Unraveling the Mysteries: What Is the True Shape of Our Universe?

Look around you. It often feels like we live in a flat world, right? You can easily navigate a new city using a map, a flat piece of paper. This might explain why some people once thought the Earth was flat. But we know now that we’re living on the surface of a huge sphere, much like a beach ball—but with more bumps and grooves.

This spherical surface is one of many possible 2D spaces. In such spaces, you can move north, south, east, or west. But have you ever thought about other 2D shapes? Picture the surface of a giant doughnut. That’s another example of a 2D space.

Mathematicians study all sorts of spaces, including 3D and beyond, through a field called geometric topology. This work helps us tackle real-world problems—from designing secure networks to deploying satellites using origami techniques. The principles of topology guide these applications.

What About the Shape of the Universe?

The universe we observe feels like a 3D space. Just as the Earth’s surface represents 2D space, the entire universe could be a complex 3D shape. Perhaps it resembles the surface of a giant beach ball, or maybe it’s something even odder.

While we still don’t fully understand the shape of our universe, mathematicians have made great strides in categorizing various 3D spaces. Understanding these shapes helps physicists and astronomers unravel the mysteries of our cosmos, sparking curiosity about where we really are within it.

When we consider time as an additional dimension, things get even more complicated. To pinpoint a comet’s position in space, you would need four numbers: three for its location and one for the time. This combination of space and time leads us into the realm of 4D spaces.

Exploring Higher Dimensions

It might seem unnecessary to think about dimensions beyond four, but physics, particularly string theory, suggests there are many more dimensions out there. Exploring higher-dimensional spaces can also solve practical problems, like programming robot movements. For instance, if you have three robots moving on a factory floor, you need six numbers to track their positions in a grid, resulting in a 6D space. As you add more robots or other complexities, the dimensions increase.

Many scientific challenges, like modeling planetary movements and analyzing large datasets, involve high-dimensional spaces. This area of study opens doors to new insights and applications.

The Intrigue of Knots

Another fascinating area of topology is how spaces can exist within each other. Think of a knotted string in your room; you have a 1D loop (the string) within a 3D space (your room). This field began with physics but now plays a crucial role in topology. Understanding knots contributes to our knowledge of 3D and 4D spaces, and researchers are still uncovering their delicate structures.

Knots have practical uses that span from string theory in physics to DNA recombination in biology and even chirality in chemistry. The study of these entangled forms reveals much about the interconnectedness of different scientific disciplines.

What’s Next in Topology?

Geometric topology is not only beautiful but also complex, with endless questions awaiting answers. For example, the smooth 4D Poincaré conjecture seeks to identify the simplest closed 4D space. Additionally, the slice-ribbon conjecture explores how knots relate to surfaces in 4D contexts.

Topology is vital to modern science and engineering. As we continue to explore the mysteries of spaces across all dimensions, we grow closer to understanding the world and solving a myriad of real-world problems.

John Etnyre, Professor of Mathematics, Georgia Institute of Technology.

This article is based on a piece originally published in The Conversation.