Sierpinski fractals hiding in Pascal’s Triangle!

Write out the first 64 lines of Pascal’s Triangle, or better still, get a spreadsheet to do the work for you. Then colour in black all the numbers that are odd. Leave the even numbers white. What do you get?

Pascal’s Triangle with odd numbers coloured black and even numbers coloured white

You get the famous Sierpinski Triangle fractal! You may have read my recent post The Chaos Game- a very surprising result, where we used random numbers to generate a Sierpinski Triangle. Here is another way to generate the fractal from the order of simple addition sums in Pascal’s Triangle.

Rather than having to calculate the numbers in Pascal’s Triangle to get this result, you can simply instruct each cell to look at the two diagonally above them and observe their colours:

Black + Black = White

White + White = White

Black + White = Black

White + Black = Black

These rules correspond to odd + odd = even etc. By following these incredibly simple four rules, a single black square at the top cascades down to form this intricate fractal.

I think this gives us a glimpse of how complex behaviours can emerge from simple rules carried out by large numbers of  ‘cells’. This could be a metaphor for the simplistic behaviour of individual brain cells collectively forming the complex behaviours of our conscious and sub-conscious brain. We see this kind of behaviour where the whole is worth significantly more than the sum of its parts, all around us. Ant colonies, financial markets, democracies and traffic flows are all characterised by complex behaviours forming from simple rules followed by large numbers of ‘cells’. The study of this kind of behaviour is called Cellular Automata.

Patterns on some animals are believed to form through similar processes. Two in particular are the Conus and Cymbiola genus of seashells.

Conus Textile sea shell patterns are formed by a cellular automata process

For further reading on this fascinating subject I would highly recommend Critical Mass: How One Thing Leads To Another by Phillip Ball.

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