Mathematics Extended Essay (Grade: A)

I. Introduction

A fractal is an object that exhibits self-similarity (Falconer 22). Self-similarity is a property that a similar shape can be identified at different scales looking at the object (Falconer 22). The more self-similar an object is, the more complex it looks (Chaos, Fractals and Dynamics). A lot of time series data, such as which are data ordered by time, display this feature and this feature commonly correlates with real-life phenomena (Kantelhardt 4). For example, in electroencephalogram (EEG) signals, if the patient is has an epileptic seizure, his EEG signal looks more complex than a non-epileptic patient’s EEG signal (Neurocomputing). To incorporate this feature of self-similarity when modeling EEG signals, functions called fractal interpolation functions can be used. These fractal interpolation functions can convert the raw data into an analyzable function that displays self-similarity (Manousopoulos 1). Not only they can model these signals, they can also measure fractal dimension, which is a measure to quantify the level of self-similarity (Barnsley 223).

By doing this, fractal interpolation functions can take data points that are not even part of the raw data points into consideration. There have been many algorithms devised in academics to measure a fractal’s fractal dimension using the raw data only, but there has not been much development in measuring the fractal dimension using fractal interpolation functions (Navascués 2)…