Recently at Cygnet U3A, June Cunningham gave us two utterly fascinating talks on fractals and the Mandelbrot set (aka the “M-set”). Thank you, June.
The piece de resistance was a 53 minute video: Arthur C Clark’s Fractals – The Colors Of Infinity. It is now available on You Tube: https://www.youtube.com/watch?v=Lk6QU94xAb8
Wonderful stuff, really worth watching.
Is the M-set “real”?
3 minutes into Clarke’s video, Prof Michael Barnsley pleads
“The Mandelbrot Set is real!”.
“It does exist” but it is “not touchable”.
Well he would say that wouldn’t he. He’s a mathematician.
Physicists are more discerning. Physics draw a distinction between mathematical existence and physical “real world” existence. Mathematicians deal with what could be the case, physicists are concerned with what actually is the case.
The Greeks like Euclid believed that their geometry explored the properties of real space, that it was a form of physics. The investigation of non-Euclidean spaces by Bolyai, Lobachevsky and Riemann in the late 19th century showed that other geometries were possible. In the early 20th Century, Einstein and Minkowsky clinched it by showing that even the real world is not Euclidean. Einstein was very annoyed with mathematical idealists who believed that you can, by thought alone, decide what the Universe must be like. Einstein retaliated with: “Space is what we measure with measuring rods. Time is what we measure with clocks”.
Perhaps the single most important theory underlying modern physics is the atomic theory: the idea that matter is not infinitely divisible but is made up of small particles called atoms and molecules. It follows that we can never construct a material Mandelbrot Set because, at sufficient magnification, we would be frustrated by the shape of the molecules themselves and unable to continue. Perhaps we could construct a field which is a Mandelbrot Set? Maybe not even that according to Hawking. Even space itself is not infinitely divisible beyond the Planck Length.
The objects depicted in the video and called fractals are not true fractals because, in the real world, at some resolution, their self similarity must break down. They are simply self-similar fractal-like objects created by a process of iteration. Mathematically speaking they only become fractals in the limit as the number of iterations tends to infinity. Like the M-set itself, fractals are a mathematical ideal. They cannot exist as material objects in the real world.
The important thing about the ferns, trees and mountains depicted in the video is not their fractal nature, not their properties “in the limit”, it is the creation of such self-similar objects by a process of iteration and which can closely resemble objects in the real world. This iterative process is analogous to the concept of auto-regression mentioned in the previous post. Instead of iterating numbers we iterate geometric shapes. In the case of the fractals depicted above, the innovation is zero and it is a perfect shape.
The image of this real world ferns is certainly not perfect. It has many irregularities which cause it to depart from the ideal state.
That’s the point. Fractals are perfect – the real-world is not.
If you get really hooked on this you may start wondering: Can it be generalized? Are there any other M-Sets?
The M-set is dependent on the addition and multiplication of complex numbers, numbers like 3+4i where, by definition, i.i = -1 (dot means multiply). That is all you need to know to multiply two complex numbers, e.g. ( remembering our high school algebra) –
(2+3i).(4+5i) = 8+10i+12i+15i.i = 8+22i-15 = -7+22i
The question boils down to “Can the complex numbers be generalized?”.
Well yes they can. They were generalized by the Irish mathematician Hamilton in 1843 who invented quaternions. A quaternion is a sort of super complex number written in terms of three imaginary numbers such as 2+3i+4j+5k. By definition, i.i = j.j = k.k = i.j.k = -1.
As before that is all we need to know in order to add and to multiply two quaternions. We can use Mandelbrot’s iterative formula to decide whether a point in 4-dimensional quaternion space is inside or outside a given area, let’s call it the Q-set, and colour it according to how fast it goes to infinity, as before.
This is what you get.
The You Tube is available at: https://www.youtube.com/watch?v=c-K4Lk98m38