The language and art of maths

For most people, learning their ABCs is much easier than learning their 123s. From when we are very young our parents spend more time talking to us than playing with numbers, making language much more familiar than maths. And it makes sense in evolutionary terms: early humans needed language as a social skill to get along with others in the ‘tribe’, while mathematics is a more recent, less natural skill.

But mathematicians will tell you that maths is also a language, having its own rules and applications, with some educators believing that if we taught maths more creatively, as we do with languages, we’d have much more success in engaging students, a critical factor in light of declining maths class sizes and therefore interest, in Australia.

University of Melbourne mathematician Dr Henry Segerman says all language is about communication, requiring the right terminology and grammar for a person to be able to express themselves and be understood. Maths is just the same. “You can’t be an engaged citizen without mathematics. You need it to understand a story in the newspaper about the economy, or to manage your finances, and you need basic statistics to understand risk and probability, of medical procedures for example,” says Dr Segerman.

“Students should be given time to understand the language of mathematics and to creatively apply it to solve problems. Often, maths classes never go beyond rote exercises, in contrast with an English class in which we would use the language to create a new idea, a story or poem.”

Dr Segerman notes a famous essay on the subject, A mathematician’s lament by Paul Lockhart, a teacher at Saint Ann’s School in Brooklyn, New York, who says students are missing out on the creative process of maths and only seeing the results of the process, such as an equation to apply like “The area of a triangle is equal to one-half its base times its height”.

The essay goes on to say that mathematics is the art of explanation. And that if you deny students the opportunity to engage in this activity – to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to be inspired, and cobble together their own explanations and proofs – you deny them mathematics itself. The art is not only in the “truth” but also in the explanation, the argument.

Taking the beauty of maths one step further, Dr Segerman has found a way to express mathematics in his 3D sculpture art. “The language of mathematics is often less accessible than the language of art, but I can try to translate from one to the other, producing a picture or sculpture that expresses a mathematical idea,” he says. Dr Segerman notes that one of Melbourne’s most famous landmarks is an example of mathematical art. The ‘pinwheel tiling’ covering Fed Square is based on the concept of five identical triangles fitting together to make a larger version of the same triangle. ”The mathematical beauty of the design is that the tiles appear at an infinite number of angles as the pattern extends outwards. This was the first known tiling pattern with this property.” Dr Segerman’s own art plays with language and 3D shapes, stemming from his work as a researcher in topology, the mathematical study of the properties of shapes that are not affected by stretching or deforming. For example, in topology, a doughnut and a coffee cup with a handle are equivalent shapes, because each has a single hole and one can be deformed into the other.

Another mathematical influence appearing in Dr Segerman’s art work is the idea of “self-reference”, that an object can describe itself.

“The concept comes up in mathematical logic, in Gödel’s famous incompleteness theorem,” he says. “An example is my work ‘Sphere autoglyph’.”

“An autoglyph is a word written or represented in such a way that it is described by the word itself. In this 3D printed sculpture, 20 copies of the word “SPHERE” fit together to form the surface of a sphere.

 ”The challenge in creating this Escher-like design was to work out how to fit the shape of the English word SPHERE together with copies of itself.

“This is both an artistic/typographical problem which asks: ‘How can the shapes of the letters be deformed to fit together but still be legible’? And a mathematical problem: ‘What are the possible ways of making a symmetrical pattern on a sphere?’.”

Explore Dr Segerman’s mathematical art:
http://www.segerman.org