Mathematics in the Modern World

• Fractals are never-ending patterns, and are infinitely complex patterns that are self-similar across different scales.
• Fractals are created by repeating a simple process over and over in an ongoing feedback loop.
• Benoit Mandelbrot (1924-2010)

  • Coined the word “fractal” in 1980.
  • Mandelbrot chose the name fractal because it reminds him of the word “fraction”.
  • This was after he realized that these self-similar shapes have the property of not being one-dimensional, or two-dimensional, or even three-dimensional, but instead, of fractional dimension.
• Where are fractals?

  • Romanesco Broccoli
  • Pinecone
  • Ice Crystals
  • 20 times magnification of dendritic copper crystals forming
  • Running electricity between two nails sunk in a piece of wet pine
  • Tree Branches
  • Bubbles
  • Leaves
• Mandelbrot Set

  • Is used to refer both to a general class of fractal sets.
  • Marks the set of points in the complex plane and to a particular instance of such a set.
• Koch Snowflake

  • Created by the Swedish Mathematician Niels Fabian Helge von Koch.
  • Shows that it is possible to have figures that are continuous everywhere but differentiable nowhere.
• Koch Edge is a line segment divided into three equal parts, with the middle part replaced with two linear segments at angles 60 degrees and 120 degrees, producing a figure with four line segments.
• Georg Cantor, the founder of set theory, is also noted for studying one of the first fractal shapes, the Cantor Set.
• The Cantor Set is formed by following the algorithm: start with a line segment, divide the line segment into thirds, remove the middle third line segment, and iterate further.
• The Sierpinski Triangle, named after Polish mathematician Waclaw Sierpinski, is one of the simplest fractal shapes in existence.
• Astrophysicists believe that the key to understanding how stars were formed is the fractal nature of interstellar gas, which can be likened to smoke trail or clouds in the sky.