Topology एक जादुई सब्जेक्ट जिससे हैरान है सभी mathematicians

Topology is a branch of mathematics, that emerged very slowly from studies of geometry. However, in the late 1800s and early 1900s, it exploded into one of the largest fields of pure mathematics, a status that remains to this day! The core question of topology is how much can we learn about shapes, spaces, and other mathematical objects by treating them like play-doh (or in posh maths -speak, up to something called continuous deformation).

Continuous deformation (described rigorously through the notion of homeomorphisms) is commonly introduced by giving a sense of what kind of things we can or can‘t do with play-doh. When playing with play-doh, we are allowed to stretch objects, make them larger, flatten them, make them spiky, but they remain the same object. A stretched, big, spiky sphere is still a sphere.
But there is one crucial rule of topology: we cannot change the number of holes in an object. We can’t punch a hole in a disc or glue up the hole in a doughnut. Such moves are “discontinuous”. There is always a point in time where a hole suddenly becomes a not-hole. Keeping track of holes allows us to answer age-old questions such as “How many holes does a straw have, 0,1, or 2?” The answer is 1, since we may continuously deform a straw by squashing it down into a ring, which clearly has just one hole! Also, one cannot find a continuous deformation from the straw into a sphere or a genus (two doughnuts glued together), so it does not have 0 or 2 holes.

Question: What’s a topologist?
Answer: Someone that can’t tell the difference between a coffee mug and a doughnut.
Standard geometry takes into account information like the distance between objects and the size of objects. For a topologist, all triangles are the same. And triangles are the same as a circle. They can all be continuously deformed to one another. So it might seem perplexing how topology can have anything useful to say when objects so obviously different are considered ‘similar’.

One advantage of this approach is that many phenomena in nature change shape continuously; strings of proteins, the sea, leaves, even the shape of our universe itself. Proteins, for example, can get themselves into tangles which make them incredibly hard to identify. However, every tangle results from a sequence of continuous deformations, so using topology, we can untangle them to get a proper look!

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Topology एक जादुई सब्जेक्ट जिससे हैरान है सभी mathematicians

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Видео Topology एक जादुई सब्जेक्ट जिससे हैरान है सभी mathematicians канала Science and myths