Fibonacci

The Mandelbrot set

Let us investigate the sequence of points in a two dimensional Euclidean space (x₀,y₀), (x₁,y₁),(x₂,y₂),(x₃,y₃),... defined by the prescription


x0 = 0, y0 = 0 and (for n≥0) xn+1 = xn2 - yn2 - 0.1 and yn+1 = 2xnyn - 0.75 The first terms of this sequence are: (0.0000,0.0000), (-0.1000,-0.7500), (-0.6525,-0.6000), (-0.3424,0.0330), (-0.0999,-0.7532), (-0.6599,-0.5997), (-0.2939,0.3667), (-0.1005,-0.7522), (-0.6556,-0.5988), (-0.2875,0.3526), (-0.1004,-0.7520), (-0.6555,-0.5990), (-0.2913,0.3520), (-0.1004,-0.7521), (-0.6555,-0.5990), (-0.2912,0.3530), (-0.1004,-0.7521), (-0.6555,-0.5990), ...

Geometrically from (xn,yn) to (xn+1,yn+1)


Here we have regarded the special case (namely with a = -0.1 and b = -0.75) of a more general prescription:
x₀ = 0, y₀ = 0 and (for n≥0)
x_n+1 = x_n² - y_n² + a and
y_n+1 = 2x_ny_n + b
In case a = -0.1 and b = -0.75 we notice that the created sequence of points are within a certain circle (with radius 1 or 2 or, if you prefer, 10) from the origin.
When we apply the latter prescription with for example a = 2 and b = 0, then this yields the sequence (0,0), (2,0), (6,0), (38,0), (1446,0), (2090918,0), ...
It is clear that this sequence of points increases to infinity.
The set of points (a,b) for which the sequence (x₀,y₀), (x₁,y₁),(x₂,y₂), ... does not eventually go to infinity is called the Mandelbrot set.
Regard the point (a,b) as a point in the Euclidean plane. Then the Mandelbrot set can be depicted (the black area in the picture on the left).



(a,b) values           = reset	(xn,yn) values


The Mandelbrot set is incredibly complex. It consists of a large kidney-like area. On the border of that area there are infinite many circular areas with strange tentacles. A circular area on its own has very complex borders. Convince yourself and zoom in as far as possible on a border or a tentacle. Sometimes you meet the ever-recurring original figure.
When you click with your mouse on a figure, the figure is magnified by factor 2. Explanation of the colors.

You will certainly have noticed that the sequence of points with which we started this story does not only remain inside a circle with radius 2, but that they also approach 3 different points in the plane. It appears (although, till now, nobody has been able to proof it) that such a sequence always approaches 1,2,3 or more points in the Euclidean plane.. The (a,b)-values from the big kidney-like area have the property, that the corresponding sequence approaches exactly 1 point in the plane. This will help us to find an equation for the border of that area. Assume that (x_n,y_n) converges to (x,y), then (x_n+1,y_n+1) will also converge to (x,y) and so
x = x² - y² + a and y = 2xy + b. Besides, for a point (a,b) on the border of the kidney-like area, the sequence (x₀,y₀), (x₁,y₁), ... converges to a point at a distance 1/2 from the origin (Proof). Therefore, the (a,b)-values satisfy the system x = x² - y² + a, y = 2xy + b and x² + y² = 1/4. Elimination of x and y from this system results in the equation ((a-1/4)² + b²)((a-1/4)² + b² + a - 1/4) = b²/4.
This is the equation of a so-called cardioid.
The points on the Mandelbrot cardioid can be constructed geometrically: Draw a circle with radius 1/4 around the origin. The point A(1/4,0) is situated on this circle. A line l through A cuts that circle in, say, point B (and in A). The 2 points of line l which lie at a distance 1/2 from point B, lie on the Mandelbrot cardioid. All points on that cardioid are defined in this way.

Click with your mouse in the big circular area (P₀) at the left side of the cardioid and after that a number of times on the right picture. The right picture shows the values (x₀,y₀), (x₁,y₁), etc. Each click adds the next point from this sequence. You will notice that there are 2 points between which the (x_n,y_n)-values swing. Now, click on the green square at the foot of the left picture (reset) and then click somewhere in the circle (P₁) at the top of the cardioid. Then, the figure on the right shows that there are 3 attraction points. for the sequence (x₀,y₀), (x₁,y₁), ....
Now choose a point in the largest circle (P₂) between P₀ and P₁. Here, the number of attraction points is 5. Now choose a point in the largest circle (P₃) between P₁ and P₂. Here, the number of attraction points is 8. Notice that the number of attraction points for P₀, P₁, P₂, ... form the sequence of Fibonacci : 2,3,5,8,13,21,... . Moreover, notice that in P₂ the 5 attraction points are situated in a circle, but that those points are not dealt with in the order they are situated. In order to deal with the 5 points you have to circle 2 time the center of the circle. For P₃ with 8 attraction points you even need 3 complete revolutions. Again the sequence of Fibonacci shows up. The number of revolutions for P₀,P₁,P₂,P₃,P₄,... are 1,1,2,3,5,8,13,... .

And that is not all there is. On top of circle P₂ (the largest circle between the large circles at the left of and above the cardioid) there is a somewhat smaller circle and on top of it there is a tentacle with 5 arms of which one is attached to the smaller circle. The number of attraction points of P₂ appears to be equal to the number of arms of the strange tentacle above it. This is a universal property: The number of arms above a circle which is situated upon a circle, on the cardioid, equals the number of attraction points of the circle upon the cardioid.
You can also regard the smallest tentacle. In case of the tentacle with 5 arms, which is attached to the circle on P₂ you see, that the second arm from the attached arm (calculated counterclockwise) is smallest. This number corresponds with the number of revolutions around the centre for P₀,P₁,... if all 5 attraction points are visited once. This is a universal property too.
Wat is the relation between the number of attraction points of a circle on the cardioid and the number of attraction points of the circle on top of that circle?
It is also interesting to investigate the behaviour of the sequence (x₀,y₀), (x₁,y₁), etc of a starting point within the cardioid in the neighbourhood of a point where a circle is attached to the cardioid.

Let us end with a special property: There exists a map f: A->B from the area in the left picture which is not colored black (thus the set of (a,b) values for which the sequence (x₀,y₀), (x₁,y₁),(x₂,y₂),... diverges to infinity) to the set of points outside the unit circle (i.e. points (x,y) with x²+y²>1). This function has an inverse f^-i: B->A so that for every point p from A and q from B f^-i(f(p))=p and f(f^-i(q)) = q. The speciality of this function is that the function does not influence angles. i.e. if p,q and r are points in A and the angle between the lines pq and rq is, say, 20^o, then the angle formed by the points f(p),f(q) and f(r) in B is also 20^o.

Form this property you can deduce that the Mandelbrot set consists of one piece and that this set does not encircle areas that are not subsets of the Mandelbrot set.
On the internet there are many sites about Mandelbrot sets. They often contain spectacular pictures.