# Koch Snowflakes

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High School

### Type:

Mathematics, Geometry

### Objective:

This experiment will explore the possibility of creating an infinite object by hand.

### Research Questions:

• Is it possible to ascertain the geometric parameters for an infinite object?
• Can an object have both finite properties and infinite properties?

### Introduction:

Although an in-depth study of fractals was not possible before computers, the theory is not totally new. Helge von Koch, a Swedish mathematician, discovered a fractal in the early 20th Century, and now you can recreate and study his “snowflake” on paper.

### Materials:

• Large piece of paper
• Ruler
• Computer
• Pencil
• Eraser

### Experimental Procedure

1. Draw a large equilateral triangle on the piece of paper. Use the ruler to make the triangle as accurate as possible.
2. Divide each line segment of the triangle into equal thirds, and erase the middle portion of each line.
3. In its place, draw another equilateral triangle. You should end up with a “Star of David” shape.
4. Repeat steps 2 to 4 until you can’t draw any more triangles. The shape should resemble a very complex snowflake. A computer could repeat this until infinity, but humans are bounded by the materials.
5. Knowing how you created the snowflake, represent its properties in an equation.
6. The area of a triangle, if s the length of a side, is (s^2(√3))/4. Using this information, attempt to figure out the area of your object.
7. Analyze this data. What is finite about the snowflake, and what is infinite? Does the perimeter continually increase? Does the area?

Concepts: fractals, Koch’s curve, geometry