The Reluctant Mathematician

I'll admit that looking at the Bridges 2019 gallery initially filled me with a bit of panic. The incredible digital art pieces drew me in, but I thought: I don't know how to recreate those. And when I looked at the mathematics behind them, some of it was beyond my current understanding. So I took a break, came back with fresh eyes, and after deliberating between a couple of pieces, I decided to recreate 'Sum of Odd Integers' by Elaine Krajenke Ellison. She was inspired by a book by Roger B. Nelson called 'Proofs Without Words: Exercises in Visual Thinking,' and this took me back to my Bachelor of Arts days when something called the Zen Proof resonated with me. The Zen Proof was essentially a visual way to prove the Pythagorean theorem—probably the first time in my life that I realized math was more than formulas on a page. I was also reminded that my BA in Liberal Studies was a weaving of all great ideas from ancient to modern times and that the ideas included all disciplines.

"Sum of Odd Integers" by Elaine Krajenke Ellison (Bridges 2019)

My recreation using scrapbook circles and hand-colored quadrants.


In deciding to recreate the quilt, I realized I had scrapbook paper to make circles I could cut out, a ruler, and some pens. What I didn't realize is how long it takes to color something with said pens, nor did I ever figure out how to recreate the border of the quilt. However, the amount of interesting mathematics I found in doing and creating was worth all the time it took. The first joyful thing was that it was absolutely relaxing. Then the rest of the excitement came from all the mathematical ideas that I found within this piece of art—and I'm saying this with a bit of a giggle in my voice because I swear I am not a mathematician. (I'm not sure I can keep saying that).




Beginning the construction: I started with the four red circles in the center, working from the corners inward.


The first thing that struck me when I looked at the artwork was its really interesting rotational symmetry. I thought: why is this artist creating these 'layer cake' quadrants, where there's 1, 3, 5, and 7, and coloring each one a different color—unless it's to point something out? As I was building, I clearly saw the rotational symmetry with each of those quadrants. I decided to work from the outside corners toward the center, thinking the center would make more sense at the end. And it did—the four-fold symmetry became strikingly clear through the act of creating it.



The four-quadrant structure revealing itself as I built outward from the corners.

The other thing that struck me was how the use of the L-shapes (called gnomons, I learned) really demonstrated how you could go from taking 1, adding 3 to get the perfect square 2×2, then adding 5 and getting another perfect square that's 3×3, then adding 7 and getting a 4×4, and so on. I could see it coming out either from each corner or happening from the center moving outwards. The pattern was there in multiple ways simultaneously.



The L-shaped gnomons became clear as the pattern filled in—each adding the next odd number (3, 5, 7, 9) to create perfect squares.


Then the next thing I noticed was that every time you added 1+3, you'd get your 4—your square made of four smaller squares. Then when you took 3 and added it to 2, you'd get your 5 (the next L-shape). I realized: these are consecutive numbers! 1+2=3, 2+3=5, 3+4=7, 4+5=9. Each L-shape is made of two consecutive numbers—the bottom row and the side without counting the corner twice. Since consecutive numbers are always one even and one odd, they always sum to an odd number. That's WHY the L-shapes are always odd numbers, and that's WHY consecutive odd numbers sum to perfect squares. The visual proof made the abstract relationship concrete in a way that formulas never had.

This was a surprisingly rich mathematical exercise for me. Like my experience at the duck pond last week where angles came alive through observation, this artwork came alive through creation. The act of building—layering colored circles, deciding where each piece went, physically constructing the pattern—forced me to engage with the mathematics in ways that passive observation never could. I found myself moving from intimidation to genuine mathematical discovery, seeing patterns I wouldn't have noticed otherwise: the rotational symmetry, the L-shapes building outward, the consecutive number relationship hiding in plain sight.


This experience makes me wonder about embodied mathematical learning more broadly. I’m looking forward to reading the experiences others had during our recreations and finding out what mathematical ideas were revealed. In thinking about our students—what shifts when we ask them to build, create, and physically construct mathematical concepts rather than just calculate them on paper?