Using spectral graph theory, mathematicians have solved a decades-old problem.
If you’ve ever drooled over a pie, cut into eight beautifully equal slices, you’re already a little bit familiar with the concept of equiangular lines—those that intersect at a single point, with each pair of lines forming the same angle. (But shoutout to those of you who always cut crooked slices of pie—we can’t all be naturals at spatial relations.)
Researchers at the Massachusetts Institute of Technology (MIT) have solved a geometry problem relating to these equiangular lines that’s more than 50 years old. The problem explores how to divide n-dimensional spaces into, well, theoretically equal “slices” of pie. The new work appears in the journal Annals of Mathematics.
Because a circle is 360 degrees, that means a pie cut into equal sixths will yield pieces that are each 60 degrees. If you cut it into eighths, that yields pieces that are 45 degrees. The lines that divide the pie are equiangular, meaning the angles between all the different lines are the same.
In this case, the math is more straightforward than cutting a pie—at least at first—in two dimensions. When you’re using a compass and straight-edge, let alone if you’re allowed the luxury of a protractor, it’s easy to draw a circle on a piece of paper and then divide it into pieces, whether eighths or sixths or thirds or whatever you want.
But what about three dimensions? Things get weirder pretty quickly. To take our simple pie cut into six slices and study its equiangular lines in three dimensions, we must turn to the icosahedron, memorialized to nerds around the world as the D20. It’s the 3D polyhedron with 20 triangular sides.
“You want to place as many lines as possible in three-dimensional space so that the lines are pairwise separated by the same angle. It turns out the best way to do that is with a regular icosahedron, which has 12 vertices, so that you can pair them together to form six diagonal lines which are separated by the same angle,” researcher Zilin Jiang, now a professor at Arizona State University, says in a statement.
Once you start to think about the three dimensional shapes in question, it’s easy to imagine, say, a starburst that has different numbers of rays representing the equiangular lines arranged around one central point. But, and we’re sorry to keep doing this, it’s about to get even messier. What happens when there are more than three dimensions?
To solve all the higher dimensional cases of equiangular lines, the researchers used something called spectral graph theory. While graph theory generalizes and studies the way nodes and edges interact in order to form mathematical units, spectral graph theory turns those graphs into matrices (grids) of eigenvalues (directional vector values). Nodes are the circles in the Tinkertoy set, while edges are the sticks.
That means an n-dimensional arrangement of equiangular lines from a fixed point can be generalized as a series of vectors that radiate from one common node. Turning the information into matrices allows mathematicians to do operations that are mind-bending when projected into four dimensional space or otherwise in the mind.
This work took years to do. “I can tell you at the beginning, we were a little bit stuck. We made some partial progress, but I guess by hitting those roadblocks we just learned a lot about what we needed at the end. And that was [a] great experience, because at least for me personally, I feel like doing research is also about the experience,” Jiang says. (And in mathematics, even failures can help move the entire field forward by ruling out approaches or ideas.)
Moreover, he was influenced by a favorite of ours here at Popular Mechanics: Po-Shen Loh, a mathematician at Carnegie Mellon University who published a new way to solve quadratic equations in 2019. “While listening to his teaching, you could feel that he just loves the subject, and it was entirely an inspirational experience,” Jiang says.
Jiang hopes to inspire the next generation. “They can break barriers,” he says.