This document discusses fractals and their properties. It introduces fractals as a relatively new field of mathematics pioneered by Benoit Mandelbrot in the 1970s. Fractals exhibit self-similarity, where parts resemble the whole, and are found everywhere in nature, such as in coastlines, mountains, and trees. Famous artists like M.C. Escher incorporated fractals into their drawings. The Mobius strip, an example of a one-sided surface, is also discussed along with its discoverer and applications.

1. Art in Mathematics Mandlebrot set M.C. Escher Mobius Strip

2. Lesson 1: Fractals A relatively new field of mathematics – pioneered by Benoit Mandelbrot in the 1970s. Above is a picture representing a Mandelbrot set.

3. What is a fractal? A shape exhibiting self-similarity (part of the whole resembles the whole) and iteration

4. Mathematical Concepts: Infinity, similarity, topology (the study of shapes), imaginary numbers!

5. Examples: Coastlines, mountains, leaves, Mandelbrot set/ ink blots (seen above), bacteria cultures, Rings of Saturn

6. Fractals in Nature Fractals are found everywhere in nature. Plants, trees, and snowflakes are just some examples.

7. M.C. Escher M.C. Escher created many fractals in his drawings.

8. The Mobius Strip Named after Ferdinand Mobius a German mathematician and astronomer who pioneered the study of topology. Discovered first by J.B. Listing.

9. Characteristics: Characteristics: Has only one-sided and one edge! Applications: Conveyer belts, continuous loop recorders, print cartridges, belts on cars and farm machinery, typewriter ribbon

10. M.C Escher and the Mobius Strip Seen above is a drawing by M.C. Escher of the Mobius Strip.