TRIPLES AND THE MANDELBROT SET
This article gives an introduction to the Mandelbrot Set and shows how it can be explained in terms of triples. Triples are an integral part of the system of Vedic Mathematics, and they are defined as a sequence of three numbers that satisfy Pythagoras theorem. For example 3,4,5 is a triple because 32+42=52. An arithmetic for combining these triples was developed by myself in the late 1970s and has an extraordinary range of application1, simplifying and unifying many areas of mathematics, including an understanding of the Mandelbrot set.
The beautiful Mandelbrot set developed by Benoit Mandelbrot in 1980 illustrates how something that you would expect to be chaotic can in fact have a lot of structure. The set is normally defined in terms of complex numbers, which those with no knowledge of this area of mathematics would not therefore understand. I will avoid this approach as it will seem very complicated to those unfamiliar with complex numbers, and those who are, probably already know how the Mandelbrot Set is defined.
The set is shown below (taken from
Wikipedia2) and has a remarkable self-similar property in that you
will see the same shapes occurring in a magnified view of the boundary no matter
how far you magnify: it has a fractal shape.
The diagram below shows the set, with the coordinate axes and scale:
Here I will explain how a point can be used to generate another point related to it, using some rule. And then from that yet another point can be found using the same rule, and so on (this cyclic process is called "iteration"). This means that the initial point moves from one place to another (in an "orbit") and its behaviour can be one of several kinds.
Let's take a simple example of generating an orbit. Suppose we start with the point (1,2), and we introduce a rule: "double the first number and square the second". Applying this rule to (1,2) we get the point (2,4) because the 1 gets doubled and the 2 gets squared.
Now repeat this using the (2,4) we just found. We get (4,16). Repeating again we get (8,256). We could plot these points (1,2), (2,4), (4,16), (8,256) etc. on a graph, and this would show the orbit of (1,2) under the given rule.
Now let's look at doubling a triple. If p, q, r is a triple then the doubled triple is p2-q2, 2pq, r2. So to double 3,4,5 we find 32-42, 2x3x4, 52, which is -7, 24, 25.
Note that we will use only the first two numbers, to simplify things. So (3,4) goes to (-7,24) under this doubling rule. And similarly (2,1) will go to (3,4).
In fact this gives us the 'rule'
we need to help us generate the Mandelbrot Set.
The rule is: double the triple and then add the original coordinates.
Suppose we start with the point (3,4). We generate a new point from this one by doubling the triple and adding on the coordinates 3,4.
And adding 3 and 4 to -7 and 24 gives -4 and 28.
So (-4,28) is the new point.
And we can repeat this with the new point: that is, we triple-double (-4,28) and add 3 and 4. In this way we generate the orbit of (3,4) and we could plot the path on a graph. However, starting with (3,4), the point will zoom off to infinity: the first 3 points are (3,4), (-4,28), (-765,-220).
The Orbit of (0.3,0.3)
Let's try the point (0.3,0.3).
We must double the triple and add 0.3, 0.3.
Doubling it we get 0,0.18.
Now add 0.3 and 0.3 to each of these numbers: we get 0.3 and 0.48.
So the point (0.3,0.3) has gone to (0.3,0.48).
If you now double (0.3,0.48) and add 0.3, 0.3 you will get to the point (0.1596,0.588).
So we now have 3 points in our orbit:
In fact if you continue this process
the point does not zoom off. You can see the orbit in the diagram below.
You can see here that the initial point (0.3,0.3) is converging onto a point. The coordinates of that point are approximately (0.1435,0.4208). [The question arises: how can we get the coordinates of the final point from the initial point without going through all those iterations?]
The Orbit of (0.4,0.4)
But if you apply this process to
the point (0.4,0.4) you will find that it does not converge on a point but diverges
- zooms off to infinity, as shown in the diagram below. So (0.4,0.4) is not
in the Mandelbrot set.
The Mandelbrot Set
If you could find all those points, like (0.3,0.3), which do not fly off to infinity, when the above process is carried out on them, you would have found the Mandelbrot set.
Clearly a huge amount of work is involved because a very large number of points have to be examined, even to get an approximate picture of the set.
1. "Triples", K R Williams, Inspiration Books, 2010