At the Mathematical Sciences Research Institute in Berkely, CA with the Klein quartic sculpture. Photo by Charles Camacho

Charles Camacho, a sixth-year PhD student in the Department of Mathematics at Oregon State University, spends a lot of time thinking about shapes. He describes his research as such: “I study the symmetries of abstract mathematical surfaces made from gluing triangles together.”

Charles explaining his thesis research at the Latinx in the Mathematical Sciences conference at UCLA. Photo by Farida Saleh from the Daily Bruin.

Charles works in a branch of mathematics called topology. Topologists think about shapes and surfaces. There’s a joke among mathematicians that a topologist is someone who can’t tell the difference between a coffee cup and a donut, and there’s some truth to that. It’s not that they can’t see a difference, but that they look past the difference to see the core similarity: both are solid objects punctured with a single hole. Topology as a formal area of mathematics is fairly recent (early 20th century). Topology’s roots go much further back, though, through the streets of Königsberg in the 1700s and to the geometry of the ancient Greeks.

Königsberg bridge problem
There’s a famous puzzle that originated in  Königsberg, Prussia in the 1700s (Königsberg is now Kaliningrad, Russia). The puzzle didn’t originate among mathematicians—but my understanding is that it’s mainly mathematicians that think about the puzzle now. Back then, there were seven bridges crossing the river Preger.

The Bridges of Königsberg (illustration by Leonard Euler, 1736).

The puzzle is this: Is it possible to cross each one of the seven bridges exactly once? (Go on, try it!) In his description of the problem and its solution, Euler said “it neither required the determination of quantities, nor did calculation with quantities help towards its solution.” He was interested in solving this superficially trivial problem because he couldn’t see a way for algebra, counting, or geometry to solve it. This goes against most people’s conception of mathematics—can it really be a math problem if you don’t fill a chalkboard with calculations?

The fact that no one yet had found a way to cross all the bridges without a repeat did not prove that it could not be done. To do that, and thus solve the problem for good, Euler had the insight to try and reduce the problem to its core. Reframing the Königsberg Bridges problem (elements of image from Wikimedia Commons, composited graphic by Daniel Watkins)
Knowing the layout of the city and all of its streets is irrelevant, so we can simplify to a map of just bridges. But even knowing that there is a river and land doesn’t really matter. All we really need is to know is represented in the network on the right (what mathematicians today call a graph). Euler’s solution was this: “If there are more than two regions with an odd number of bridges leading into them, then it can safely be stated that there is no such crossing.” It didn’t matter where the bridges were, it just mattered how many of the possible paths led to each landmass.

With collaborators at a summer research workshop on graph theory. Photo copyright American Mathematical Society

Being a mathematician, Euler wasn’t satisfied just stating a solution to the Königsberg problem. He went further, and generalized: he came up with rules and a solution that would work for any city with any number of bridges. All you have to do is look at the crossings, and note whether there’s an odd number of ways to get there, or an even number of ways. Euler’s method was developed by later mathematicians into graph theory, a branch of mathematics focusing on sets of points and the paths connecting them. Graph theory has a reputation for having many problems that are simple to state, but incredibly difficult to solve conclusively. In this sense, graph theory has a lot in common with geometric toy blocks.

Platonic solids
Charles has a set of magnetic toys in familiar shapes: triangles, squares, pentagons. These shapes are known as regular polygons, which just means that they are shapes composed of straight lines, each of which has the same length. Playing with these, one can hardly help but to arrange them into three-dimensional shapes. Playing with the triangles, you can quickly form a triangular pyramid: a tetrahedron. With six squares, a cube. With eight triangles, an octahedron. And with twelve pentagons, a dodecahedron. Surprisingly, there are only five shapes that can be made this way! Why is this the case? And must this always be the case?

The Platonic Solids: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron. Image copyright Daniel Watkins.

You might notice some other interesting things about these shapes. If you turn a cube while holding the middle of a side, you will see that it looks the same after each turn. It has rotational symmetry. Each of these shapes has multiple axis of symmetry. They can be rotated holding them in different ways and still show symmetry.

As a mathematician, Charles thinks about ways to generalize these ideas. We know that the five Platonic shapes are the only solids that can be formed from regular polygons, but what shapes could be formed if you used slightly different definitions? What if, for example, you used arcs of a circle to form the lines? What can we say about different kinds of surfaces? These shapes are defined on flat planes, like a piece of paper, but we know of lots of other surfaces, like the world we live on, that aren’t perfectly flat.  What kind of symmetry do polygons in these geometries show? “Specifically, I wanted to know all the ways that such surfaces can be rotated a given number of times. I generalized previous research on counting symmetries and discovered a formula describing the number of these rotational symmetries,” Charles said.

A topological representation of a four-holed surface with a twelve-fold rotational symmetry (blue arrows indicate which edges are to be glued to make the surface. Graphic copyright Charles Camacho

Tune in to KBVR Corvallis 88.7 FM on Sunday March, 10 at 7 PM to hear more about Charles’s research, his inspirations, and his path to research in mathematics. Stream the show live or catch this episode as a podcast.