What is the dimension of each geometric object?

Mathematical Interpretation of Dimension via Self-similarity

  1. Notice that the line segment is self-similar. It can be separated into 4 = 4^1 "miniature" pieces. Each is 1/4 the size of the original. Each looks exactly like the original figure when magnified by a factor of 4 (magnification or scaling factor).

  2. The square can be separated in to miniature pieces with each side = 1/4 the size of the original square. However, we need 16 = 4^2 pieces to make up the original square figure.

  3. The cube can be separated into 64 = 4^3 pieces with each edge 1/4 the size of the original cube.

In these simple cases the exponent gives the dimension:

4 = 4^1pieces

16 = 4^2pieces

64 = 4^3pieces

Therefore, N (the number of miniature pieces in the final figure) is equal to S (the scaling factor) raised to the power D (dimension).

N = S^D

(c) Copyrighted 1994,1995,1996,and 1997 by Mary Ann Connors. All rights reserved. If you wish to use any of the text or images in Exploring Fractals please contact its author Mary Ann Connors at the following address. Thank you.

Dr. Mary Ann Connors
Department of Mathematics & Statistics
Lederle Graduate Research Tower
University of Massachusetts
Amherst, MA 01003