Every Museum Is a Museum of Symmetries

Editor’s note: Laurence Kirby was among the first cohort of the The Baruch College-Rubin Museum Project, the brain-child of Stan Altman, Professor in the Baruch School of Public Affairs. The BC/RM Project was established on the principle that exposure to and participation in the arts enriches students’ college experiences and greatly enhances their abilities to learn, understand, and function across a range of critical literacies. Information on the project and subsequent conference can be found here. This post is the first in a short series that explores the use of visual culture in disciplines other than art history.

Teaching mathematics courses for liberal arts students at a large, diverse, urban university, I have found a need to engage and motivate students. One successful approach has been to incorporate museum visits into the course. The mathematics of symmetry provides an interesting and enlivening bridge between mathematics and art (or art history), subjects which on the face of it seem so far apart. I wonder whether art history teachers could not also take advantage of this bridge to enrich their courses, especially those for a similar population of non-specialist students. It might seem far from obvious that math could enliven an art course but, as we shall see, the mathematics of symmetry can lead to a whole new way of looking at works encountered in museums, or indeed at streetscapes.

My adventure in this way of integrating the arts into math courses came about through an institutional collaboration between Baruch College and the Rubin Museum of Art (rmanyc.org). The Rubin, a lovely museum of Himalayan art, welcomes class visits and interdisciplinary explorations of this sort, and the Himalayan mandala provides a particularly rich and complex exploration of symmetries.

First we spend a couple of class meetings on the basics of the mathematics of symmetry, including an analysis of the seven types of border pattern, which as this illustration suggests are to be found in almost any culture:

7 border pattern symmetry types

Then I give my students a mini-tour of the museum, looking in the light of our knowledge at artworks such as the Rubin’s Mandala of Krishna Yamari:

Mandala of Krishna Yamari

Mandala of Krishna Yamari, Nepal, 1400-1499

The mathematical concept of symmetry extends beyond our everyday parlance which refers to mirror-image symmetries or more generally to balance or elegance. Mathematically, there are many other kinds of symmetries such as the rotational symmetries manifested in this mandala: 4-fold rotations as well as reflections leave this mandala unchanged, in its basic form at least, and that’s what characterizes a symmetry.

The Rubin Museum also provides plenty of border patterns to analyse as this detail of a mandala suggests:


My students’ homework assignment requires a further individual visit to the museum to enjoy the artworks and answer some mathematical questions about certain items on display: questions that necessitate a close-up study of these artworks with their amazingly intricate layers of detail. In fact, I think these assignments are worthwhile, apart from anything else, as an exercise in close looking (a skill that tends to get lost in our fast-moving, image-saturated culture).

In addition, the students experience something of the subtleties of applying an abstract mathematical theory to real-world objects. Which features of the artwork are we considering, and which should we ignore or abstract away, when we classify it?

There are also opportunities for further exploration, via photography, a class blog, and creative responses. And we extend the discussion to talk about symmetry in music: another Baruch College initiative is its Ensemble in Residence, the Alexander String Quartet, and we share a class session with them and learn how practicing musicians see (or hear) symmetry and how it influences their performance.

Significant symmetries are to be found in the architecture of street and campus, in students’ own homes and places of worship. This topic naturally bursts the classroom walls. We took it to the streets of Manhattan, which provide a veritable compendium of architectural influences. Any block will have border patterns such as this one:

Manhattan border pattern

At Baruch College we have one of the most diverse student bodies in the country. So I ask my students for examples of symmetries from their own backgrounds. From some Hindu students, for example, I learned about the rangoli, traditionally made by women:


Every culture delights in symmetries, and any museum is a museum of symmetries and could be a venue for such an exercise. Analysing these symmetries can illuminate not only the mathematics but also the culture, and point up contrasts between cultures – how important and how varied are symmetries in a culture? Which symmetries does it seem to prefer? In which art forms does it explore them?

More than this, as I have experienced for myself and noticed in a significant minority of students, the lens of symmetry can give the mind a whole new way of looking at art –  and at the world. “Now everywhere I go I see symmetry,” one student told me.

I think that we naturally structure the world, and the stories we tell about it and the things we fill it with, through symmetries. And just a little mathematics can go a long way in helping us appreciate this. Many writers have explored the mathematics of symmetry in the arts and in the natural world – one should begin with Hermann Weyl’s classic book “Symmetry” – but it would be interesting to see more development of this from a pedagogical point of view, taking advantage of the way that direct contact with artworks can bring the subject to life.

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