Fractals are new on the mathematics scene, however they are in your life everyday. Cell phones use fractal antennas, doctors study fractal-based blood flow diagrams to search for cancerous cells, biologists use fractal theory to determine how much carbon dioxide an entire rain forest can absorb.
Fractals are in the mountains, clouds, coast lines, central nervous system, flower petals, sea shells, spider webs… they’re everywhere! And the really nifty thing about fractals is that they are not only cool, they’re super-useful in our world today.
Many mathematicians today are building on the work pioneered by Karl Weierstrass (1872), Helge von Koch (1904), and Waclaw Sierpinski (1915) to figure ways of using the ideas behind fractals. One of the most interesting parts about fractals is that many ideas about fractals were first thought up of in our lifetime. Many different fields, including medicine, business, geology, clothing fashion, art, and music use ideas about fractals.
Fractals are beautiful (there is something hauntingly stunning about the computer generated images of objects such as the Mandelbrot set, Julia sets, the Koch snowflake). But that’s not all – they are useful in our technology world. However, you’ll find that many research mathematicians still roll their eyes at the mention of the word “fractal’, mostly because the discussions you’ll find out there concerning fractals are missing the most important element – the mathematical content! This is why you’ll often find both students and teachers thinking that fractals are reserved only for art and video games, when that’s only one side of a multi-faceted concept.
There is solid mathematics behind the pretty pictures – in fact, with a good program, most kids can create their own fractal images after starting with the mathematics (which is often more beautiful than the images themselves!)
I’m going to help you unravel some of the mystery of fractals while having a lot of fun doing it. There are lots of easy-to-teach topics involving ideas from fractal geometry.
Download triangle grid paper here.
Download Student Worksheet & Exercises
Chaos Fractal Game
Once you’ve gotten a little more familiar with fractal geometry by watching the above video, take a peek at how you can play a simple fractal game that gives you the Sierpinski triangle!
Download Student Worksheet & Exercises
Going Further: Fractal Generators for your Computer
If you’re really into fractals and you really want to make your own, here’s a list of programs on the internet (most are free or low cost):
- Free Double Fractal Generator
- Trial version of Ultra Fractal
- Easy Fractal Generator for PCs and Macs
- An inexpensive Fractal Science Kit Program
Exercises : fractal
- What are the best structures that can describe the structure of human hand, flowers, and peaks of mountains among others?
- What are the building units of fractals?
- Identify the name of the following figure.
- From the above lessons, what is the best type of three-sided figure that can model fractals. How many triangles are you able to identify in the following figures?
- Where is the structure commonly used?What are the basic procedures that are used in making the following fractals in the following two examples?
Exercises : chaos fractal game
- How many players can comfortably play the fractals game at a time?
- How many ink pen colors are required to effectively play the game?
- When are the first dots made on the grid?
- What are the building units of the fractal grid that is used in the game?
- Where specifically are the first dots placed?
- Apart from the pens and the grid, what else is required to play the game?
- What is the probability that at least one player will play in a given turn?
- What is done on the die to ensure equal chances for each player?
- In the subsequent plays, where is the dot corresponding to the player positioned?
- What is the name of the resultant figure that results if the player goes through 100 turns?