The image is red, gray and black pixels, representing a stereogram ("magic eye") with a hidden Scotty Dog

When Abstract Problems Take Shape

Mathematician Florian Frick looks at problems as shapes, revealing solutions invisible from a single perspective

By Amy Pavlak Laird Email Amy Pavlak Laird

Heidi Opdyke

Magic Eye posters from the 1990s hid three-dimensional images in what looked like a swirl of dots and colors. At first glance, you see only noise. Change your focus, and suddenly a shape appears.听

无码专区 mathematician Florian Frick studies something similar in mathematics. His work explores how problems related to fairness, arrangements and patterns can reveal hidden geometric structure when viewed from the right perspective.听

鈥淢athematics is a kind of mapmaking for abstract worlds,鈥� Frick said. 鈥淚 enjoy the process of taking an abstract problem and translating it into something that I can reason about visually.鈥澨�

Frick, an associate professor of mathematical sciences, works at the intersection of geometry, topology and combinatorics. Geometry is the study of shape. Topology looks at the features of shapes that stay the same when they are stretched or bent. Combinatorics studies discrete structures like arrangements, selections and patterns. By bringing these fields together, Frick turns problems that don鈥檛 seem geometric into questions about spaces, symmetry and overall structure. He describes it as geometry in action.听

This idea is already familiar in everyday language.听

鈥淧eople talk about a 鈥榩olitical landscape鈥� to describe a complicated arrangement of views, alliances and tensions. The phrase suggests that you may want to understand the terrain as a whole, because it may not be enough to look at one position at a time,鈥� Frick said.听

Frick鈥檚 research puts this concept into mathematical terms. For many problems, all possible solutions can be organized into a kind of geometric space. Once that space is built, its shape can reveal information that is not visible from any single point of view.听

The Shape of Fairness听

Consider the problem of fairly dividing rent among roommates. Imagine several roommates are sharing an apartment with different-sized bedrooms. Each room has its own advantages, and each person may value those features differently. The question is: Can the rent be divided among the rooms so that everyone gets a room they prefer without anyone feeling envious?听

At first glance, this sounds like a practical problem. But if you look at it mathematically, it takes on a geometric shape. If a point represents one way of assigning prices to the rooms, in a three-roommate case, the possible rent divisions form a triangle. If there were four roommates, it would form a tetrahedron. More generally, they form a higher-dimensional simplex.听

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Three roommates share an apartment. The total rent is $1,500, and the rooms each have pros and cons. How can they divide the rent fairly among the rooms? Mathematically, each possible rent split is represented as a point in a triangle. The price to achieve rental harmony lies somewhere between the prices at the three corners. The black dot marks the point in the triangle that corresponds to the rent division indicated on the right.

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Instead of checking one rent split at a time, mathematicians study the entire space of possibilities at once. 鈥淎 single proposed rent split may fail. Many proposed splits may fail. But the overall shape of the space of possible splits may force a fair solution to exist somewhere,鈥� Frick said.听

Frick and his collaborators explored a twist on this classical problem: what if the preferences of one roommate are unknown? They showed that a fair rent division exists, even without knowledge of the preferences of one of the roommates. Their proof is constructive, providing a recipe for how to actually compute an envy-free rent division. Frick鈥檚 work was featured in an episode of PBS鈥檚 Infinite Series called Splitting Rent with Triangles.听

Geometry Beyond the Visible听

This type of problem reflects a broader theme in Frick鈥檚 research.听

鈥淲hen I look at a problem, I parameterize the space of all possible solutions. I use the word space because it is a geometric object,鈥� Frick said. 鈥淎nd on this geometric object I use geometric and topological tools to find an actual solution. This is particularly powerful because geometry keeps track of global phenomena.鈥澨�

Numerous problems across mathematics and its applications are global in this sense, ranging from data science to economics. Frick develops topological and geometric methods to tackle problems further afield. He studies questions about how shapes can be divided, how high-dimensional objects must intersect, how discrete structures can be partitioned, and how local constraints can force global behavior. His work connects fair division, combinatorics, convexity, topological methods, and related questions in theoretical computer science. A through line in this work is not that the problems come from physical geometry, but that a geometric viewpoint reveals hidden constraints within them.听

In that sense, geometry acts like a shift in focus. By giving abstract problems a geometric form, Frick鈥檚 work helps uncover the hidden shapes that govern them 鈥� and shows how much those shapes can tell us.