By Albina Jambulatova Muratovna
Mathematics often leads us on a voyage into the unknown, where seemingly abstract ideas turn out to be glimpses into the underlying principles of our world. For example, the pine cone is a natural wonder that exhibits a mesmerizing spiral pattern. This spiral arrangement is a striking example of the Fibonacci sequence’s manifestation in the natural world. Professor Jake Levinson from the University of Montreal recently gave a talk in which he examined the enigmatic world of curves and polyhedra. Just like the pinecone example, there are hidden mathematical rules that underlie this natural object’s form. So, while curves and polyhedra might sound like subjects best left to the mathematical classroom, they actually have the potential to substantially influence how we see the world.
At the heart of the lecture was the question: what does it mean for a surface to be curved? It’s easy to dismiss this question as purely theoretical, but Levinson brought it to life with a simple example. Triangles superimposed on a surface can provide us with a key insight. If the sum of their internal angles is 180 degrees, the surface is flat. However, venture onto the surface of a sphere, and the sum exceeds 180 degrees. For example, on a sphere, consider a path from the north pole to the equator, following a quarter turn along the equator, then back to the north pole. This gives a triangle with three 90-degree angles, for a sum of 270 degrees. Therefore, a fairly straightforward calculation can tell us a surprising amount about curvature.
Levinson then introduced us to polyhedra — solid shapes with polygonal faces (i.e., two-dimensional geometric figures that have a finite number of sides), like cubes and octahedrons (polyhedrons with eight faces). Simply put, we can connect flat faces to create different 3D shapes. The challenge lies in defining curvature for such polyhedra, especially at their corners. Curvature represents the degree to which a curve differs from a straight line or a surface differs from a flat plane. It’s not as simple as measuring the angles between the faces because as we rotate a square, the bend in the path between faces doesn’t count as a corner. It’s in the vertices where curvature happens.
To examine this predicament, on a polyhedron, as soon as a triangle encloses a vertex, the triangle’s angles might no longer add up to 180 degrees. Levinson then presented a theorem: every such triangle’s angles will sum to the same value — a constant that depends on the enclosed vertex, but which may not be 180 degrees. This led us to the notion of curvature as a mathematical quantity, denoted as K. The curvature of a polyhedron at a point is defined as K(P, p) = 360 – (sum of the angles directly adjacent to p) where K is the function for curvature of the polyhedron P at the point p. When the curvature is zero, it means the point is not a corner. Meanwhile, negative curvature hints at the hyperbolic surface, which “curves away from itself at every point” (source link).
Connecting Topology to Curvature
The lecture then ventured into the intriguing world of topology, which explores properties preserved under continuous deformation, or distortion. Here is where the relationship with curvature becomes even more fascinating. When we tried to transport a spherical triangulation onto a torus, which is doughnut-shaped, it turned out to be impossible. Ultimately, this can be explained by the fact that the Euler characteristic of a sphere (two) is different from that of a torus (zero), where the Euler characteristic is a number describing the topological space’s shape according to its vertices, edges, and faces.
Note: The College will offer its next topology course in Spring 2025, which has been listed in the past as MATH-455, so keep an eye out!
On the whole, in Levinson’s lecture, we found more than just math; we uncovered a narrative that brings even the most abstract concepts to life. Curvature and polyhedra, seemingly confined to the world of mathematicians, actually offer profound insights into our reality and the hidden properties that govern it.
To reinforce some of the concepts discussed here, check out this article from mathematician and tutor Nicolas James Marks Ford.