## What is the dimension of each geometric object?

**Mathematical Interpretation of Dimension via Self-similarity**

- 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).

- 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.

- 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

Email:

mconnors@math.umass.edu

*mconnors@math.umass.edu*