Topology is Impossible Without These 7 Things
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- Опубліковано 17 тра 2024
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🎥 Discover the origins of topology and how it revolutionized mathematics! 🌍🔍 In this video, we dive into the key milestones that shaped this fascinating field, from Euler's solution to the Königsberg Bridge Problem 🌉 to Poincaré's algebraic topology 🌀. Learn about the pivotal discoveries and concepts like manifolds, homeomorphisms, and more! Perfect for math enthusiasts and anyone curious about the abstract beauty of topology. Don't miss out! 🚀✨
📅 Timeline:
1️⃣ Geometry & Calculus: The 17th Century Transformation 📐
2️⃣ Euler's Königsberg Bridge Problem 🌉
3️⃣ Listing's "Vorstudien zur Topologie" 📚
4️⃣ The Möbius Strip and Continuous Deformation 🌀
5️⃣ Riemannian Manifolds and Higher Dimensions 🌌
6️⃣ Poincaré and Algebraic Topology 🧩
7️⃣ Modern General Topology and Knot Theory 🔗
📌 Why Watch?
- Understand how topology connects with various mathematical fields.
- Explore the historical evolution of topology.
- See real-life examples and applications of topological concepts.
👍 If you enjoyed this video, give it a thumbs up, and don't forget to subscribe for more exciting math content! 🔔
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#Topology, #MathHistory, #Euler, #Riemann, #Poincare, #MathEnthusiast, #EducationalVideo, #STEM, #LearnMath, #MathDiscoveries, #MathRevolution, #GraphTheory, #AbstractMath, #Mathematics, #HistoryOfMath, #MathConcepts, #KnotTheory, #MathExploration, #MathEvolution, #MathOrigins, #Calculus, #Geometry, #Manifolds, #Homeomorphism, #MathEducation, #MathematicalJourney, #MathResearch, #AdvancedMath, #MathInspiration, #MathFacts, #MathGenius, #MathKnowledge, #MathPrinciples, #MathCulture, #MathCommunity, #STEMEducation, #EducationalContent, #MathGeek, #MathematicalThinking, #MathNerd, #MathTalk, #MathScience, #MathVideo, #MathLife, #MathLove, #MathLearning, #ExploreMath, #MathFun
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Image credits:
Boy’s Surfacecommons.wikimedia.org/wiki/Fi...
Coffee to doughnut
commons.wikimedia.org/wiki/Fi...
Sphere Eversion
commons.wikimedia.org/wiki/Fi...
Topological Construction
commons.wikimedia.org/wiki/Fi...
Mobius Strip
commons.wikimedia.org/wiki/Fi...
3D Mobius Strip
commons.wikimedia.org/wiki/Fi...
Torus
commons.wikimedia.org/wiki/Fi...
Untangling Knot
commons.wikimedia.org/wiki/Fi...
Topological Data Analysis
commons.wikimedia.org/wiki/Fi... - Наука та технологія
Of course Euler was the first one who did something close to modern topology.
Yeah, Euler was often the first one to tackle important math fields
Also, one fascinating thing about general topology or specifically point set topology is how historically it was an extension of topics in classical analysis like continuity, connectedness and compactness ( mentioned in the video) to arbitrary spaces beyond the real line like Complex Plane, Finite Fields etc. ( like how Heine-Borel theorem for compactness fall short when we're dealing with non-Euclidean spaces ) . Classically the aforementioned 3 topics are engendered from the notion of metric spaces, but with Topology it is generalised to Topological Spaces.
Thank you for your comment! 😎 You're right, general topology indeed extends classical analysis concepts like continuity, connectedness, and compactness to more abstract spaces. Actually, it is the shift from metric spaces to topological spaces that allows us to explore properties in more general settings, such as the complex plane and finite fields. Your point about the Heine-Borel theorem shows the limitations we face when transitioning from Euclidean to non-Euclidean spaces, which is fascinating for me haha. Topology is able to broaden our understanding beyond traditional bounds, and that’s why it is so important
Aaaah, looks like my days of liking and subscribing to excellent indie math videos are coming to a middle
Glad to see that you subscribed 😎🤙🏻
I dropped out just when vector calc was starting to get complex. Now I don't have time to be a ft student and this is the ideal way for me to continue!
@@MattHudsonAtx awesome! Let us know what topics you’re interested in!!
Number theory and geometric algebra top my interests lately
But really I'm there for the history. Little else lights up the subject for me like the original context.
man being one of the first few for a banger vid feels great. a whole different genus
Thanks for the nice comment!!! Topology is really interesting, we’re glad you liked it 😎🤙🏻
There’s so much to love about math
It is true!! So much to learn ❤️
Agree. It's unfortunate that many people have negative opinions about math.
@@JohnDoe-tt6tr yeah… but why do you think some people are so negative about math? What’s your guess?
WHOA!!! WHERE'D YOU GET THAT GAME-OF-LIFE ON THE SURFACE OF A TREFOIL?!?! I'm a knot theorist and you guys absolutely did us justice! I especially love that you go through the HISTORY and not just the mathematics. Subscribed, liked, can't wait for more! (and to know where the trefoil surface Conway cellular automaton came from!!)
Thanks for the nice comment! I think knot theory is super interesting too!! We will make a video on it 😎🤙🏻
Also here’s the link to the image commons.m.wikimedia.org/wiki/File:Trefoil_knot_conways_game_of_life_without_background_and_fitting.gif
I thought topology is purely an abstraction of open sets and the geometric interpretations were just the result of the theory.
Topology indeed started as an abstraction involving open sets, and many geometric interpretations are the results of applying these abstract concepts. But topology also emerged from practical problems that required a different way of looking at the properties of spaces, as we showed in the video. It’s a combination of theoretical abstraction and geometric application that makes topology so interesting. I hope you liked the video 😎🤙🏻
@@dibeos Thank you for the clarification. I love your videos. ur Liked and Subscribed 👍👍
@@irvinep thanks 🙏🏻😎
@@dibeos I have a request for you. I need the geometric interpretation of Riemann Steljis integral. You can just make a video on origins of integration and include this geometric interpretation, as in which area is represented by this integral or does it even have a geometric interpretation. While at it you can even discuss what breaks when the integrator is not of bounded variation. Regards.
@@irvinep yes, great idea. I will study it, create a visual representation of it and include in one of our next videos
This channel deserves 1M+ subscribers.
Thank you!!!!!!!!!!!! Hopefully one day ❤️
Wow, it’s awesome to see the video idea I referenced come to life, I appreciate the visuals and also the explanations style. Towards the end of the video I started to be in more my territory, though I haven’t done topology in six months the delta epsilon continuity definition brought back memories of writing that down to solve problems. I appreciate this video. Once again superior quality Luca and Sofia❤️
We are glad the video was helpful. We want to make more videos like that where we help people to “visualize” the math. And thanks for the awesome comment Riley, it really motivates us to keep going 😎
Great video, you guys are very good in making complex subjects sound easy. Muito bom!!!
Thanks Peruvian God, may the other gods bless you 😎🤙🏻
Great visualisations and explanations! I bet next time I try to open a textbook it will make more sense. Also love the historical presentation of it!
Hahhahaha I can totally relate, really. I have an approach that always works though: I ALWAYS start with the (non-rigorous) intuition of a subject, just to get used to the core concepts. This is what you called pop-sci explanations. Then I gradually move to the rigorous, textbook, definitions, theorems, corollaries, etc. it works well for me, and it’s no different with topology 😎
Very insightful 👍
Thanks for the nice comment 😎🤙🏻
1735 is probably Euler and the bridges
Excellent video. Where can I find more info on the animation of the orange figure at 4:40?
Thanks 😎 there you go:
en.m.wikipedia.org/wiki/Alexander_horned_sphere
Let us know if you need anything else 😉
Geometry was studied long before the greeks. In fact, it is believed that Euclid's book did not contain his work but also all the works that preceeded him.
Wow that’s cool, I didn’t know that. Where did you learn it?
@@dibeosI personally learnt it in serveral courses on the history of Mathematics, wikipedia has a subsection on it, citing Proclus saing that Euclid collected and improved on results from Eudoxus and Theaetetus, also Pythagoras is a likely source for a lot of the planar geometry done in books I and II
Geometry was studied long before the geeks.
@@feraudyh oh yeah, and it is studied to this day by geeks
I guess Descartes was INTEGRAL to the later development of calculus relying on analytic geometry.
Oh yeah 😎
So you can turn a sphere inside out by passing planes of itself through the same plane of itself . And this is useful how ? Unless you've found what ghosts are made out of in what world is that sphere demonstration useful in anyway ?
Very good question! The surprising fact is that a sphere can be turned inside out in a smooth way/transformation. This is interesting because many natural phenomena (in physics for example) are continuous and smooth. I do not know of any application for this mathematical fact, and I do not believe there is nowadays. However, almost all of mathematical results that are extremely useful in applied sciences now were first discovered as something useless, and only later their amazing applications were found. So, in a sense, I think we should continue looking for ghosts… they may very well turn out to be useful at some point 😬🫥 at least that’s what history has taught us so far 🤷🏻♂️
So good the video
So good the comment! Thanks Felipe 😎
At 6:58 why did you show the map of Gothenburg? Good video
Thanks 😎 just because I was talking about how maps, in general, can be flat (not take into account the Earth’s curvature) since it is a local representation
@@dibeos Ah yes I understood that general point. I was curious about why Gothenburg specifically? Are you a student of the university there?
@@satiremuch2643 no… it is just a random map we found haha are you a student there or do you know somebody who studies there?
@@dibeos Not a student. But I do know 2 persons that have studied there! Interesting that you chose that map.
Of course Euler had to be in there.
Yeah, the guy was good hahah 😎
Thoughts on using Maxim for future videos?
Sorry, do you mean the Manim library?
@@dibeos lol yes, my bad.
@geekoutnerd7882 I started doing some animations there, but the problem is that using keynotes already takes us a week to make the entire video. With Manim, the animations are (probably) better but it would take us longer and thus we publish less… so for now I’m learning, after I get really skilled at it I want to use it more often for sure.
@@dibeos that makes a lot of sense. I look forward to watching more regardless of what y’all use!
What exactly is yalls (formal) background with topology?
My master’s thesis was in an intersection between topology and dynamical systems
Nice vid!
Thanks Paul, let us know what you liked, so that we can double on it! 😎
@@dibeos small concepts linked together to a bigger picture. Emergence of the whole.
Very interesting video. It is somehow sometimes hard to watch tho
Why is it hard to watch? Let us know how to improve it please
Although it is hard to understand but interesting❤
That’s awesome! Let me know what was hard to understand and I can explain it to you if you want 😎🤙🏻
Sir... The real fact is I am in class 12th in india and i have read calcus, limits, 3d and vector yet not having the knowledge of topology. But i have heard about Topology, real analysis, complex analysis, Differential Eqn which are the higher mathematics taught in PG and Ph. D from my brother who is doing Ph. D in mathematics 😃
Sir I want from u to make a vedio on Relation and Function 🙏🏼🙏🏼
@@ashutoshsahu654 that’s great! The sooner you start getting used to these subjects the better
@@ashutoshsahu654 I’m actually preparing a video about functions, but let’s say that I have a loooong list of ideas hahaha there are just so many interesting things to talk about in math and physics that in my opinion are not usually explained in a clear way
The vast majority of usable objects in civilization are deformed spheres to three holed fidget spinners 😂
You missed a donut!!
Yeah, that’s funny, but true 🤣
@@keeperofthelight9681 🍩
Really good
Thanks!!! What did you like about the video and what would you like to see and learn more about? 😎
@@dibeos tbh math is a subject I'm generally not a fan of so I don't know lol tbh i don't know a lot about math but I'm trying to overcome this fear and your video is helping
@@gerardlabeouf6075 I’m happy to hear that!! We try to make it as simple as possible 😎👌🏻
what does graph theory has to do with topology I didnt understood
I think it was how Euler had to think in a topological way to break the problem down to its smallest immutable components. So instead of islands, a river, and bridges- he thought of something that was the same problem, just using used circles and lines. Since the two problems had the same basic properties, you can get the answer to the more complicated/confusing one by solving the simpler one. They are intrinsically linked, and if you think of sentences in the same way you think of shapes, you can mold the problem from one into the other without adding or removing any critical information.
@@migsy1 you nailed it! 😎
Hi! Yeah, I would explain in more detail here but @migsy1 ‘s explanation says everything. Their link is in the simplification of the problem, i.e. in ignoring everything that is irrelevant
There's so much to HATE about math. (Any math past the 5th grade, that is.)
Why do you say so?
Greeks learned maths and geometry from Egyptians and Mesopotamians.
Yep, the Greeks were heavily influenced by earlier civilizations like the Egyptians and Mesopotamians in their mathematical and geometrical studies. This cross-cultural exchange was very important in the development of many foundational concepts in these fields 😎
Do the Numbers Theory
Everybody is asking for it! I'm convinced that the one about Number Theory will be a success, so we will do it soon! (Probably right after the one about black holes that we will publish Saturday) 😎
@@dibeos tnx for answering!)
You the best!)
I will be waiting!)
@@SobTim-eu3xu There you go, as promised 😎
ua-cam.com/video/56x3hzBg48I/v-deo.htmlsi=YQjxnP6gcZVKKSJl
@@dibeos thanks, I so happy, now I go watch it 😇
Ah yes. topologie
Do you like topology? We were thinking about making another video on a specific subject inside topology. Let us know if you’d prefer that or another area of Mathematics (or Physics) 😎🤙🏻
The problem you are referring to is known as the "Seven Bridges of Königsberg." It is a historically notable problem in mathematics that was solved by Leonhard Euler in 1736. The problem laid the foundations of graph theory and prefigured the idea of topology.
In Königsberg, a city in Prussia (now Kaliningrad, Russia), there were seven bridges connecting four land masses and two islands. The challenge was to find a path that would cross each bridge exactly once and return to the starting point.
Euler realized that the problem could be solved by representing the land masses and bridges as a graph. He proved that it was impossible to find a path that crossed each bridge exactly once if more than two land masses had an odd number of bridges connected to them.
This problem was significant because it marked the first use of graph theory in solving a real-world problem. Euler's solution laid the foundation for the study of networks and paved the way for the development of modern graph theory.
Yes, chatGPT, you are correct! 😎👌🏻
Haha, what if its real person, although the writing style is similar to gpt @@dibeos
@@Grateful92 that’s Sofia’s dad 😂😂😂😂 👨🦳
Always welcome !)) Very cool story !
Hmmm nice video
Thanks! Please tell us what content you’d like us to post 😎
@@dibeos can u make a machine to change time? Like one to filter years and years of chaotic data down to a constant live feed of proofs in maths with the stories of the world put in to help with personal analogies to help get the point across of how it all might work ?
@@codybarton2090 can you explain better the idea, please? We might do it…
@@dibeos u know the steel ball plinko experiment and the laser split experimental as well as newtons refraction of light can all be used to describe chaotic spread of information over time but in a world of no resistance some things bounce back and some get stuck how would follow that flow of information even if it was correct or not it still might be valuable in time
Like a lot of apps and websites and live feeds connected to ur phone connected to WiFi Bluetooth etc but connected to an outside computer to filter and grab the information before it gets stuck or bounces back or “lost” in the hypothetical sense
Topological logic went too far and almost crossed the boundaries 😏😳😂😂folk she knows what I mean..
7:30 "The shortest path is along the great circle route" I think this is untrue. The geodesic (segments) on S^2 are (segments of) great circles, but geodesics are not necessarily distance-minimizing curves on S^2, and also, distance-minimizing curves on S^2 are not necessarily segments of geodesics. For example, consider two points with the same latitude.
Yes, you are correct and I noticed this right before publishing. But even though it is not completely true, it illustrates pretty well the concept 😎
I correct, his original statement was correct. You are confusing great circles with latitude lines. Only the equator is a latitude line that is also a great circle. Great circles are the circles whose radius is the same as the radius of the sphere, by definition, and geodesics are the minimizing distance curves, by definition. This is a classic and basic example in an introductory calculus of variations course.
@@andrewkarsten5268 I am well aware of what a great circle is. You are wrong about the definition of a geodesic, it is NOT a distance minimizing curve: it is merely locally distance minimizing. For example, for non antipodal points on a sphere, there is a unique great circle containing them both. But there is a “long way” around and a “short way” around, which are both geodesic segments as segments of a great circle, yet plainly cannot both be distance minimizing.
Please review your basic introductory course, because you seem to have several embarrassing misconceptions. Goodbye.
@@tmjz7327 you seemed to have misread my comment, I did not say every path on a sphere between two points which is the arc of a great circle is distance minimizing, which you are implying I claimed. It is known however, that if a path on a sphere between two points is distance minimizing, then it is the arc of a great circle. You’re conflating the direction of the implications.
Also, in the context of this problem on S², the geodesic is a path with the minimal distance. Again, you are the one who is wrong.
@@andrewkarsten5268 You are right on the first point, a distance minimizing curve indeed is an arc of a great circle, I misinterpreted that.
For the second point, again, you are incorrect. Firstly, what does "the" geodesic mean? Between two points there need not be a unique geodesic. Secondly, if you meant the statement "a geodesic is a path with the minimal distance" then that is just not true, I don't know how to make it simpler for you. Like I already painstakingly laid out for you, just take two non-antipodal points on a sphere and consider the great circle containing both. Going the "long way around" is a geodesic, but not distance minimizing.
Do not respond again with another misconception, because my patience is growing thin.
Calculus... developed by Renee Descartes...
How tf are you gonna mention Descartes over Newton and Leibniz with respect to calculus?
Well, you are correct. Newton and Leibniz are the true “developers” of Calculus. What we meant is that Descartes gave very important contributions to Newton’s and Leibniz’s “toolbox” of mathematical tricks. In fact we have another video here in the channel where we talk about Newton and Leibniz and how they developed Calculus 😎
sorry i couldn't continue, this format kills me (you explaining to someone else)
Why don’t you like it? Let us know how to improve the format
I will spare you guys my criticism this time. One reason is because this theme is kinda complicated. Another reason is because the Möebius strip hat was so wholesome it made me smile.
Thanks!!! Finally a positive comment 😂 (just teasing you Samuel)
@@dibeos hahahahaha. You may be teasing, but it is probably true. I demand a lot.
@@samueldeandrade8535 that’s good. Keep on demanding from us. We do not know everything, so I’m pretty convinced that in some of the next videos there will be wrong explanations, despite the fact that we really do our best to deeply research each topic (that’s why we can only publish once a week). But when you see something wrong, please correct us, this way we learn and improve more and more 😎🤙🏻