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Day 4 | Advent Calendar

After a day of playing with snowflakes it is the right time to play with the Koch Snowflake - fractal curve.

The Koch Snowflake

The Koch snowflake is a fractal curve (also known as the Koch curve, Koch star, or Koch island) and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch.


Creating a Koch Snowflake

The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed from adding outward bends to each side of the previous stage, making smaller equilateral triangles. When creating your posts you can:

  1. divide the line segment into three segments of equal length;

  2. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward;

  3. remove the line segment that is the base of the triangle from step 2 (green line from 2nd drawing);

  4. repeat as many times as you want to.


Other activities involving the Koch Snowflake

  1. Find the area and perimeter of the 3rd image above (for some help on the area, take a look at this article).

  2. What happens with the area or perimeter in the 4th iteration?

  3. Can you find a generalized formula for both area and perimeter?

  4. As the number of iterations tends to infinity, what is the limit of the perimeter and area?

We would love to see your solutions or ideas to these questions. Comment down bellow with what you think.


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