studied heavily and are helping in many ways.  One such way is leaves, based on the fractals of leaves it can be determined how much carbon is in a given tree. gives the following extensive list of fractals in nature...

  • clouds
  • river networks
  • fault lines
  • mountain ranges
  • craters
  • lightning bolts
  • coastlines
  • Mountain Goat Horns
  • Animal coloration patterns
  • Broccoli and Cauliflower
  • Heart rates
  • Heartbeat
  • Earthquakes
  • Snowflakes
  • Crystals
  • Blood vessels and pulmonary vessels
  • Ocean waves
  • DNA
  • Soil pores
  • Psychological subjective perception
For those that are interested in fractals and in viewing more fractals, is a good source.  Some fractals that I found pretty cool were the Julia set (, Karperien Strange Attractor 200 (, Uniform Triangle Mass Center grade 5 fractal (, Lichtenberg figure (, and the Mandelbrot set (

My information was either previously known information or found from

1 comment:

  1. So those examples are fractals, or are best modeled by fractals? While examples of fractals predate Mandelbrot, he was more significant than just the namer. (Do you know what the B in Benoit B. Mandelbrot stands for? Benoit B. Mandelbrot.) A relevant term here would be self-similarity. It's okay to write for a non-mathematical audience, but you don't want friendly language to cross over into inaccurate statements. (perpetuates the same image over and over) And this would benefit greatly from some illustrations. Try wikimedia, or GeoGebraTube.