Koch Snowflakes.

Koch Snowflakes are a well known of fractal often used in Geometry and proposed by Swedish mathematician Neils Fabian Helga Van Koch in 1904.  Its also known as the Koch curve, Koch star, or the Koch Island.

To get a Koch Snowflake you start with a line that is divided into three equal parts and turned into an equilateral triangle.  The process is repeated off sides until you get the final product.

In detail, the math behind it is quite interesting.  The number of sides equals three times four to the a power where a represents the iteration.  When a = 0, n = 3.  When a = 1, n = 3.  When a = 2, n = 48 and when a = 3, n =. 192.

In regard to the length of the sides, they are always one third the previous length or mathematically it is x * 3^-a where a is iteration.  Thus for the first four iterations where a = 0, 1, 2, 3, you get length = a, a/3, a/9 and a/27.

All though this the perimeter is the same no matter the number of iterations.  The formula for the perimeter is n * the length or p = (3 *4^a) * (x *3^-a) simplifying to p = 3a (4/3)^a.  Simple sweet and easy.

Now if you are teaching a math class where students are not as advanced in the math department, can still draw a decent representation of the Koch snowflake without calculating the math for each iteration.

Step 1.  Draw your first equilateral triangle in the middle of the paper.

Step 2. Draw one  equilateral triangle off the middle third of each side so now you have what looks like two triangles, one behind the other.

Step 3. Draw one equilateral triangle of the middle third of each side of every triangle so it looks like there are three layers of triangles.

Repeat until you have the fractal done. Normally you'd only draw the outside of the triangles but I used color so you could see what was happening.

So with just a few triangles, you have a fractal. Let me know what you think, I'd love to hear.  Have a great day.