Prime numbers

It is from these recursive equations that some mathematical wonders are created. We begin with plane filling curves or fractals, which are curves that fill planes without any holes. The first such curve was discovered by Guiseppe Peano in 1890. Other mathematicians who used difference equations in their work with plane filling curves include David Hilbert (1862-1943), and Niels Fabian Von Koch (1870-1924). The relevant work all three will be discussed in the following. As will the work of Emile Picard (1856-1941) and Martin Kutta(1867-1944), both of whom used recursive equations in solutions to differential equations.

There are curves that fill a plane without holes. The first such curve was discovered by Guiseppe Peano in 1890 and the second by D.Hilbert (1862-1943). Calling them Peano Monster Curves, B. Mandelbrot collected a series of quotations in support of this terminology.

Fractals burst into the open in early 1970s. Their breathtaking beauty captivated many a layman and a professional alike. Striking fractal images can often be

obtained with very elementary means. However, the definition of fractals is far from being trivial and depends on a formal definition of dimension.

It takes a few chapters of an Advanced Analysis book to rigorously define a notion of dimension. The important thing is that the notion is not unique and even more

importantly, for a given set, various definitions may lead to different numerical results. When the results differ the set is called fractal. Or in the words of B. Mandelbrot, the father of fractals:
`` A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension."
Visit Mandelbrot set

The topological dimension of a smooth curve is, as one would expect, one and that of a sphere is two which may seem very intuitive. However, the formal definition

was only given in 1913 by the Dutch mathematician L. Brouwer (1881-1966)).

A (solid) cube has the topological dimension of three because in any decomposition of the cube into smaller bricks there always are points that belong to at least four

(3+1) bricks.

The Brouwer dimension is obviously an integer. The Hausdorff-Besicovitch dimension, on the other hand, may be a non integer. Formal definition of this quantity

requires a good deal of the Measure Theory. But fortunately for a class of sets Hausdorff-Besicovitch dimension can be easily evaluated. These sets are known as

the self-similar fractals and, because of that ease, the property of self-similarity is often considered to be germane to fractals in general. The applet below illustrates

the idea of self-similarity.

Initially there is an equilateral triangle in the middle of a straight line segment. This combination of four segments form a pattern which serves as a description of an

algorithm to be used later on. The secret meaning of the pattern is this:

Start with a straight line segment. Proceed in stages. On every stage traverse the curve obtained previously. Every time you encounter a straight line segment, replace it with the four-segment pattern as was first done with the original line segment. As with the Hilbert curve, which is similar in construction, it's possible to formally prove that thus obtained curves converge to a limit curve. This limiting curve is self-similar because it consists of four smaller copies of itself which, in turn, also and naturally consist of four smaller copies of themselves and so on.

To see how the algorithm works try the following applet (click on hilber.scr top of the page) or try this movie.

To understand the notion of the similarity dimension, first observe that, if the initial line segment was 1 unit in length, then the second stage curve that consists of four

segments each one third of the initial line, is 4/3 units in length. On the third stage the length becomes (4/3)2 and on the fourth it's (4/3)3 and so on. Since 4/3>1 the

farther you go the longer the curve results. Finally, the limiting curve will have infinite length. Which is somewhat surprising because of it's being generated by such a simple procedure in a very limited space from the curves of finite length.

The similarity dimension of the snowflake curve is finite. This is arrived at in the following manner. If it were a straight line we could split it into to smaller segments

each half the length of the "parent" line. The length of the line would be the sum of the two smaller segments. If we were talking about areas, then taking a square and

splitting it into 4 smaller squares with areas 1/4 of the "parent" square. We would observe that the four smaller areas sum up to the original size. Notice that when the

side of a square is halved, its area decreases by the factor of 4 which is (1/2)2. For a cube, acting similarly, decreasing its size by a factor of 2, results in smaller cubes each with the volume 1/8=(1/2)3 of the "parent" cube. We can detect a commonality in these three examples. Given a shape of size S. It's split into N similar smaller shapes each with the size (S/N) so that N*(S/N)=S. In each of the three cases we used a different function S. If a is a linear dimension of the shape we have S(a)=a for a line segment and S(a)=a2 and S(a)=a3 for the square and cube, respectively. Thus, N*(S/N)=S can be rewritten as N*(a/M)D=aD where a is the

"linear size" of the shape, M is the number of linear parts, and N is the total number of the resulting smaller shapes. This gives N*M-D=1 or N=MD. In all three cases we took M=2 and D was successively 1, 2, and 3. We see that D=log(N)/log(M) is what we would call the dimension in all three cases.

This quantity D is known as the similarity dimension. It applies to shapes that are composed of a few copies of themselves whose "linear" size is smaller than that of

the "parent" shape by a factor of M. Returning to the snowflake, we have N=4 and M=3. In this case D=log(4)/log(3) is somewhere between 1 and 2.

The Koch's snowflake has no self-intersection and is obtained from a line segment as an image of a continuous function. By one of Brouwer's theorems this function

preserves the topological dimension of the segment (which is, of course 1). Finally, the curve has topological dimension 1 whereas its Hausdorff-Besicovitch dimension is log(4)/log(3).

Note that for the Hilbert curve that (disregarding small links) consists of 4(=N) copies of itself each twice (M=2) as short as the whole, we get D=log(4)/log(2)=2.

Which is not surprising as the curve covers the whole of the square whose dimension is 2. From the above mentioned Brouwer's theorem it follows that the

limiting curve must have self-intersections. For otherwise it would be a topological (in particular, dimension preserving) transformation. But a line is of (topological)

dimension 1 whereas a square has dimension 2.

Fractals and plane filling curves are two of many applications of difference equations. Another important application is for the numerical solution of differential equations. Two of the most important are called the Runge-Kutta method, based on the work of Martin Kutta(1867-1944), and the method of successive approximations, based on the work of Emile Picard (1856-1941). We begin by discussing the Runge Kutta Method, which is quite simple to use. This method for solving initial value problems of the form

The Runge-Kutta formula involves a weighted average of values of f (t,y) at different points in the interval [t_n,t_n+1].
It is given by

Successive application of this formula yields increasingly more accurate results. This method is of two orders more accurate (the results oscillate less) than the Improved Euler Formula and the three term TaylorMethod.

We can write an ordinary differential equation as a fixed point formula. If we have a starting guess, we can iterate the equation to find an approximate solution to the original equation. For example, suppose we have the first order ordinary differential equation

with initial condition y(x₀ )= y₀ . This equation can be written as an integral equation in the form

Then, knowing y₂(x) we could form y₃(x) by the same technique. We can continue this process indefinitely, each time using the formula

Difference Equations

and

Recursive Relations