Of course Euler was the first one who did something close to modern topology.

@dibeos

Ай бұрын

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.

@dibeos

Ай бұрын

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

There’s so much to love about math

@dibeos

Ай бұрын

It is true!! So much to learn ❤️

@JohnDoe-tt6tr

Ай бұрын

Agree. It's unfortunate that many people have negative opinions about math.

@dibeos

Ай бұрын

@@JohnDoe-tt6tr yeah… but why do you think some people are so negative about math? What’s your guess?

man being one of the first few for a banger vid feels great. a whole different genus

@dibeos

Ай бұрын

Thanks for the nice comment!!! Topology is really interesting, we’re glad you liked it 😎🤙🏻

Aaaah, looks like my days of liking and subscribing to excellent indie math videos are coming to a middle

@dibeos

Ай бұрын

Glad to see that you subscribed 😎🤙🏻

@MattHudsonAtx

Ай бұрын

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!

@dibeos

Ай бұрын

@@MattHudsonAtx awesome! Let us know what topics you’re interested in!!

@MattHudsonAtx

Ай бұрын

Number theory and geometric algebra top my interests lately

@MattHudsonAtx

Ай бұрын

But really I'm there for the history. Little else lights up the subject for me like the original context.

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!!)

@dibeos

Ай бұрын

Thanks for the nice comment! I think knot theory is super interesting too!! We will make a video on it 😎🤙🏻

@dibeos

Ай бұрын

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.

@dibeos

Ай бұрын

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 😎🤙🏻

@irvinep

Ай бұрын

@@dibeos Thank you for the clarification. I love your videos. ur Liked and Subscribed 👍👍

@dibeos

Ай бұрын

@@irvinep thanks 🙏🏻😎

@irvinep

Ай бұрын

@@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.

@dibeos

Ай бұрын

@@irvinep yes, great idea. I will study it, create a visual representation of it and include in one of our next videos

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❤️

@dibeos

Ай бұрын

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 😎

This channel deserves 1M+ subscribers.

@dibeos

Ай бұрын

Thank you!!!!!!!!!!!! Hopefully one day ❤️

Great video, you guys are very good in making complex subjects sound easy. Muito bom!!!

@dibeos

Ай бұрын

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!

@dibeos

Ай бұрын

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 😎

1735 is probably Euler and the bridges

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.

@dibeos

Ай бұрын

Wow that’s cool, I didn’t know that. Where did you learn it?

@TheLuckySpades

Ай бұрын

​@@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

@feraudyh

29 күн бұрын

Geometry was studied long before the geeks.

@dibeos

29 күн бұрын

@@feraudyh oh yeah, and it is studied to this day by geeks

Very insightful 👍

@dibeos

Ай бұрын

Thanks for the nice comment 😎🤙🏻

Excellent video. Where can I find more info on the animation of the orange figure at 4:40?

@dibeos

Ай бұрын

Thanks 😎 there you go: en.m.wikipedia.org/wiki/Alexander_horned_sphere Let us know if you need anything else 😉

I guess Descartes was INTEGRAL to the later development of calculus relying on analytic geometry.

@dibeos

Ай бұрын

Oh yeah 😎

The vast majority of usable objects in civilization are deformed spheres to three holed fidget spinners 😂

@keeperofthelight9681

Ай бұрын

You missed a donut!!

@dibeos

Ай бұрын

Yeah, that’s funny, but true 🤣

@dibeos

Ай бұрын

@@keeperofthelight9681 🍩

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 ?

@dibeos

Ай бұрын

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 🤷🏻‍♂️

Of course Euler had to be in there.

@dibeos

28 күн бұрын

Yeah, the guy was good hahah 😎

So good the video

@dibeos

Ай бұрын

So good the comment! Thanks Felipe 😎

Greeks learned maths and geometry from Egyptians and Mesopotamians.

@dibeos

Ай бұрын

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 😎

Thoughts on using Maxim for future videos?

@dibeos

Ай бұрын

Sorry, do you mean the Manim library?

@geekoutnerd7882

Ай бұрын

@@dibeos lol yes, my bad.

@dibeos

Ай бұрын

@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.

@geekoutnerd7882

Ай бұрын

@@dibeos that makes a lot of sense. I look forward to watching more regardless of what y’all use!

At 6:58 why did you show the map of Gothenburg? Good video

@dibeos

Ай бұрын

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

@satiremuch2643

Ай бұрын

@@dibeos Ah yes I understood that general point. I was curious about why Gothenburg specifically? Are you a student of the university there?

@dibeos

Ай бұрын

@@satiremuch2643 no… it is just a random map we found haha are you a student there or do you know somebody who studies there?

@satiremuch2643

Ай бұрын

​@@dibeos Not a student. But I do know 2 persons that have studied there! Interesting that you chose that map.

What exactly is yalls (formal) background with topology?

@dibeos

Ай бұрын

My master’s thesis was in an intersection between topology and dynamical systems

Very interesting video. It is somehow sometimes hard to watch tho

@dibeos

Ай бұрын

Why is it hard to watch? Let us know how to improve it please

Although it is hard to understand but interesting❤

@dibeos

Ай бұрын

That’s awesome! Let me know what was hard to understand and I can explain it to you if you want 😎🤙🏻

@ashutoshsahu654

Ай бұрын

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 😃

@ashutoshsahu654

Ай бұрын

Sir I want from u to make a vedio on Relation and Function 🙏🏼🙏🏼

@dibeos

Ай бұрын

@@ashutoshsahu654 that’s great! The sooner you start getting used to these subjects the better

@dibeos

Ай бұрын

@@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

Nice vid!

@dibeos

Ай бұрын

Thanks Paul, let us know what you liked, so that we can double on it! 😎

@pauldruhg2992

Ай бұрын

@@dibeos small concepts linked together to a bigger picture. Emergence of the whole.

Really good

@dibeos

Ай бұрын

Thanks!!! What did you like about the video and what would you like to see and learn more about? 😎

@gerardlabeouf6075

Ай бұрын

@@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

@dibeos

Ай бұрын

@@gerardlabeouf6075 I’m happy to hear that!! We try to make it as simple as possible 😎👌🏻

There's so much to HATE about math. (Any math past the 5th grade, that is.)

@dibeos

Ай бұрын

Why do you say so?

Do the Numbers Theory

@dibeos

Ай бұрын

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) 😎

@SobTim-eu3xu

Ай бұрын

@@dibeos tnx for answering!) You the best!) I will be waiting!)

@dibeos

27 күн бұрын

@@SobTim-eu3xu There you go, as promised 😎 kzread.info/dash/bejne/Z2qslcqzcsqaaKw.htmlsi=YQjxnP6gcZVKKSJl

@SobTim-eu3xu

27 күн бұрын

@@dibeos thanks, I so happy, now I go watch it 😇

what does graph theory has to do with topology I didnt understood

@migsy1

Ай бұрын

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.

@dibeos

Ай бұрын

@@migsy1 you nailed it! 😎

@dibeos

Ай бұрын

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

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.

@dibeos

Ай бұрын

Yes, chatGPT, you are correct! 😎👌🏻

@Grateful92

Ай бұрын

Haha, what if its real person, although the writing style is similar to gpt ​@@dibeos

@dibeos

Ай бұрын

@@Grateful92 that’s Sofia’s dad 😂😂😂😂 👨‍🦳

@olegsirotkin48

Ай бұрын

Always welcome !)) Very cool story !

Ah yes. topologie

@dibeos

19 күн бұрын

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) 😎🤙🏻

Topological logic went too far and almost crossed the boundaries 😏😳😂😂folk she knows what I mean..

Hmmm nice video

@dibeos

20 күн бұрын

Thanks! Please tell us what content you’d like us to post 😎

@codybarton2090

20 күн бұрын

@@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 ?

@dibeos

20 күн бұрын

@@codybarton2090 can you explain better the idea, please? We might do it…

@codybarton2090

20 күн бұрын

@@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

@codybarton2090

20 күн бұрын

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

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.

@dibeos

Ай бұрын

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 😎

@andrewkarsten5268

Ай бұрын

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.

@tmjz7327

Ай бұрын

@@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.

@andrewkarsten5268

Ай бұрын

@@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.

@tmjz7327

Ай бұрын

@@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?

@dibeos

Ай бұрын

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)

@dibeos

28 күн бұрын

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.

@dibeos

Ай бұрын

Thanks!!! Finally a positive comment 😂 (just teasing you Samuel)

@samueldeandrade8535

Ай бұрын

@@dibeos hahahahaha. You may be teasing, but it is probably true. I demand a lot.

@dibeos

Ай бұрын

@@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 😎🤙🏻