Painting By Numbers

Flame fractals were discovered by Scott Draves in 1992. I am not a mathematician, so I do not fully understand what they are, or how they work, and cannot explain them in any way that would stand up to scrutiny. So without apology, and for all the other non-mathematicians out there, here is my take on the matter:

Imagine a game:

The game is played outside, on one of those giant chessboards that you can walk around on. I have labelled the squares on the board with grid numbers. The square at lower left is 1 Across, 1 Up, or, as we we will refer to it (1,1). The square at top right is 8 Across, 8 Up, or, (8,8).

I give you a six-sided die cube, and a phone. The die cube has colours on its faces instead of numbers, and the colours are red, blue, yellow, purple, green and orange. The phone has contacts for the six assistants who will help you play the game. The contacts have codenames which are Red, Blue, Yellow, Purple, Green and Orange. I also give you six pots of paint and a brush. The paint colours are (you guessed!) red, blue, yellow, purple, green and orange. I ask you to pick a square on the board at random, and go stand there.

Now I explain the rules:

Each turn, you will roll the die cube, and then phone the contact whose codename is that colour. You tell them where you are by quoting the two grid numbers – for example if you are standing on 4 Across, 3 Up, and if you rolled red on the die, then you would call your Red contact and say “I am at (4,3)”. The red contact will then tell you which square to go to next, by quoting two numbers back at you (for example,”6,5″). You will walk to that new square, and paint it with the colour of the contact who instructed you to go there. When you have painted that square, you roll the die cube again, check the colour, phone that contact, and get new instructions, and so on.grid1

The contacts each have a set of mathematical rules which they follow rigidly. Each contact has a different rule set, but they all do the same job – they convert the two numbers that you provide into two different numbers.

What happens:

At first it’s a very boring game. You walk, you paint, you walk, you paint… Of course, as time goes on, you eventually begin to revisit squares that you have already painted. When you paint over, you notice something mildy interesting. These are obviously special paints – instead of obscuring the previous colour, each new colour blends with whatever colour is already there, producing a new, mixed colour. Mmm… You carry on walking and painting.

grid3After a long, long time, you are getting very bored and very tired. But then you start to notice something very odd. Many of the squares have by now taken on a muddy brown appearance – as you expected they would – but there are some areas of the board which are definitely tending towards one particular colour. This piques your curiosity. You begin to focus on the contacts and try to remember where they send you – but, although you think that you detect certain trends, there is absolutely no repetition, and it is clear that none of the contacts are sending you repeatedly to the same square. Nevertheless, you seem to be returning time and again with the same paint colour to “somewhere near” the existing concentration of that colour. It is almost as if some strange force is attracting that colour to that area.

Eventually you give up out of sheer axhaustion. As you step back from the board, you realise to your amazement that some kind of weird pattern or design has arisen out all those numbers!

Computing power:

When rendering flame fractals on a computer, there are some differences from the imaginary game:

  • The grid has millions of squares, or pixels.
  • The software equivalent of the die cube has many ‘faces’, and is unevenly weighted.
  • The mathematical functions that replace two numbers by two other numbers are calculated in software.
  • Multiple transformations can be applied.
  • The colour blending is done by a mathematical formula.
  • The colour schemes are selected for aesthetic effect.
  • There are additional ways of tweaking and adjusting the vividness of the resulting image.

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