MATHINFINITY

FRACTAL

The Polish-born mathematician, Benoit Mandelbrot, created the term Fractal to describe a broken stone - fragmented and irregular. The important features of a fractal are its structural self-similarity and indefinite resolution. Simply speaking, a self-similar object is similar to a part of itself. For example, the coastlines look irregular from a distance. However, if it is closely observed, part of it is similar to the entire shape. Infinite resolution means that when observing objects on a small scale, its unique and detailed structure can be seen.

Fractal Geometry is geometry that studies irregular geometric shapes. It is used to describe, model and analyze complex forms, non-uniform shapes and rough edges found in nature. Hence, it is also known as geometry of nature.

Interactive Game

If you still find fractals difficult to understand, you may look at the screen carefully, and see if you can get some insights from the Fractal Tree and Koch Snowflake.

1) According to the researches by Benoit Mandelbrot and Michael Frame, the construction of a Binary Fractal Tree is defined recursively by symmetric binary branching. The trunk is split into two branches, each having the same angle with the direction of the trunk. Each next branch then splits by the same rule. Continuing this process as many times as we wish, the tree is the set of branches together with their terminal points.
2) The Koch Curve introduced by the Swedish mathematician Helge von Koch is a kind of fractal curve. It is also called Snowflake Curve due to its appearance. It can be constructed by first dividing each side of an equilateral triangle into three segments of equal length. Then remove the line segment in the middle and replace it with an equilateral triangle pointing outwards but without base. It will become a hexagram. Repeat the same procedure for each side of the shape. After each repetition, the length of its perimeter will increase by one-third. Therefore, the snowflake curve has self-similarity and infinite length, but the area is still limited.