Quite naively, when I originally decided to to analyse mathematical beauty, my goal was to develop a set of criteria to describe it as objectively-as-possible. Of course I knew then that aesthetic responses are within the eyes of the beholder, but I still believed that it would be possible to at least provide common aesthetic features of mathematical ‘objects’ and concepts. Research into this highlighted common criteria quite quickly (I’d already found a decent set of criteria within 30 minutes), so then my goal changed to dig deeper into the common features of mathematical beauty to find more meaning within each feature, but also between features. I’m going to highlight the common criteria/features/dimensions here using a very helpful ATM article written by Paul Ernest at the University of Exeter called “Mathematics and Beauty”. This will provide the foundations to help build many of the following posts.

Ernest (2015) reminds us initially how beauty in mathematics is perceived. To quote:

“the beauty of mathematics is not a response to something perceived through our sense organs, as with paintings, music or even landscapes… it must be experienced cognitively, through reason, the intellect, intuition, and affect (feelings).”

This is an incredibly important aspect of mathematical beauty which we will come back to in a future post, given that a direct consequence of this results in an amendment to the common phrase mentioned above. The phrase “beauty is in the eyes of the beholder,” transforms into **“beauty is in the mind of the beholder,” from a mathematical standpoint. **Experiencing mathematical beauty therefore necessitates some level of knowledge and understanding… (to be continued in an upcoming post with a discussion on basic and performative aesthetic responses).

After this, he provides a set of seven dimensions of mathematical beauty, noting that are ‘significant differences’ between the opinions of mathematicians. That implies that a mathematical object may only relate to one of these dimensions and be considered more beautiful than another object which relates to three of them, for example. Individual judgement and taste matters immensely, just as it would if you asked a group of people what their favourite food is.

*1. Economy, simplicity, brevity, succinctness, elegance *

The compression of a formula or a theorem of wide generality or an argument (proof) into a few short signs in mathematics is valued and admired.

*Note: In relation to simplicity, many mathematicians would argue that the right balance of simplicity and complexity is key to promote a positive aesthetic response. If something is too simple, it is deemed as obvious and unremarkable. *

*2. Generality, abstraction, power**
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The breadth and scope of a generality or a proof is one of the key characteristics of mathematics and evokes appreciation.

*3. Surprise, ingenuity, cleverness *

Unexpectedness, like wit, is appreciated and valued when it reveals a new knowledge connection, method or short cut in solving a problem.

*Note: Unexpected simplicity in complexity, unexpected connections and unexpected applications all excite positive aesthetic responses. It is has thus become my belief over time that unexpectedness is one of the most important elements of mathematical beauty for the mathematician.*

*4. Pattern, structure, symmetry, regularity, visual design ***
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The discernment of pattern in its various and abstracted forms is the closest the values
of mathematics come to those of art and general aesthetics in the visual eld, although in mathematics these properties are largely abstract. Nevertheless, mathematics is the science *par excellence *for elucidating the meaning of structure and pattern.

*Note: See my previous posts on pattern – specifically posts 2, 3 and 4. Seeking and analysing pattern cold be considered the major pursuit of mathematics, and hence mathematics as a domain could be considered inherently beautiful.*

*5. Logicality, rigour, tight reasoning and deduction, pure thought ***
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The development of logical reasoning to its ultimate forms of rigour and purity of thought is a valued part of mathematics and the steps in a well constructed mathematical proof evoke admiration like a gold necklace with well forged links.

*Note: Look no further than Euclid’s Elements for verification of this dimension. See Michael Enciso’s book, “The True Beauty of Math” for more on logic , reason and mathematical beauty.*

*6. Interconnectedness, links, unification**
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The evidence of connections between different concepts and theories within mathematics is intellectually exciting and attractive. It combines economy, generality, ingenuity and structure and so it could be argued that it is reducible to these first four dimensions of beauty. Or it can be seen as sufficiently valuable in its own right so as to deserve independent listing, as I have done here.

*Note: Gian Carlo Rota famously wrote about mathematical beauty being misunderstood in his popular paper on “The Phenomenology of Mathematical Beauty”. He referred to mathematical beauty as being just a moment of enlightenment, in which the mathematics suddenly becomes well placed within the topic and the mathematical domain generally, and also more deeply understood. There have been a number of rebuttals to this work, but connections/unification/deepened understanding – i.e. enlightenment – nevertheless stands as an important aspect of mathematical beauty.*

*7. Applicability, modelling power, empirical generality ***
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Like metaphors in poetry the capture of empirical situations in mathematical models and more generally in applied theories and concepts is something appreciated both within and outside mathematics as a demonstration of its power and ‘unreasonable effectiveness’ in the physical world (Wigner 1960), as opposed to the world of pure mathematics.

*Note: Paul Ernest rightly points out that Hardy did not include applicability in his famous set of criteria on the beauty of a mathematical proof, mostly because of his bias towards pure mathematics. I will refer to Cedric Villaini’s thoughts on this in a future post. *

So there you have it – a relatively strong set of criteria for mathematical beauty. There are many others who have attempted to develop criteria, such as Doris Schattschneider’s beauty criteria for mathematical proofs, but I am happy that Paul Ernest’s dimensions are foundational enough to help us springboard into deeper and more complex discussions on mathematical beauty.

The key for public understanding is to somehow explain all of this with appropriate examples to non-mathematicians. That will come soon enough…