Rivers, coastlines, seashells.

Clouds, hurricanes, mountains.

Trees, ferns, cauliflowers.

Blood vessels.

Snowflakes.

What do these have in common..?

Fractals.

What is a fractal?

Fractals are never ending patterns. They are infinitely complex and have a recursive nature.

Fractals have two key properties: self similarity and non-integer dimension.

Self similarity means at the, for example, image can be magnified many times, and after every step of magnification, you will eventually see the same shape as the start.

A non-integer dimension is hard to imagine, but as the name suggests, it is between integer dimensions. For example, in classic geometry we meet 0 dimensional, 1 dimensional, 2 dimensional and 3 dimensional objects. A straight line is a one dimensional object whereas a fractal curve is between 1 and 2 dimensions.

Famous Fractals

**Sierpinksi Gasket**

Start with an equilateral triangle.

Take the midpoints of each side of the triangle and join them together with a line.

Repeat infinitely (or for as long as you can concentrate for)

**Von Kock snowflake**

Start with an equilateral triangle.

In the middle of each side, add a new triangle one third of the size of the triangle you drew before.

Again, repeat infinitely (or for as long as you can concentrate for)

**Mandelbrot set**

Visit *https://mathigon.org/world/Fractals* and scroll to the Mandelbrot Zoomer.

Have fun 😀

Some fractals in3D spaces include the Menger sponge and the Mandel bulb.

Note: there is no restriction regarding the number of dimensions a fractal may exist in, these are just examples

How can we create fractals?

Recursion.

Recursion is a method of problem solving, as well as programming, where the problem is broken down into smaller pieces of the same problem. These smaller problems are similar to the bigger problem, thus can be solved in the same way as the big problem.

Fractals are useful when creating realistic landscapes and planetary surfaces in the film industry. An algorithm can be created to formulate to the complex visual imagery to mimic the fractals in nature to create the computer-generated images of, for example, rivers.

It was also suggested that perhaps fractals could rival jpeg compression, but, quite clearly, that wasn’t a big hit.

However, more successful uses of fractals have been in the architecture of the networks that make up the internet, describing financial markets, understanding astrophysics and DNA as well as even diagnosing some diseases!

**“I believe that scientific knowledge as fractal properties, that no matter how much we learn, whatever is left, however small it may seem, is just as infinitely complex as the whole was to start with. That, I think is the secret of the Universe.” – Isaac Asimov**

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Ada Knowe 🙂

*References:*

* https://fractalfoundation.org/resources/what-are-fractals/*

*http://whatis.techtarget.com/definition/fractal*

*http://www.fractal.org/Bewustzijns-Besturings-Model/Fractals-Useful-Beauty.htm*

*https://mathigon.org/world/Fractals*

*(link above has an interesting Mandelbrot zoomer that is mind blowing)*

*http://www.bbc.co.uk/news/magazine-11564766*

*https://cstheory.stackexchange.com/questions/16965/algorithms-as-fractals*

*(will explore fractals in algorithms in a later blog post. Links below are a useful start for recursive programming)*

*http://www.cs4fn.org/graphics/casinoroyalefractal.php*

*http://www.cs4fn.org/graphics/clubfractal.php*

http://www.cs.ox.ac.uk/geomlab/

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