# 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 '19 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 '19 at 21:23
• I took a look at APL, Coq, Matlab, Maple and wxMaxima, but sadly I found no relevant icons. – joeytwiddle Apr 15 '19 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 '19 at 9:42

example of some stuff I worked on a while ago, try going abstract if you can't think of something that would fit your exact needs...

• Could you provide some details about how you end up with the specific iconography and the methodology you used, especially for abstract concepts ? – Brice Nov 27 '19 at 9:49

Having provided a previous comment for this question, I have gone back and looked at some of the more complex mathematical applications (i.e. natural language processing or highly visual graphical interfaces), and interestingly enough some of the more popular programs don't seem to need much in the way of icons. In fact I was quite surprised to see that Wolfram Alpha was actually mostly text based.

As designers we should not be afraid to question conventions, as often we seem to follow standards or guidelines without really understanding the end-users. This is a hypothesis, but given the need for mathematics to be precise, one might assume that the users expect the same degree of clarity if any visual representations of concepts were to be used. That is, they should be unambiguous and universal if they are to be effectively applied. The fact that these don't exist (or at least you are not aware of them) means that either no effort has been spent looking into this area because the need isn't great enough, or that there are still too many design issues that needs to be solved first.

Having said that, I believe there are cases where icons are applicable and should be promoted (ideally combined with text) in contexts with shared audiences, such as:

• Statistics, which is applied in many different disciplines
• 3D modelling and visualization, which seems to be logical because of the visual connection
• Basic arithmetic, such as those seen in calculator apps
• The calculator apps is something I have not thought of, but is clearly an area where icons make sense - indeed, Texas Instruments have some icons they use on the calculators, for different areas of math. – Per Alexandersson Oct 9 '19 at 6:49