Fractals

Fractals are wonderfully fantastically amazing mathematical patterns! As anyone who knows the math fairy knows, I love fractals. I have fractal jewelry (boutiqueacadamia.com) , fractal screen savers and I love telling people about fractals. I think it’s because they’re so mathmagical! But what are they, you ask? Well, in simplest terms a fractal is a pattern.  At Fractalfoundation.org they describe fractals as never-ending patterns, which many others are too, so why are fractals so special? I think there are several reasons. First of all, fractals have what is called self-similarity. This means that you can zoom in on the picture and zoom in again and it will always be the same pattern. Broccoli is a great example of a fractal found in nature with obvious self-similarity. Take a head of broccoli. Break off a branch. See how it looks just like the big bunch, except smaller? Now break off another, smaller branch. Do you see it again? Look around you in the world. Can you find other fractals in nature? I think ferns are a beautiful example but see what you can find. A second reason that fractals are so special is because they are essentially mathematical equations that make pictures. You may have done some coordinate graphing? Where you draw a picture on an x,y grid using coordinate points? They’re fun. Kind of like a dot-to-dot for big kids. Well, fractals are equations (and this is a very simplified math fairy version of what can be a very complicated mathematical exercise) where certain answers have been assigned certain colors. Fractals like the Mandelbrot Set are created from an iterative equation (I love that word! Iterative!), which means that you use the answer from the equation as the value of the variable in the next solving of the same equation.

mandelbrot setHere is a Mandelbrot set. Isn’t it beautiful?! And it is so mathmagical that this image was created using a computer and a math equation. On fractalfoundation.org they have a great explanation of fractals: http://fractalfoundation.org/fractivities/WhatIsaFractal-1pager.pdf

I really like how they explain fractals using pictures. Check it out! You can google fractal images and find all kinds of wonderful creations. On the fractalfoundation.org page you will also see Pascal’s triangle, which the mathfairy has talked about on other pages.

As well as finding fractals in nature and exploring online images, making a fractal is a great way to really understand how they work. You can find some really cool apps and java applets to use in making your own fractals on the computer. I especially like making tree fractals. I invite you to see what you can find online! Have fun!

In the meantime, it’s also really fun to draw a fractal. Today we’re going to make a Koch Snowflake in honor of the 2013 Disney movie Frozen. In the movie they even sing about fractals!

You will need: a pencil, maybe a ruler, crayons or colored pencils and definitely some triangular graph paper. (For finding triangular graph paper I recommend two websites: www.dr-mikes-math-games-for-kids.com or http://math.rice.edu/~lanius/frac/koch.html. This second website, Cynthia Lanius’, is where I have found lots and lots of wonderful fractal activities. Check it out!)

First, draw a triangle in the center of your paper. You want it to be an equilateral triangle that is big enough to break up the sides into smaller lengths but not so huge that you don’t have room for the fractal to grow. I counted 9 triangle edges per side in this example.

koch snowflake 1

Second, divide each side into thirds and draw a new triangle, making a star. Make sure that everything is the same size. In my example, you’ll notice that every piece is now 3 lengths.

Koch snowflake 2

Now, keep going! Keep dividing up each side of each side length into thirds, draw a little triangle off that third. Keep going until you can’t draw anymore!

Koch snowflake 3

Then, color and cut out your beautiful fractal. You have created your own never ending pattern with self-similarity.

koch snowflake 4

Have fun with math every day!