This is my first attempt at blogging. Please excuse my simple language – I am a fledgling. I will be sharing thoughts and ideas during my fairly young life as a thinker of all things mathematics. I’m hoping this will include topics that don’t immediately bring maths to mind…
And that brings me neatly on to the subject of this first post, which is one of the main things that got me seriously thinking about a career in maths (other influences will be presented in due course). The subject is Fractals. These are patterns that have the same structure when we zoom into them and keep on zooming. The way they are created is through recursive mathematical equations, starting from a simple initial seed. Thus, I have deliberately entitled my post as a fractal of sorts – starting from H, we add letters (one at a time) such that the next string of letters resembles the word “Hello” more and more accurately.
The first project I carried out was as a final year MMath student, and it covered this topic. I was hooked – even at this stage of my life, when I should’ve felt comfortable doing maths, it still felt daunting. Could I really put up with these never-ending equations that don’t really look appealing? Fractals, on the other hand, literally showed me how equations can be beautiful.
So that’s where my real interest started. I was thrilled – not only did fractals provide a fantastic link between maths and art, they were actually vital towards research. For example, chemical reactions in the body take place in crowded environments that are conveniently modelled by fractals. Reaction-diffusion equations can then be studied in the fractal approximation to give us a better idea as to what is happening in the real reactions. The pictured Sierpinski gasket is a nice fractal to use for this purpose. I carried out a bit of this work in Manchester after completing my PhD there. Indeed, part of my PhD also involved the creation of fractal patterns using cellular automaton models.
I don’t want to delve too much into this topic here as (i) this is meant to be a short introductory post, and I want to encourage any reader to look fractals up themselves and gain their own perspective and (hopefully) appreciation, (ii) I still don’t have a sufficient grasp of fractals (though that keeps me interested), (iii) I could never do justice to the power and beauty that such simple equations and patterns possess.
I will finish by noting (quite excitedly!) that the default title picture that WordPress have selected for me may be classed as a fern, which is also a fractal. Starting from an initial state (which could simply be a straight line), iterating a computer model of a simple rule that adds straight lines in appropriate places will produce such a fern. Try it yourself and produce nature!