## Fractals: What are they?

**Introduction**

Hi my name is Catherine Bruce and I’m a first year PhD student in Pure Mathematics at the University of Manchester. I study the geometry of fractals, which are objects deemed too irregular for traditional geometry such as straight lines, area, etc. (They have an infinite perimeter!) During the final year of a 4 year undergraduate degree I realised that I really enjoyed research and was lucky enough to be offered the opportunity to do it full time.

**How I got here**

**How I got here**

After leaving secondary school I did A-levels in Maths, Further Maths and French. I was also interested in Politics and took it as an extra AS-level in my second year at college. I then applied to do a four year undergraduate degree in Mathematics at the University of Manchester. This meant I graduated with an MMATH degree which is called an integrated Masters. I enjoyed my whole degree but realised only while undertaking my final year project that research was for me and applied for a PhD. I took a year out to go travelling after graduating and returned to Manchester in September 2016 to start my research.

**In Depth**

Fractals are too irregular for their size and structure to be measured using classical geometry. The main tool of fractal geometry is dimension, which has many forms. A lot of research is done into finding the dimension of different fractals and the image of these fractals under different functions. Fractals are often very beautiful – they have detail at all scales which means no matter how much you zoom in to one part of the fractal it will always have an interesting structure, which is not true for classical geometrical objects like 2D and 3D shapes.

An important property of fractals is self-similarity. This vaguely means that an object looks like it’s made up of lots of smaller versions of itself. Notice that each branch of this fern looks like a smaller fern, and each smaller branch looks like an even smaller fern. This is a simple example of self-similarity.

Many things in nature have a fractal-like structure: clouds, mountain skylines and forked lightening. Fractal geometry can be applied to many different things in the real world. Examples that I know of are lasers, cancer treatments and fracking. However, I do not deal with any of these applications as I (along with all other pure mathematicians) study the theoretical side of mathematics.In the first few months of my PhD I have been reading a lot to learn what everyone else in my field of research already knows. When I have done this I will be able to start answering questions that no one has answered before, and coming up with brand new research. This is the whole point of a PhD and is an exciting if not scary prospect!

**Going Further**

**Going Further**

Learn more about what fractals are:

http://fractalfoundation.org/resources/what-are-fractals/

Have a go at constructing your own fractal:

https://sciencevsmagic.net/fractal/#0060,0090,1,1,0,0,1

Have a look at the exciting research that’s going on in dynamical systems at the University of Manchester:

(http://www.maths.manchester.ac.uk/our-research/research-groups/analysis-and-dynamical-systems/)