Fractals: Gods Artwork, Part III

Returning to this series is simply an excuse to include the most marvellous of images that are after all simply contained within the simplest of non-linear iterative equations, coded into the very maths we live in. Whereas the previous two parts in this series focussed on the most common of 2D fractal sets, the Mandelbrot set, and the rather complex boundary that is in a state of wrenching around it, this post extends out another couple of dimensions into the realms of fractal quaternions.

In a normal(?) complex number Z = x +iy where i x i = -1.  Quaternions introduce  j and k extensions into 4D such that i2 = j2 = k2 = -1. Using the same method as before of taking a starting point, putting it through the same equation again and again then seeing if that points bounces away to infinity (i.e. starting point not inside the set) or converging to a point (i.e. starting point is inside the set) one can create 4D fractal set geometries using this dimension extended complex number approach.

The tricky thing about 4D geometry is that you can’t visualise it easily. Our evolution didn’t necessitate our ability to see such objects as these objects, apart from them not existing, do not pose a threat to our food supply or general health and welfare (“Oi, quaternion, stop chatting up my girl”) therefore we’ve never evolved the ability to see them. All we can do is to plot a 3D ‘slice’ of the 4D object onto a 2D screen. Even with such a reduction you get some lovely output (these from the quaternion form of the Julia set, an adjunct to the Mandelbrot set):

julia_2

julia_3d_2c

julia_3d_1

If you’re interested, all the above were created with the superb tool Chaoscope.

jgI’ll come back to this series at some later stage. In the mean time I’ve bought Edward Lorenz’s book “Essential Chaos” to brush up on my knowledge of the butterfly effect and sensitive dependence to initial conditions generally. Once we’ve covered that, and maybe touched on largest Lyapunov exponents, you’ll be more than match for Mr. Chaos-scientist himself Jeff Goldblum from off of Jurassic Park.

20th August 2009 Ross-on-Wye

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This article first appeared on the Siemens Digital Industries Software blog at https://blogs.sw.siemens.com/simulating-the-real-world/2009/08/20/fractals-gods-artwork-part-iii/