PopularScience: Topology - What is It? Balls, Bagels and Loops...
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!
Previous article about Genetic Algorithms
Interesting post, when you get down to the math of things, reality often gets pretty freaky
@justtryme90 I think the math puts things in order in my head. It's necessary to apply math correctly.
Wow Science is so amazing... As an aside: Some are saying that the Earth is actually a torus! upvoted
ahah) the earth is actually an ellipsoid, not a torus)
@magdalenaruth follow my articles and learn many new things!)
Ah, you mean something like a Flat(ter) Earth Theory?
Ellipsoid - 3-d figure. Sphere is special case of the ellipsoid.
Definitely food for thought!!! Thank You puhoshville!
I love topology! Was one of my favorite college math classes. I love having my students cut the Möbius strip in half to reveal it stays one strip, as you foreshadowed above. And it absolutely BLOWS their mind when they cut it in half again!
Oh, yeah! It is not clear the first time) Anyway you need to understand that there is dependence on the number of spins. By the way, look at this task! What do you think?)
I like it! Don't want to give away the answer, but it's a fun exercise in supplementary angles and isosceles triangles. I am big on using movement or goofy phrases to help people remember seemingly useless facts. Two years ago, when Uptown Funk by Mark Ronson and Bruno Mars was popular, I had students remember "Up/Down supp you up" for consecutive angles in traps. To this day, I still have them come up to me and tell me they remember. I would like to think if I pointed them to this task they would all get it right ;)
You're really pretty good teacher!) If I may ask, which school (university) do you work?
I'm based out of the Midwest in the United States. Math dept. chair at my our high school and adjunct university professor of Calculus through Indiana University. Loving your posts on math! It is cool seeing so many other people here dig its beauty as well.
@team-leibniz, unfortunately, comments do not support a depth of more than 5. So I write here. I've been looking for people like you. I have an interesting idea that you might like. I wrote to you in steemit.chat. Please check incoming messages.
I always like to ask myself "so once I have tricked myself into thinking I have understood or learned something, what benefit do I get?"...I always get the same answer: "no benefit from being programmed to belief something". I'd rather stay free from beliefs and ideas and so stay mentally and physically free. To me, to 'under-stand' something is to stand under informational belief and be a slave to it. So I take no information seriously and only entertain it for fun.
you don't need to believe in gravity to obey it.
If I don't believe in gravity then obeying its 'laws' doesn't come into my mind. My mind does not exist anywhere in particular, it is not real, only experience beyond mind is true, is real.
just because you don't think about it doesn't mean your brain isn't calculating it. virtually every movement your body makes obeys the laws of gravity, if it didn't, you would not be able to move with any accuracy.
what you call your mind is your idea about yourself, and maybe, peripherally, you grasp that it is your attention. computers have attention too, though their circuits are simpler and less smooshy. in a computer system the focus of attention is in a register called the 'program counter' which tells it where to get its next instruction from. This instruction could be part of a string that forms the function 'measure the temperature' or 'copy the input from a camera buffer into a file on disk'. It is still attention. Attention, aka consciousness, is simply what your mind's scanners are focusing on. In the human brain this focus can actually be seen by special machines that show it as an area of increased electrical activity. our brains don't have only one centre of focus, just as modern computers no longer do either. the video processor has its own focus of attention, the cpu has one, the chip that shuffles memory in and out of the ram chips, from the hard drives, pci slots and so forth, it has a focus of attention also.
it's absurd nonsense to say that this does not exist somewhere. it exists inside your head, and i know it does, and anyone can see it does, because if you chopped off your head, it would stop happening.
consciousness is an imaginary idea, and if I am not aware of it, it is not happening
my brain is also a imaginary idea, because I would die if I tried to cut it open and see it
I have never seen a persons brain, so it is just a story, something imaginary, not true, a lie
@onesunbeingnow very interesting point of view!) I understand that math can be not interesting for you... But to deny the information as a whole - very strong.
I don't deny information, I just accept that it is just information, matter that has not taken form yet, so I can pick and choose whether I care about it or not. I only care about some basic information that is required to keep me at a high level of joy and fun. If it gets too complex the joy and fun have to be sacrificed for some perceived future gain. What is really happening here is that one shuts everything else and focuses really hard to see the hologram change before oneself into what one wants to see.
this is an effective way to have a nice dream but it isn't a practical way of doing something productive. maybe sometimes it is productive to have a nice dream, but not always. most of the time, actually, you have to do something productive, even if it's just alleiviating discomforts your body goes through on a regular basis, the urge to wee, the urge to take a dump, the urge to fuck, and the urge to eat. in between that, somewhere the food has to find a way into the mouth and it won't do it by itself, you are the only person who cares about it, as much as you do, and if you didn't, then why would anyone else care either?
dreaming is to think my imagination is real
In science, what is assumed to be the theory of today may be an approximation of the theory of tomorrow. Knowledge is not static. It evolves with time and postulates may change over time.
One thing however, science is not a matter of beliefs. We have facts, proofs, predictions, assumptions, etc. Now, if one does not accept proofs, it is another issue (and I am wondering what would be wrong with a proof not to accept it).
I nevertheless strongly agree with you on one point: it is always good to ask ourselves questions. :)
[ I know I already said that in other posts. ]
Knowledge ceases as soon as it is not on my mind. It does not exist if I do not give it attention.
My proof is my experience, and my level of experience is higher than mere words. Words, thoughts etc don't validate my experience, experience allows words to think as though they provide something called 'validation'. When really all they do is invalidate my experience by adding useless nonsense to the raw intensity of a direct experience. Direct personal experience is what matters, is what is true, and words come after the fact to label things as this or that, when its just light.
To be honest, the question/answer loop is for slaves who like staying trapped.
The topology of networks is especially interesting to me. The more holes it has the more resistant it is to collapse. pure stars, which are equal to spheres, are the most brittle network shape. When you take out the centre (or head) it falls into pieces.
You are talking about discrete science - graph theory. Interestingly, the graphs are often described by the surfaces on which they lie.
Thanks a lot fpr this nice article. This remains me my old time when I was more attracted by maths than by physics :)
Actually, I am a engineering designer using topology optimization. That is why this post is so interesting to me.
Cool article!