How to make beautiful and accurate mathematical illustrations: 5 ways
Have you ever noticed that most illustrations in math articles look the same? The same basic style, the same LaTeX’ed labels, similar lineweights and the same arrowheads? Why is this? Similarly, have you ever noticed that every single math article is written with a single font, Computer Modern (it IS a nice font, but still!)? Do you think it is odd that our disciplinary conventions of presentation are so very rigid?
Mathematical figures in journal articles serve an expository purpose. They are there to explain, not to prove. Proof by figure is not a concept mathematicians are fond of. In fact, algebra was effectively invented to do away with geometry, or at least to formalize it to try to avoid some misleading errors of accuracy. And it seems that mathematicians remain suspicious of figures, with the guiding principle that fewer is better. However, perhaps one of the reasons that figures are used so sparingly is that nice mathematical figures are hard to make! There does not seem to be one mathematical figure maker solution, and every mathematician has a different way of doing this.
So what are these different ways of making mathematical images? In this post I will discuss a few different methods to make (and not to make) mathematical pictures.
Analog
What beats paper, pencil, scissors and glue? Not much to be honest, and usually this is the starting point (at least paper and pencils, maybe not so much the glue!). The method is immediate, repeatable, and easy. The ultimate WYSIWYG methodology (What You See Is What You Get). Although these figures usually do not form part of a final published journal article (why not?!), they are an important part of the process, and they help to refine what it is that needs to be illustrated.
This method needs little explanation. Pick up a pen, draw a thing. Taking this a step further though, if you do wish to level up your hand drawings, here is a hack for you: it is possible to make nicer looking pictures by drawing things really big and then shrinking them down. It’s true! It works every time. The figure above was made using a drawing tablet, circa 2003.
Postscript
This maybe falls into the category of What Not To Do. Postscript is a page description language developed in the 1980s. Usually Postscript is written by another program, and the postscript is then interpreted by a printer. However, it is also possible to write programs directly in Postscript! If you are feeling brave, give this a try. Check out Bill Casselman’s book on the topic “Mathematical Illustrations”.
Be ready for a fairly steep learning curve. And like all procedural methods of making images, this requires some forward planning and patience. But sometimes the guess and check method can be a lot of fun, even when it spits out some unexpected results.
Despite the arduous nature of this method, I did produce over 140 illustrations using Postscript for my Masters Thesis about Penrose Tiles. I become one with the machine.
TikZ
This will likely meet the needs of most mathematicians except those who are really creating some unusual figures. All of the figures in my PhD dissertation were created with this methodology. Although even the name of the package is a warning (“TikZ is kein Zeichprogramm”), for many things, TikZ is just right. It compiles inline in your latex document, so it is well positioned to create figures that flow with the text. Arrows, basic graphs, nice labels: on these scores TikZ has you covered. TikZ was developed by Till Tantou, the same individual who brought you Beamer (the presentation software that is now ubiquitous at math conferences everywhere). I’ll confess that I have no love for Beamer, despite the fact that it does make giving a LaTeX heavy talk very simple. Its aggressive aesthetic style also has the effect of making all math talks look *exactly the same*! Which is tragic and very boring.
Keynote
Back in 1988, our family got our first computer. It was an apple computer, and my absolute favourite thing about it was MacPaint (so I guess you could say I’ve been doing computer graphics for over thirty years!). MacPaint was a drawing program, with a state of the art palette of grayscale fill patterns. I drew many many many smiley faces. The funny thing about it is that no matter how hard I tried to do something “different this time”, it always ended up looking kind of the same (yet another smiley face). This of course was a reflection of the tools available, which were quite limited.
Present-day Keynote reminds me of MacPaint, but in the best possible way. I love using Keynote to make figures. There is a limited palette of shapes, colours and line weights. But this is advantageous when we want to make figures that have a uniformity to them, which is often the case when formally illustrating mathematics.
To add nicely typeset labels to your Keynote figures, checkout the mini equation editor LaTeXiT. It is a simple and easy way to create tiny pdfs that you can drag onto your Keynote slide.
Rhino + Grasshopper
Now that I work in the design industry, I find that I often need to create figures of 3D mathematical concepts. By far my favourite method of doing 3D illustrations is in Rhino (a 3D modelling software), together with the parametric modelling overlay Grasshopper. Grasshopper has a nicely curated set of tools, and makes it easy to get 3D geometry on screen, and do fun operations on that geometry. Because of its ease of use and visual interface, it is a tremendous tool for exploratory work.
Aesthetics, Accuracy and Easy of Use
So how do all these methods compare? Are they accurate or beautiful? Exploratory or expository? Were they easy or dreadful to make? I made a chart for that:
Conclusion
Mathematical figures are not necessarily easy to make. But a well made illustration can be incredibly illuminating. If I’m trying to decide whether or not to read a mathematics paper, I’m usually skeptical of those without illustrations.
I’ve learned some nice things from architects while working with them at MESH. And one thing I really appreciate about architecure is the range of accepted visual styles. Even though, like mathematics, architecture has very technical communication needs, it seems understood that it is possible to communicate precisely without sacrificing aesthetic freedom. I quite like this idea, especially when I look at the formality of some of images I produced for my doctorate (using TikZ). They were appropriate for the time and the type of publishing I was doing at the time in mathematics journals. But did they need to be like that? It reminds me a little of Piper Harron’s amazing and discipline-busting approach to writing mathematics that she assumed for her doctorate. I’d like to see some examples of folks doing this for mathematical illustration — if you know of any, let me know.
The content of this post formed the basis for an invited talk at “Illustrating Geometry and Topology”, a workshop at the Institute of Computational and Experiment Research in Mathematics (ICERM) at Brown University, September 2019.
