• Question: What is the circumference of a fractal?

    Asked by ainsley.harriott on 13 Jan 2020. This question was also asked by printer.
    • Photo: James Bentham

      James Bentham answered on 13 Jan 2020:

      I’m not sure – that’s the sort of question I’d find a mathematician (not a statistician) to answer!

    • Photo: Daniel Bearup

      Daniel Bearup answered on 14 Jan 2020:

      Many fractals are constructed in such a way that, if they were completed, they would have infinite circumference.

      An example of this is the Koch curve. We start with an equilateral triangle with edges of length 1. We then cut the middle third of each edge out and replace it with an equilateral triangle with edges of length 1/3. The length of each edge of the new shape is 4/3 (we get 2/3 from the sections of the triangle we didn’t cut out and 2/3 from the outer edges of the triangle we have inserted). If we repeat this process, cutting out the middle third of each edge and replacing it with a (scaled down) triangle, the length of the edge will increase by a factor of 4/3 each time. Since 4/3 is greater than 1, if we did this infinitely many times (to form a complete fractal) we would have an infinite perimeter.

      Of course this comes from the fact that to have a mathematical fractal we must extend the generating process to infinity. Fractal structures in nature cannot extend to infinity and thus will not have infinite perimeters.

      Whether it is possible to construct a shape that has fractal properties but does not have infinite perimeter when the construction process is extended to infinity is an interesting question … . I haven’t been able to do so off the top of my head.

    • Photo: Samantha Durbin

      Samantha Durbin answered on 16 Jan 2020:

      As Daniel says, many fractals have a perimeter which is technically infinite.

      There are others where the fractal is inside the shape rather than on the edge – like the Sierpinski triangle: https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle

      Fractals are really interesting shapes which seem to break all the rules – for example, the Koch Snowflake (made up of 3 Koch curves, like Daniel describes, put together in a triangle shape) has infinite perimeter but a bounded area (i.e. the area of the shape will never get bigger than some fixed value). In addition, fractals have a non-whole-number dimension – so instead of being 2D, 3D etc, when you use some methods for calculating their dimension, it gives a decimal number.

      You can explore more here: https://nrich.maths.org/4757

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