What is the shape of the universe?

When you look at your surroundings, it seems you live on a flat plane. After all, that’s why you can use the map to browse new cities: a flat piece of paper that represents everything around you. This may be why some people in the past thought the earth was flat. But now most people know that this is far from true.

You live on the surface of a huge sphere, like the size of a beach ball, adding some bumps. The surface of the sphere and the plane are two possible 2D spaces, which means you can walk in two directions: north, south or west.

What other possible spaces are you in? That is, what 2Ds are there around you? For example, the surface of the giant donut is another 2D space.

Through a field called “geometric topology”, mathematicians like me study all possible spaces in all dimensions. Whether trying to design a secure sensor network, mining data or using origami to deploy satellites, potential languages ​​and ideas can be topological.

The shape of the universe

When you look around the universe you are in, it looks like a 3D space, just like the surface of the Earth looks like a 2D space. But, like the Earth, if you want to look at the universe in general, it could be a more complex space, like a giant 3D version of the 2D beach ball surface, or something more exotic than that.

While you don’t need topology to determine that you live on a huge beach ball, knowing all possible 2D spaces will be useful. More than a century ago, mathematicians figured out all possible 2D space and its many properties.

Over the past few decades, mathematicians have learned a lot about all possible 3D spaces. While we don’t fully understand it like 2D space, we do know a lot. With this knowledge, physicists and astronomers can try to determine the 3D space in which people actually live in.

While the answer is not entirely clear, there are many interesting and surprising possibilities. If you think of time as a dimension, the options become more complex.

To see how this might work, note that to describe the location of something in space (such as a comet), you need four numbers: three numbers to describe its location, and one to describe how long it was at that location. These four numbers form a 4D space.

Now you can consider possible 4D spaces and which of these spaces you live in.

Higher dimension topology

At this point, there seems to be no reason to consider spaces larger than four, because that is the highest imaginary dimension that may describe our universe. But a branch of physics called string theory shows that the universe has more dimensions than four dimensions.

Considering higher dimensional spaces, such as robot motion planning, there are also practical applications. Suppose you are trying to understand the movements of three robots moving on the factory floor in the warehouse. You can place the grid on the floor and describe the location of each robot on the grid by its X and Y coordinates. Since each of the three robots requires two coordinates, you need six numbers to describe all possible locations of the robot. You can interpret the possible location of the robot as 6D space.

As the number of robots increases, the size of space increases. Consider making the space more complex in other useful information, such as the location of obstacles. To study this problem, you need to study high-dimensional spaces.

From modeling the motion of planets and spacecraft to trying to understand the “shapes” of large data sets, there are countless other scientific questions that emerge from high-dimensional space.

Binding

Another type of problem topology is how one space sits in another.

For example, if you hold a knotted rope loop, we have a 1D space (a string loop) in the 3D space (your room). Such a cycle is called a mathematical knot.

The study of knots is first derived from physics, but has become the central area of ​​topology. They are crucial for how scientists understand 3D and 4D spaces, and have pleasant and subtle structures that researchers are still trying to understand.

Furthermore, junctions have many applications, ranging from string theory in physics to DNA recombination in biology to chirality in chemistry.

What shape do you live in?

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

For example, the smooth 4D Poincaré conjecture asks what the “easiest” enclosed 4D space is, while the Slice-Ribbon conjecture aims to understand how knots in 3D space are related to surfaces in 4D space.

Topology is currently useful in science and engineering. Revealing more mysteries of space in every way is priceless for understanding the world we live in and solve the real world.

John Etnyre, professor of mathematics at Georgia Tech

This article is republished from the conversation under the Creative Sharing License. Read the original article.