When you look at your surrounding environment, it may seem that you live on a flat plane. After all, that’s why you can navigate a new city using a card: a piece of flat paper that represents all the places around you.
This is probably why some people in the past believed that the earth was flat. But most people now know that it is far from the truth.
You live on the surface of a giant sphere, like a beach ball the size of the earth with a few added bumps. The surface of the sphere and the plane are two possible 2D spaces, which means that you can walk in two directions: the North and the South or the East and the West.
What other possible spaces could you live on? In other words, what other spaces around you are 2D? For example, the surface of a giant donut is another 2D space.
Through an area called geometric topology, Mathematicians like me Study all possible spaces in all dimensions. If trying to design Secure sensor networks,, Mines data or use Origami to deploy satellitesThe language and the underlying ideas are likely to be those of topology.
The form of the universe
When you look around the universe in which you live, it looks like a 3D space, just as the surface of the earth looks like a 2D space. However, just like the earth, if you look at the universe as a whole, it could be a more complicated space, like a giant 3D version of the surface of the 2D beach ball or something more exotic than that.
Although you don’t need topology to determine that you live on something like a giant balloon, knowing that all possible 2D spaces can be useful. Over a century ago, mathematicians understood All possible 2D spaces And many of their properties.
In recent decades, mathematicians have learned a lot about all possible 3D spaces. Although we do not have a complete understanding as we do for 2D spaces, we do know a lot. With this knowledge, physicists and astronomers can try to determine this 3D space people really live.
Although the answer is not fully known, there are many Intriguing and surprising possibilities. The options become even more complicated if you consider time as a dimension.
To see how it could work, note that to describe the location of something in space – say a comet – you need four numbers: three to describe its position and one to describe the time when it is in this position. These four numbers constitute a 4D space.
Now you can consider which 4D spaces are possible and in which these spaces live.
Topology in higher dimensions
At this stage, it may seem that there is no reason to consider spaces that have dimensions of more than four, because it is the most imaginable dimension that could describe our universe. But a branch of physics called string theory suggests that the universe has much more dimensions than four.
There are also practical reflection applications on higher dimensional spaces, such as Robot movement planning.
Suppose you try to understand the movement of three robots moving in a factory board in a warehouse. You can put a grid on the ground and describe the position of each robot by their X and Y coordinates on the grid.
Since each of the three robots requires two coordinates, you will need six numbers to describe all possible positions of robots. You can interpret the possible positions of robots as a 6D space.
As the number of robots increases, the dimension of space increases. The impact in other useful information, such as the locations of obstacles, makes space even more complicated. In order to study this problem, you must study large spaces.
There are countless other scientific problems where large spaces appear, modeling of Movement of planets and spaceship to try to understand the “Form” large data sets.
Knot
Another type of study of problem topologists is how a space can be inside another.
For example, if you hold a tied chain loop, we have a 1D space (the chain loop) inside a 3D space (your room). These curls are called mathematical knots.
THE nodes It first came out of physics but has become a central domain of topology. They are essential to the way scientists include 3D and 4D spaces and have a delicious and subtle structure that researchers are Always trying to understand.
In addition, nodes have many applications, ranging from string theory in physics for DNA recombination In biology at chirality in chemistry.
What form do you live?
Geometric topology is a magnificent and complex subject, and there are still countless exciting questions to answer about spaces.
For example, the Smooth 4D poincaré conjecture asks what is the “easiest” closed 4D space and the Ribbon conjecture by edge aims to understand how nodes in 3D spaces are linked to surfaces in 4D spaces.
The topology is currently useful in science and engineering. Decreasing more mysteries of spaces in all dimensions will be invaluable to understand the world in which we live and solve problems of the real world.
John EtnyMath teacher, Georgia Institute of Technology
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