The Koch snowflake

Niels Fabian Helge von Koch (1870-1924) was a Swedish mathematician famous for his discovery of the Koch snowflake curve. The Koch snowflake is built-in iterations, in a sequence of stages.

Starting with an equilateral triangle, you create another, smaller, equilateral triangle in the centre of each side. This is repeated indefinitely, and during this, a snowflake shape would be created. The Koch snowflake is an example of a fractal curve (in fact, it was one of the first fractals to have been described), a shape that has a similar pattern at any magnification. For example, if you look at one part of the shape in its third iteration, you would be able to see a very similar structure at a later iteration, when the snowflake is magnified.

The Koch snowflake has quite unique properties: it has an infinite perimeter, but a finite area. As iterations happen infinitely, the perimeter and area keep increasing. With every iteration, the perimeter of the shape increases by a factor of 4/3. As the number of iterations tends to infinity, the perimeter keeps on increasing, reaching infinity. As the number of iterations increases, the triangles added get smaller and smaller. With the area, the snowflake would never exceed the area enclosed by the orange hexagon, as the triangles being added are so small. For this reason, the Koch snowflake has a finite area.

Following the same concept, many other mathematicians created variants of the Koch snowflake, using different initial shapes, angles, and planes. After many iterations, intricate and beautiful shapes can be created.

Tessellations can also be created using snowflakes of the same or different sizes. In the gaps, the same snowflake is created, demonstrating its repetitiveness and versatility.

The Koch snowflake is used to illustrate that ‘it is possible to have figures that are continuous everywhere but differentiable nowhere’. Fractals are used to understand important scientific concepts, for example, the way in which bacteria grows and brain waves.

By Samantha and Saachi

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