Posts Tagged ‘fractals’

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Where art, neurons and fractals collide.

November 12, 2012

One of the things I have always appreciated in its many forms is pictorial art. As the use of my hands has reduced, I’ve been able to produce an awful lot less of it, but my appreciation of other people’s art has not changed.

This gallery of images(go through the slideshow) is a case in point. Dunn noticed the similarity between blowing ink across paper and neuron form. He is a neuroscientist, so the comparison would be foremost in his mind, and the art he has produced is striking both aesthically and in familiarity.

I, being a mathematician who has done a fair amount of neuronal modelling, see neuronal forms when playing around with other mathematical tools. Take for example this simple fractal flame I made with Apophysis:

Neuron Flame

It is simple to see the somas and dendrites within this fractal. There is a reason for this. Neuron growth can be described in fractal terms. There is a touch of pareidolia to seeing things in such fractals as that flame, and a lot of people see fractals in virtually everything living, but that paper shows that neuronal growth can be modelled quite closely using fractal techniques.

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Of Sponges, Carpets and Blog Headers

November 2, 2012

I thought I’d start with explaining what the header image to my blog actually is. It is a zoomed in image of a type of fractal called a Menger Sponge, generated by an excellent program called Mandelbulber and smoothed a little in Photoshop.
I shall leave an explanation of what a fractal actually is until a later post, except to say that generally they’re mathematical sets that tend to stay just as complex the closer you look at at a representation of them. Also they had a tendency to make an appearance on techno and dance music CD sleeves in the mid to late 90s. Of course this is a massive simplification of what they are, but I’m trying to keep the tone of this blog somewhere between pop-science and wading through obscure corners of the literature.
The Menger Sponge is a three dimensional version of the more commonly seen Sierpinski Carpet. As for the names, it’s easy to see that they follow the trend of <Discoverer’s Name> <Something everyone has heard of>. Looking at a Sierpinski carpet makes its name quite sensible.

Sierpinski Carpet
It is calculated in a very simple manner. You start with a square, and divide it into 3 parts on each side, making 9 squares. You then cut out the square in the middle. The fun bit comes next. You then take all the squares that are still there, and do the same to them as you did to the original square. You can then repeat this to make smaller and small features forever, or until you get bored, whichever comes first.
A Menger Sponge takes this idea, but starts with a cube, and each side of the cube is treated in the same way as for the Sierpinski carpet, but the whole goes all the way through the cube, making lots of sub-cubes, which you do the same to. After a few hundred repetitions of this you end up with something like this:

Menger Sponge

Which you then colour in, smooth a bit and set at the top your blog with some writing on!

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