When you think of a mathematics textbook you probably imagine a series of intimidating pages with a few words and a bunch of strange (often Greek) symbols. I don’t think I’m alone in thinking that the fact that a lot of modern mathematics is only presented in this form is a bit of a crime. For instance, Tristan Needham expresses a similar feeling in the pre-amble to his text “Visual Complex analysis.” Professional mathematicians can usually get some idea of what is going on in the pages of a paper or textbook. However, anyone who hasn’t had as much training in their past loses out. Particularly since, to the untrained eye, there is no way to associate these abstract symbols with anything visual or otherwise.
Reading a mathematics textbook is not like reading a novel–it can be a slow and arduous process. Despite this, someone with mathematical knowledge might eventually be able to understand what is going on. To draw an analogy with computers – it is as if you need the “correct software” installed in your mind to process the text. The same concept applies to reading novels written in other languages – if you don’t have the “correct software” installed in your mind then all that you will see is a series of random symbols. An education in mathematics allows you obtain this “software”, once you have this you can “speak” about things you have never spoken about before.
The important thing to draw from above is that the symbols are simply placeholders for various ideas and concepts just as they are in any other written language. In my opinion adding visuals or graphics to a piece of mathematics significantly helps us to tie down what the symbols are trying to suggest (even if it is just a crude analogy). On the other hand, it is probably true that even illustrations and graphs on their own are probably not enough. Without the proper context, graphs or illustrations may simply appear as static creations with no further meaning. I think additional understanding can be achieved by playing around with the image in your head (or by sketching variations with a pen and paper). This playful approach to imagining visualising mathematical concepts no doubt inspired artists such as M. C. Escher (M. C Escher was a Dutch graphic artist who is well known for his often mind boggling and mathematically inspired work).
Nowadays we can go significantly further than Escher with the power of computer graphics. As an example I have listed a few of my the coolest looking pages and blogs related to visualising mathematics and mathematical concepts below:
- http://www.graphonaute.fr/ A selection of animations and images created by French engineering student Hugo Germain. Makes me think of what Escher’s work may have been like if he were born into the digital age!
- http://blog.matthen.com/ Lots of nice visualisations and a few cool visual proofs of well known mathematical theorems as well. The code for the visualisations is also available allowing anyone to play around/ learn how to create their own!
- http://imaginary.org/ Imaginary is an interactive platform which designed to showcase mathematical media content. The site contains plenty of pictures, videos and interactive demonstrations!
- http://visualizingmath.tumblr.com/ A tumblr page full of math related visualisations.
- http://geometric-aesthetic.tumblr.com/archive Tons of geometrical patterns and fractals.
In summary, the “beauty” of mathematics may be something “cold and austere” (as Bertrand Russell puts it), however, I believe everyone can gain if we do more to visualize the concepts involved. As well as helping our understanding, it allows us to think up strange new worlds (such as those depicted in other M.C Escher’s work – and potentially Einstein’s general relativity). Given the amount of maths out there I’m sure there is a lot of potential for mathematically inclined artists out there!
References and Further Reading
 This post is a continuation on themes I previously wrote about in about Maths and Visualization