Fractals

• History
  • "Discovered" by Cantor in 1883 - his "set" has dim~0.631
  • Other examples given by von Koch (snowflake curve) in 1905, Weierstrass (no-where differentiable curves)
  • Named by Mandelbrot in 1975
• Definition(?): A set with non-integer dimension.
• Definition: A self-similar set, formed of N pieces with linear stretching factor S, has similarity dimension D, where S^D=N.
• Examples:
  • Segment: S = 2, N = 2, D = 1
  • Square region: S = 2, N = 4, D = 2
  • Cube: S = 2, N = 8, D = 3
  • Cantor set: S = 3, N = 2, 3^D=2, D=ln(2)/ln(3)