Fractals And Number Theory: Mysteries Of Pure Mathematics

What do trees, waves, mountains, and a Romanesco cabbage have in common? These natural objects are all considered fractals. A fractal is a geometric figure whose structure is independent of scaling. In other words, we can “zoom in” or “zoom out” of the structure, and it would appear the same, or at least for natural objects, very similar. Therefore, fractals consist of infinite patterns which are repeated infinitely.

A tree, for example, is primarily made out of a trunk. This trunk is then divided into branches, which are divided into smaller branches, and so on. At some point, the smallest branches will produce leaves, which consist of a main petiole, which is divided into veins, which are divided into smaller veins, and the process continues infinitely as long as the tree grows.

It is also possible to create artificial fractals using mathematics. In fact, there are much more possibilities of artificial fractals than there are of natural fractals. These geometric figures exhibit certain properties that are exclusive to them. One of these properties is the fractal dimension. The fractal dimension (also called Hausdorff dimension) is a measure of roughness in mathematics, and it serves as a measure of the local size of a space. For example, a point has dimension 0, a line segment has dimension 1, a square has dimension 2, and a cube has dimension 3. However, most fractals have non-integer Hausdorff dimensions (i.e. a non-finite number defines its dimension). This is the case of the Koch curve.

A Koch curve is first formed by a line segment of length 3. At the first step, the middle-third of the segment is replaced by two line segments of length 1, forming an equilateral triangle. In total, we now have four segments. At the second step, each middle-third of each four line segments is now replaced by two line segments of length 1/3. The process is repeated at each step. In total, for n steps, there will be 4n-1 new line segments. Therefore, the perimeter of the curve is infinite.

In terms of fractal dimensions, there are various methods to calculate the Koch curve’s dimension. One of the goals of the project is to verify each method. For simplicity, it has been demonstrated that the Koch curve’s dimension is equal to ln(4)/ln(3), or 1.26…

The main goal of the project is to demonstrate the relation between the Koch curve’s fractal dimension and its perimeter, area, and the overture at each step. What happens to the dimension of the Koch curve when the overture is not equal to one? Can we calculate the curve’s dimension if each triangle is not divided proportionally (i.e. the triangles are not self-similar)? What value of the overture will produce the largest area under the curve? What happens to the curve’s dimension if we randomly choose some steps to form inward pointing triangles, and other to form outward pointing triangles? Can we apply the Koch curve pattern to various applications (e.g. the length of a shoreline)?

The mysteries of randomness and self-similarity are infinite.

Image Source: Flickr


Benjamin Piche
Contributor

Originally published in Bandersnatch Vol. 47 Issue 10 on March 14, 2018