Hello everyone, this is puhoshville and today I publish my second post in the series "Popular Science" (1st post here). I urge everyone to learn new interesting things and watch the beauty of science.
Today we will talk about the topology. Try to understand what it is and touch main objects from this field of knowledge.
Topology is the science about surfaces properties that are preserved under continuous deformation. In the simple term, it is some kind of surface deformation, that satisfy some simple rules:
- shouldn't let the surface get breaks
- shouldn't let a different points to be in touch (two points in touch - one point)
These rules are given us definition of regular deformation. You can imagine a piece of plasticine or list of paper.
So, let's give a definition of topologically equality of surfaces. Two surfaces are topologically equal if exist the regular deformation that is reshape surfaces into each other. For example, typical mug and torus (bagel) are topologically equal:
As you can see, the mug is gradually transformed into the bagel and then back to the mug. The origin of two different items the same!) This article is only about a two dimensional surfaces that live in three-dimensional space. Unfortunately, the consideration of higher dimensions is not clear and requires a strong mathematical apparatus for working with them.I have prepared for you some interesting examples.
Topological man. A man from plasticine.
There is an interesting problem that will excite your imagination. Can a person untie the hands?
Ordinary people are certainly not able to do this. A man made of plasticine will be able! Look here:
Everything here is clear. Did you come to this solution?)
Now complicate the task... put on a topological man's dress watch. This is what will happen if we try to do the same.
So a small detail can affect the essence of the problem.
Sphere with handles
One of the most basic types of surfaces is the sphere with handles. In the example above, you can see that the man considered as a sphere with two handles. There is a theorem stating that every two-dimensional oriented surface without boundary is a sphere with some number of handles. The sphere without handles - just a sphere, the sphere with one handle is a torus, sphere with two handles is a double torus.
(sphere)
(torus or sphere with 1 handle)
(double torus or sphere with 2 handles)
(triple torus or sphere with 3 handles)
According to my calculations, the human is a sphere with 4 handles.
Non-oriented surfaces
There is also a class of surfaces, which are arranged in a different manner than a sphere with handles. Oriented surfaces divide the entire space into external and internal parts. An non-oriented surfaces does not have this property.
(Klein Bottle)
In this video you can see that the concepts of "inside" and "outside" do not exist in the case of the Klein Bottle.
Dissecting the Klein bottle results in Mobius Strips.
Half part of a Klein Bottle is topologically equal to Mobius Strip:
This surface with boundary is notable for the fact that it is very easy to glue by yourself. Enough to take a rectangular sheet of paper and glue the opposite side in that way:
Try to dissect this one and get a 2x longer strip. 2nd dissection will get us two Mobius strips of same length. Be sure to try to check it yourself! For lazy guys here's a video on youtube.
I hope you like the article. Comment, ask questions, and follow me!
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