# Icons for mathematical concepts

With the advent of mobile devices and small screens, symbols are very commonly used for abstract concepts. As I type this, the MathOverflow editor has symbols for bold font, links, code, lists etc. We are all also familiar with the common images used for bluetooth, wifi, and battery level indicators. Even my bank app uses symbols on buttons to represent concepts, such as transactions and savings.

We mathematicians have a long history of using symbols for abstract concepts, such as infinity, $\infty$ the integers, $\mathbb{Z}$, summation and integration, which are internationally accepted.

It is not that difficult to come up with symbols for subfields of mathematics, such as calculus $\int$, logic (wikipedia uses $\forall$), programming language concepts $\lambda$, and so on. Many more examples can be found on wikipedia. Even arxiv uses icons for platforms such as twitter, email, reddit, and mendeley.

However, there are some very common concepts that I have been thinking about for a while, which I would like to have good symbols for (immediately recognizable, or at least make sense), for concepts such as definition, conjecture (wikipedia uses a question-mark), theorem, example and proof.

This would be useful when designing web pages or math-related apps, that deals with research-level mathematics.

I have been looking around a bit, but it is not that easy to come up with interesting results.

I am perhaps asking several related questions:

• What would be good symbols to use for definition theorem, proof and all our favorite LaTeX environments?

• What are good examples of usage of icons for mathematical concepts? This is specifically for interfaces in books, text, and mainly online. Bourbaki's dangerous bend symbol comes to mind as an example of such a symbol for a specific concept.

• What should one for sure avoid / what would be bad usage of icons?

More motivation: Personally, I am working on an overview of symmetric functions, and I have been thinking hard on what type of symbol to use to represent the concepts "symmetric function", "quasisymmetric function", the concept "is a basis" etc, but I feel that is too specialized. One might wonder about the motivation, but when the list of functions grows to over 70 such functions (and I have not even started on the shifted symmetric functions, and the quasi-symmetric ones yet), it would be helpful to have some icons (with alt-text) rather than text to indicate properties. Also, it just looks great if done correctly.

• +1 I think this is a good question to ask, and I thought you might want to look up places like The Noun Project to see what exists. Perhaps we can get some answers from people who have worked on applications in this area :) – Michael Lai Apr 13 at 20:59
• @MichaelLai That was a nice suggestion! The symbol for Pythagoran Theorem is quite nifty, and almost universally recognizable. – Per Alexandersson Apr 13 at 21:23
• I took a look at APL, Coq, Matlab, Maple and wxMaxima, but sadly I found no relevant icons. – joeytwiddle Apr 15 at 9:31
• For a symmetric function I suggest ⇄ or an f with a mirrored f sitting right next to it. Proofs often use ∴ or ⇒ for each step, and end in a ⯀ square. So a proof could be an arrow into a square, or ☑ a square with a tick, as opposed to a partial square with a question-mark for a conjecture. Some words have a single symbol in Chinese or Kanji, although not immediately recognisable to all. Brainstorm a bunch of symbols then test! – joeytwiddle Apr 15 at 9:42