Microsoft / Wkimedia Commons
Example of a mathematical node
The geometric topology of the universe itself is returning mathematicians to astrophysics. What is the form of everything around us?
When it looks at the surrounding environment, it may seem to be living on a plane. After all, that’s why you can navigate a new city using a map: a flat piece of paper that represents all the places around you.
We live on the surface of a giant sphere, like a beach ball the size of the earth with some added protrusions. The surface of the sphere and plane are two possible 2D spaces, which means we can walk in two directions: north and south or this and west.
What other possible spaces can we be living? That is, what other spaces around you are 2D?
Through a domain called geometric topologymathematicians study all possible spaces in all dimensions. Whether we are trying to conceive of secure sensor networks, extract data or using Origami to put satellites, language and underlying ideas will probably be those of topology.
The form of the universe
When it looks at the universe in which it lives, it looks like a 3D space, just like the earth’s surface looks like a 2D space. However, like the earth, if it looked at the universe as a whole, it could be a more complicated space, like a giant 3D version of the beach ball 2D surface or something even more exotic than that.
Although you don’t need topology to determine that you are living in something like a giant beach ball, knowing all possible 2D spaces can be useful. More than a century ago, mathematicians discovered all possible 2D spaces and many of their properties.
In recent decades, mathematicians They learned a lot about all possible 3D spaces. Although we don’t have a complete understanding as we have for 2D spaces, we know a lot. With this knowledge, physicists and astronomers can try to determine people really live.
The options become even more complicated if we consider time as a dimension.
Note that to describe the location of something in space, for example, a comet, we need four numbers: three to describe our position and one to describe the time we are in this position. These are four numbers that constitute a 4D space.
Now we can consider that 4D spaces are possible and in which of these spaces we live.
Topology in higher dimensions
At this point, it may seem that there is no reason to consider spaces more than four, since this is the maximum imaginable dimension that can describe our universe. But a branch of physics called suggests that the universe has much more dimensions than four.
There are numerous other scientific problems in which spaces of high dimension arise, from the modeling of the movement of planets and spaceships to the attempt to understand the “Shape” of large data sets.
Wrapped in us
Another type of problem that topcologists study is the way one space can be within another.
For example, if we hold a wire bow with a knot, then we have a 1D space (the wire loop) within a 3D space (our room). These ties are called mathematical us.
O study of nodes It initially emerged in physics, but became a central area of topology. They are essential to the way scientists understand 3D and 4D spaces and have a delicious and subtle structure that researchers are still a.
What format do we live in?
There are still numerous questions to answer about the spaces. For example, the conjecture of Poincaré over the 4D Smooth space asks what is the “simplest” closed 4D space and the conjecture gives fita fatia It aims to understand how nodes 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 understand the world in which we live and solve real world problems.
