Finding fractal dimensions

1
Finding fractal dimensions

• Goals for today
• understand what the fractal dimension tells us
  about the shape
• find a way of computing fractal dimension without
  the replacement rule

2
Examples

• Can we create the replacement rules for a
  fractals that have dimension
• log(4)/log(3) 1.26.
• log(5)/log(3) 1.46
• log(6)/log(3) 1.63
• Create these at http//illuminations.nctm.org/math
  lets/fractal/index.htmldirections
• What happens to the fractal as we increase
  dimension?

3
Finding the dimension

• Given your broccoli picture, determine how many
  of these measurements are needed to trace the
  curve as the crow flies.

4
Finding the dimension

• For each pair of rows, note that the scale factor
  is 1/2.
• Find the ratio of the number of copies (small
  measurement/large measurement).
• Use logs to solve the equation for dimension for
  each pair of rows.

5
Building fractals from 2D

• Current method start with one-dimensional
  objects
• Alternate method start with two- dimensional
  objects
• Example Sierpinskis triangle and carpet
• Find the dimension of both objects.