This experiment will explore the possibility of creating an infinite object by hand.
- Is it possible to ascertain the geometric parameters for an infinite object?
- Can an object have both finite properties and infinite properties?
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.
- Large piece of paper
- Draw a large equilateral triangle on the piece of paper. Use the ruler to make the triangle as accurate as possible.
- Divide each line segment of the triangle into equal thirds, and erase the middle portion of each line.
- In its place, draw another equilateral triangle. You should end up with a “Star of David” shape.
- 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.
- Knowing how you created the snowflake, represent its properties in an equation.
- 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.
- 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
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