As you observe your surroundings, you may think you’re living on a flat surface. This is why maps, which are flat pieces of paper, can help you navigate unfamiliar cities. Historically, some people believed the Earth was flat due to this appearance. However, it’s now widely accepted that this is not the case.

In reality, you reside on the surface of a massive sphere, similar to a giant beach ball with some added bumps. Both the surface of the sphere and a flat plane are examples of 2D spaces, allowing movement in two directions: north-south and east-west.

What other possible 2D spaces could you be living on? For instance, the surface of a giant doughnut is another example of a 2D space.

Through the field of geometric topology, mathematicians like myself study all possible spaces in various dimensions. Whether designing secure sensor networks, mining data, or using origami to deploy satellites, the underlying language and concepts often involve topology.

The shape of the universe

When you look at the universe, it appears to be a 3D space, similar to how the Earth’s surface seems 2D. However, just as the Earth is more complex than a flat surface, the universe could be a more complicated space, like a giant 3D version of the 2D beach ball surface or something even more exotic.

While you don’t need topology to determine that you live on a giant beach ball-like surface, understanding all possible 2D spaces can be useful. Over a century ago, mathematicians discovered all possible 2D spaces and many of their properties.

In recent decades, mathematicians have made significant progress in understanding all possible 3D spaces. Although we don’t have a complete understanding like we do for 2D spaces, we have made substantial advancements. With this knowledge, physicists and astronomers can attempt to determine the 3D space we actually live in.

While the answer is not yet fully known, there are many intriguing and surprising possibilities. The options become even more complex when considering time as a dimension.

To understand this, note that describing the location of an object in space, such as a comet, requires four numbers: three for its position and one for the time it occupies that position. These four numbers make up a 4D space.

Now, consider what 4D spaces are possible and which one you might inhabit.

Topology in higher dimensions

It may seem unnecessary to consider spaces with dimensions larger than four, as that’s the highest imaginable dimension for our universe. However, string theory, a branch of physics, suggests that the universe has many more dimensions than four.

There are also practical applications for thinking about higher-dimensional spaces, such as robot motion planning. Suppose you’re trying to understand the motion of three robots moving around a factory floor. You can place a grid on the floor and describe each robot’s position using x and y coordinates. Since each robot requires two coordinates, you’ll need six numbers to describe all possible positions, which can be interpreted as a 6D space.

As the number of robots increases, the dimension of the space also increases. Factoring in other useful information, such as obstacle locations, makes the space even more complicated. To study this problem, you need to examine high-dimensional spaces.

Countless scientific problems involve high-dimensional spaces, from modeling planetary motion and spacecraft to understanding the “shape” of large datasets.

Tied up in knots

Another type of problem topologists study is how one space can be contained within another.

For example, holding a knotted loop of string represents a 1D space (the loop) inside a 3D space (your room). Such loops are called mathematical knots.

The study of knots originated in physics but has become a central area of topology. Knots are essential to understanding 3D and 4D spaces and have a subtle structure that researchers are still trying to comprehend.

In addition, knots have many applications, ranging from string theory in physics to DNA recombination in biology and chirality in chemistry.

What shape do you live on?

Geometric topology is a complex and beautiful subject, and there are still many exciting questions to answer about spaces.

For example, the smooth 4D Poincaré conjecture asks what the “simplest” closed 4D space is, and the slice-ribbon conjecture aims to understand how knots in 3D spaces relate to surfaces in 4D spaces.

Topology is currently useful in science and engineering. Unraveling more mysteries of spaces in all dimensions will be invaluable to understanding the world we live in and solving real-world problems.

John Etnyre, Professor of Mathematics, Georgia Institute of Technology

This article is republished from The Conversation under a Creative Commons license. Read the original article.