Lecture on Fractals for Middle School Students

Question

I'm going to have a one-hour lecture for middle school students next Monday. It will be about fractals. The students know virtually nothing about this subject. I'll show some fractal images and a few short films which I've found on youtube, discuss on the ways a fractal can be constructed, and introduce a software. But I'll need other things, too:

1. Some serious mathematical content.
2. Some questions to propose to the students for further study. Could you please help me with these?

Thanks in advance.

Answer 1

Your task is both a challenge and an opportunity: they will be unfamiliar with complex numbers, but perhaps you could motivate the utility of complex numbers. I might try to introduce them to the computation of a Julia set, at first entirely computationally, showing them how $z$ grows under repeated computation of znew = zold² + c, all in terms of coordinates and distance from the origin (without mentioning complex numbers). They need not know any programming language to understand a simple iterative loop. Once they see how some starting points $z$ scoot off to infinity, and others hang around the origin, they can appreciate it would be natural to color each point according to its scooting-to-$\infty$ speed. And then they could understand how to make a Julia set:
     
     _{(Image from cgtutor)}

With this understanding secured, you might be able to introduce complex numbers.

For motivating applications, you could easily connect to the use of fractals in computer graphics in movies (Lord of the Rings; The Hobbit, etc.):
  
  _{(Image from LifeInWireframe)}

Answer 2

You should see http://mathforlove.com/2011/02/sierpinski-triangle-talk/ It was given to the exact range of grades to whom you are referring!

Answer 3

The Digital Sundial is a neat application of fractals. It is also related to some fairly important mathematics in Geometric Measure Theory -- specifically the the so-called "Structure Theorem".

Answer 4

A very accessible application of fractals can be found in Richard Taylor's "Order in Pollock's chaos" (an article in the November 2002 issue of Scientific American), which is based on the Richard Taylor, Adam Micolich, and David Jonas's "Fractal analysis of Pollock's drip paintings" (a paper in a June 1999 issue of Nature). The Nature paper has the following abstract:

Scientific objectivity proves to be an essential tool for determining the fundamental content of the abstract paintings produced by Jackson Pollock in the late 1940s. Pollock dripped paint from a can onto vast canvases rolled out across the floor of his barn. Although this unorthodox technique has been recognized as a crucial advancement in the evolution of modern art, the precise quality and significance of the patterns created are controversial. Here we describe an analysis of Pollock's patterns which shows, first, that they are fractal, reflecting the fingerprint of nature, and, second, that the fractal dimensions increased during Pollock's career.

From the Scientific American article:

The painting is scanned into a computer. It is separated into its different colored patterns, then covered with a computer-generated mesh of identical squares. The computer analyzes which squares are occupied and which are empty. This is done for different mesh sizes. The patterns were found to be fractal over the entire size range.