What is a fractal?

From coasts to consciousness to colour..

Fractals are beautiful. Most examples we see are self-similar no matter how far you zoom in. However, fractal doesn’t mean self-similar; even squares are self-similar. Fractal means having a dimension that is not a whole number. I can tell you that this shape is about 1.465-dimensional. That sounds crazy, but when we say a line is 1-dimensional, a square is 2-dimensional, and a cube is 3-dimensional, how are we defining dimension here?

The answer is that when you scale a line up by a factor of three, you get three identical copies of the original, of course, which is 3¹.

When you scale a square by a factor of three, you get 9 copies of the original, or 3².

And when you scale a cube by a factor of three, you get 27 copies of the original, or 3³.

Notice that the powers of 1, 2 and 3 all match the dimension. That’s how we define ‘dimension’. But when we scale the fractal in the image above by a factor of three, we get five copies of the original. But we can’t represent 5 as a whole number power of three. Instead, we would have to take three to the power of around 1.465. The coast of the UK is also a fractal. If you measure it with 1-metre measuring sticks, it would be way longer than if you used 100-metre measuring sticks. But how do we define the dimension? We put the coast of the UK on a square grid. Count how many squares are touching the coast. Scale the coast by a factor of 2 and measure it this way again and test it over a wide range of zoom levels. We have determined that the coast of the UK is about 1.21-dimensional. Sure, self-similar fractals are beautiful, but it’s also important to only view them as a low-information toy-model of fractals. The prototypical examples are in nature, all around us. Snowflakes, trees, cracks, certain other plants, etc. How does this map onto the brain? Emotions are infinitely complex, fractal connections and associations. Every emotion is a UNIQUE fractal that, no matter how far you zoom in on it, never becomes smooth. And indeed, colours, may come from those emotions. I think that spiritual consciousness and colour is fractal.

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