‘The Library is the Laboratory’

In the late 19th and early 20th centuries, mathematicians realized they had a problem: Their expanding field of study was quickly evolving from the concrete to the theoretical.

Imagine, if you will, the arc of evolution from the abacus to the abstract, from early scratchings in cuneiform to inscrutable scribblings running across blackboards.

How to respond to this dilemma?

Here’s a thought: Could visual examples help students better comprehend these ideas? Why not create three-dimensional models to demonstrate various principles across a wide range of mathematical concepts?

And so, these problem solvers went to work with string and plaster, glass and beads, cardboard and aluminum, fashioning renditions so beautiful that the artist Man Ray looked to them as inspiration for his work.

Nearly 400 of these stunning models have been the prized possession of the University of Illinois Department of Mathematics since the turn of the century—counted as one of the largest such collections in the world. Some models were purchased from the German company Martin Schilling, others made by U. of I. math professors. Over the decades, students had been encouraged to not only observe but physically handle the items—now more than 100 years old—turning and examining them, much like a modern-day Rubik’s Cube, admiring the architecture and the angles to grasp a concept captured concretely.

For them, and for many, seeing is believing.

What’s old is new again
A century later, as part of a class, young mathematicians on the Illinois campus brought the models back to life.

Having been carefully stashed away during the Altgeld Hall renovation, the models became the star exhibit in a research endeavor open to undergraduates. Working under the auspices of the Department of Mathematics and the Illinois Mathematics Lab (IML), three sophomores and a junior spent the fall of 2024 learning how to suss out equations, conduct research, and present their findings in an academic paper, with the contents of the Mathematics Library proving an essential element of their work.

“Some mathematicians are really good at solving the problem,” but lack writing experience to explain concepts, said Sarah Park, head of the Mathematics Library. “So [this endeavor] is about encouraging them to engage in this research process.”

One such sophomore was Darya Reutenko, who had specifically chosen Illinois because of its research opportunities, as had fellow sophomore Eva White. Joined by undergraduates Dadi Feng and Cathy Guo, the foursome spent a semester on the project, guided by Park and IML Director Karen Mortensen, analyzing and documenting the unique collection, as well as writing a historical overview.

“I really like being able to see math or being able to visualize it better,” Reutenko said. “This was a project that allowed for making a representation of something that is also difficult to imagine in your head, but it’s also at the same time possible if you make a model of it.”

Each student took on 10 models from the collection, working collaboratively and separately to divine the models’ meaning and provide a description. While some models had equations attached to them, others did not, and the students had to puzzle out what concept the physical presentation portrayed.

Students also consulted research materials, such as catalogs and historical literature, to assist them in their work. As an added asset, White used her fluency in German to better comprehend the texts that accompanied the models from that nation.

“It gives a really interesting insight into the way that people thought about math before we had computer modeling,” White said of the models’ origin.

“It’s difficult for us to imagine that people didn’t have that kind of thing, and people really just had to think about these equations and try to determine what they would look like in their own heads.

“That’s inspiring to me.”

The historical aspect also appealed to others in the group. “It’s just really cool to study something from the past that not many people had studied,” Reutenko said, “and to bring it back out and analyze it and then have future generations be able to study from it as well.”

White appreciated that the project would continue to help make mathematics more accessible. “A visual stimulus should help people become more interested in math,” she said. “It’s not just equations and problem solving—[it] can also be turned into something really beautiful, like an entire collection of models.”

Park, the librarian, echoed those sentiments. She viewed the project as recognizing the efforts of former members of the math department who realized the value of the models and collected them in the first place. “We feel like we have to celebrate their efforts,” she said, and noted the long-held tradition of using the models as teaching tools, still effective all these years later.

Students modernized the models collection by using Mathematica software to create 3D digital images. These online renderings, as well as 3D models constructed by another cohort of students a semester later, bring the models to an even wider audience.

Nevertheless, the old-school physical renditions continue to impress.

“Actually having the physical copy there is so different,” White said. “Because looking at it online, no matter how perfect the reproduction is, will never really give you the same kind of thrill as having the model there” in front of you.

As Park put it, “Connecting the math and the beauty [of the models] together is just fascinating.”

Rewarding efforts
As the students labored at their task, their efforts were not unnoticed. In spring, their work received the first Nancy D. Anderson Mathematics Research
Award.

Anderson, who headed the Mathematics Library at Illinois from 1972 to 2000, was highly regarded for her work in the field of mathematics libraries, as well as for her success in elevating Illinois’ collection to worldwide acclaim.

The monetary award, which is funded by the Nancy D. Anderson Mathematics Library Endowment Fund, evaluated three areas: what students learned, presentation of their findings, and use of library resources. The students’ project was notable, judges said, for its extensive use of the Mathematics Library collection.

And who knows that better than Park? “The library is the laboratory for mathematics,” she said, referring to the accumulation of mathematics information from centuries past to today’s discoveries.

And while the project provided a valuable addition to library resources, it was personally rewarding to its participants as well.

“The experience was more than an academic project,” Reutenko wrote in a LinkedIn post. “It was a turning point that deepened my passion for mathematics and research. I discovered the value of applying mathematical theory to open-ended, realworld problems, where solutions are not predefined. It highlighted the importance of creativity, persistence, and interdisciplinary collaboration in mathematical inquiry.”

An outcome, perhaps that may just be what the original model-makers had in mind.