Isabella Bautista | November 13, 2022

Imagine you are standing on the surface of a geometrical sphere, much like we do in real life on the planet Earth. Can you tell that you are standing on a sphere? You probably can’t. Think about it: when we go outside and look around, we don’t see the curvature of the Earth’s spherical shape, despite satellites telling us the Earth is spherical. Instead, a relatively flat surface appears to stretch in all directions. 

Therefore, we could say that, from our point of view, the Earth’s surface resembles a coordinate plane, or two-dimensional space. However, since humans also have height in addition to length and width, we can more accurately say that we appear to live in three-dimensional space. In geometry, these three dimensions form the basis for Euclidean space.

Now imagine you are very small and standing at a point on a coordinate plane, in the center of a shaded circle. From where you are standing, you are likely only able to see the immediate space around you. Although you can see shading going out to the horizons of your vision in all directions, you don’t know what the shape you are standing on is because your vision is localized to a limited area.

The figures in these hypothetical scenarios are examples of manifolds. Mathematically, a manifold is a space or shape that locally resembles Euclidean space near each point, but globally may be more complicated. First-year Vanderbilt graduate student Denali Relles explained the concept of a manifold at the Vanderbilt Undergraduate Seminar in Mathematics. In his talk, Relles described the different types of manifolds. For instance, manifolds with boundaries are those that have an edge, such as a sheet of paper. Other manifolds can be constructed by “gluing” two manifolds together along their boundaries—if you were to glue two sheets of paper together at their edge, the resulting figure would also be a manifold.

Some complex examples of manifolds. From top to bottom: the Möbius strip, the Klein bottle, and the real projective plane.

Relles’s talk encompassed manifolds in several dimensions, ranging from zero-dimensional manifolds (a point or a set of points) to three-dimensional manifolds (a sphere). A manifold is called a 2-manifold when its surface appears to look relatively flat, but in reality is not. Thus, a sphere—and therefore the Earth—is a 2-manifold. 

The fact that the Earth is a 2-manifold explains why it can never be represented by a two-dimensional map. Every attempt at a world map that shows all countries and continents is distorted in some way. Most often, far-north regions like Greenland are made to appear greater in size relative to other countries than they actually are. In some world maps, Greenland looks larger than South America and nearly as large as Africa. In reality, Greenland’s area is about 840,000 square miles, drastically smaller than both South America (area ≈ 18,000,000 square miles) and Africa (area ≈ 12,000,000 square miles). These reasons are why the best representation of Earth so far is a three-dimensional globe—it does not force us to attempt to depict a spherical figure two-dimensionally. Manifolds are all around us, and learning about them can help us understand the logic behind the assumptions that we presume to be true, such as the spherical nature of Earth.

To learn more about advanced mathematical topics like manifolds in a casual, easy-to-understand conversation, come check out the Undergraduate Seminar in Mathematics. It is held biweekly on Tuesdays from 5 to 6 p.m. in Stevenson 1320. Whether you are about to complete a degree in mathematics or never plan on taking a mathematics course again, the seminar is the perfect place to learn to wrap your head around complex concepts presented in a simple way.

Sources

Scharping, N. (2020, April 17). Finally, a world map that doesn’t lie. Discover Magazine. Retrieved November 12, 2022, from https://www.discovermagazine.com/environment/finally-a-world-map-that-doesnt-lie.

2007 Schools Wikipedia Selection. (2007). Manifold. Retrieved November 12, 2022, from https://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/m/Manifold.htm.