By Tim Chartier

Within the intersection of mathematics and art lie beautiful creations to delight mathematicians and artists alike.  Mathematics may be the muse that inspires art or serves a more fundamental role alongside a brush, lathe, or loom.  To gain greater context on this field, we interview Dr. Edmund Harriss, a mathematician, artist, and assistant professor at the University of Arkansas. He discovered the Harriss spiral and created the construction toy Curvahedra. He is the coauthor of Hello Numbers, What Can You Do? and the coauthor and illustrator of two mathematical coloring books: Patterns of the Universe and Visions of the Universe.

Tim Chartier: You don’t just produce mathematics or art, you hold a position at the intersection of mathematics and art within your university.  Can you describe your position?

Edmund Harriss: In my own work I do not see a strong distinction between the mathematics and the art; they very much work hand in hand.  The artistic thinking feeds the mathematical enquiry, and the mathematical ideas support the artistic investigation. The goal of the position is to highlight those connections at an institutional level. On the art side we are working to ask how to bring back an art and artisan model of technology that is more curiosity driven. With the role that computers can play as tools (from digital manufacturing, through virtual reality to AI), mathematics opens possibilities for interesting exploration different from the usual mathematical approach of creation through engineering.

On the mathematics side, we investigate how art considers and critiques how images, objects and experiences are perceived. This gives a framework to ask how to reverse classical applied mathematics. Instead of asking and testing how we can understand the physical world mathematically, we can ask how we can understand mathematical ideas through physical experiences. This requires us to consider how the physical model is perceived and how that might mislead us when translated into the mathematical world.

Tim Chartier: Can you give us some examples of your work?  

Edmund Harriss: I have been fortunate enough to work on many different sorts of projects. One of the most surprising was to make two coloring books of mathematical images, created with Alex Bellos. These have sold over 100,000 copies worldwide, revealing the beauty of mathematics to people who may not think to engage with math otherwise. Through those I was also able to create a counting book Hello Numbers, What can you do? that seeks to give a sense of the mystery, play, and exploration possible in mathematics from the first learning experience. This does mean that it can be a little challenging to parents reading it, as the concepts are designed to be more open ended (even though the book only goes to 5). Reviewers like to point out that it purports to have run out of pages and then goes on for several more, but those final pages are so important.  They are the invitation to play and explore for yourself following the ideas that start in the book. 

A more formal piece as an artist is my recent sculpture (installed fall 2021). This began with a course in our honors college “Place in Mind.” This was a perceptualist approach to thinking about a space on campus, the courtyard outside our honors college.  I co-taught the course with my friend and colleague Carl Smith, who is a landscape architect. We asked the students to explore and analyze the courtyard in a variety of ways inspired from mathematics and drawing. From that they were able to form opinions on the courtyard and understand its failings as a place. Proposals to change the courtyard came out of that investigation, including a sculpture based on Curvahedra, a toy I created that explores geometry and the curvature of surfaces. David Gearhart (a former chancellor) and his wife Jane gave a donation to turn this into reality.  However, this came with a challenge: how to make a bent metal sculpture on a generous, but not vast budget.

I was able to bring in Emily Baker who is an architect and metal worker (also at the University of Arkansas). We combined mathematical ideas of framing curves together with plasma cutting and welding to create a system we called “Zipform” that can turn flat cut surfaces into curved beams. That technology has now been used on several more projects including developing techniques to create optimized concrete beams in the global south.

Pleasingly, when the sculpture was installed, it immediately became a place for students to read and study.  Even the Honors Dean uses the space for her office hours. This directly addressed the issues with the space that the students had identified.

Tim Chartier: Can you give us examples of how you use mathematics and art in your teaching?

Edmund Harriss: Teaching provides interesting challenges as both math students and art students often (though certainly not exclusively) made their choice of field in partial opposition to the other subject. So making math majors create images or making art majors do algebra can have its challenges. On the other hand, I think both work well to challenge and develop student experiences. Having an artistic space in a mathematics course, even something as simple as making a drawing using ruler and compass, gives students the ability to feel ownership of the mathematical ideas. In an art course a more mathematical approach, especially the way concepts, procedures and software can interact, can help students see how to abstract the process or creation and find new ways to explore ideas.

Tim Chartier: In 2015, The Guardian reported on some of your work, a beautiful new curve inspired by the golden ratio, aptly named the Harriss spiral (seen in Figure 1).  What is the curve?  How is it made? And, how did it come about?

Edmund Harriss: There were several motivations behind the spiral. The first was an interest in the spiral patterns of both Celtic and Islamic art, and a desire to create a similar effect with as few rules as possible. The second was a desire to have something to show people who had some interest in the golden ratio that would lead them deeper into the world of algebraic numbers. The spiral achieves both of these goals.  It is a set of nested spirals similar to the inspiration images built instead of the “plastic ratio” that is the real root of x^3-x-1=0. It is constructed by taking a rectangle with height 1 and length the real root of x^3-x-1=0. With this rectangle you can cut off a similar rectangle, and then a square to get another similar rectangle. You can repeat this construction on the smaller similar rectangles, in each case getting a square and two more similar rectangles. (This process is seen in Figure 2.) Now simply adding an arc of a circle to each square (in the right way) gives the spiral, or more correctly the nest of spirals. (This is seen in Figure 3.)

The connection between the motivation and the actual work was surprisingly tenuous, however. It felt more like I had the idea, began exploring other things, and those things managed to link back to the goal. This seems to happen often in my work.  I have ideas that I don’t pursue directly. Instead, I explore things rather broadly and see what might be possible. In other words, I try to balance being open to new ideas while also having a goal that I’m working toward. 

I have been very fortunate in my successes, but it did not come without hard work. When we only talk about the successes, it can seem like good fortune.  However, there have been many other occasions where things I have put a lot of effort into have not come together. That can be dispiriting, especially when there are many people, both mathematicians and artists, who are skeptical about a combination of mathematics and art. The interesting thing is that sometimes it is precisely those failures that allow me to see or take advantage of a later opportunity.

Tim Chartier: The Harriss Spiral is named after you, of course. It’s essentially seen as yours. What are your thoughts on this?

Edmund Harriss: That is an interesting example of whether mathematics is created or discovered. Many times when I am creating work using algorithmic methods the results outshine the amount of design that goes into them. It thus feels a little wrong to be credited in that way. I think in general the topic of naming mathematical objects after people has its problems. There are the obvious ones that the names do not necessarily credit the right person or that the names we use are predominantly European men, but I feel there are issues with the idea of using names at all.

The first is that it helps to strengthen the idea that mathematics is the work of singular geniuses, as opposed to communities working together. This makes the creation of mathematics seem more alien and special which I think can discourage many from engaging in it. In addition, the names can be rather confusing. “Mean curvature” gives a clue as to what the concept might be, whereas “Gaussian curvature” is only informative if you know about Gauss’s work. In the case of Gauss, though, Gauss’s breadth of work makes the connection very unclear!

I am very proud of the work creating the “smallest PV spiral,” as I had named it, where PV is short for Pisot-Vijayaraghavan, two other names that do not give a clue to the meaning. I do feel honored that my name was attached, but also a little conflicted!

Tim Chartier: If not with names how might credit be given for mathematical objects and art?

Edmund Harriss: Although naming does over-credit individuals it is also one of the few ways that we acknowledge the human creation of mathematics. I think it would be better to tell both a little more of the history of a mathematical object and identify the people involved. This can help to bring the broader cultural aspects of mathematical development into greater awareness.

One area where naming conventions definitely need more work is in visualizations of mathematics. Visualizations can be difficult to create, and often people are less likely to  be curious about their origins. For example several of the images I created for my coloring books with Alex Bellos have since become “standards” for their topic. That means that people will use them, citing the creator of a specific version of the image, but not questioning where the idea for the illustration originated. I am not trying to complain here, and certainly the creators of a specific artwork deserve that credit, but I think a more subtle approach to crediting and being aware of the origins of ideas could help us value a wider collection of approaches to mathematics. 

Tim Chartier: For readers who may be new to creating mathematical art or using it in their teaching, what tips, hints and suggestions do you have? 

Edmund Harriss: I think the key is to embrace failure. Try things to see what happens, and then pursue the things that succeed, rather than trying to create reliable systems.

Edmund Harriss is a mathematician, artist, teacher and maker at the university of Arkansas, where he holds the first joint appointment in the Department of Mathematical Sciences and the School of Art.