An Amazing Theorem for Tangents (Fabricius-Bjerre Theorem)

[@wargreymon2024]

Your simple curve differs from Jordan curve which is*the* simple curve.

[@admiralbananas]

You are totally right. I'll expand on this a bit more for those reading the comments: my definition of a "simple" smooth curve is what meets the definition of "simple" for THIS proof. In analytical geometry, "simple curves" are often not allowed to have double points (or "self intersections", as they're often called). Since Jordan curves are the subject of extensive study, and they have very nice properties, Jordan curves get to keep the title of "simple curves". So in most papers a "simple smooth curve" is a smooth Jordan curve.

[@zaitzerzazza2830]

Excellent selection of a both novel and appropriatly challenging topic. Your articulation is commendable, and the animations were neatly incorporated. I appreciate that you didn't romanticize mathematics excessively and maintained a direct and well-paced presentation.

[@admiralbananas]

Thank you! I'm glad you liked it.

[@WannesMalfait]

I had not heard about this theorem before. Neat result and proof. You did a great job presenting it in a clear and concise way!

[@admiralbananas]

I'm so glad you liked it!

[@oddlyspecificmath]

I'm here due to the YT recommender (i.e., I started + stayed because I like it). In the context of SoME (but not officially reviewing), the volume's a little low for me, but the way you naturally carry the story and pair it with appropriately-timed animations is very nice. The trading/inversion? of types around 10:52 was a nice moment and the conclusion is well done. Very comfortable; thank you for putting this up.

[@admiralbananas]

Yeah I wish that audio was better too. I'm glad you were still able to resonate with the content. I'll invest some time into audio fidelity for the next video.

[@thecalculusofexplanations]

This is exactly what SoME is for, something I've never heard of, with a simple, intuitive explanation beautifully presented, and gets a new creator to commit and make their first video / piece of content. Well done, if I was judging this video you'd do well. Crazy good for a first video

[@admiralbananas]

My heart swells with your kind words. I'm glad you enjoyed it.

[@wqferr]

The moment at 1:48 got a good laugh out of me, good job on the video. You might also want to add the #SoME3 hashtag to the title, as it helps discoverability.

[@jensknudsen4222]

Frederik Fabricius-Bjerre taught at my alma mater, the Technical University of Denmark (back when it was named Den Polytekniske Læreanstalt). This was long before I went there, so I didn't get to enjoy any lectures by him, but we were still using the excellent textbooks he wrote for Linear Algebra and Differential Geometry. Oh BTW, great video! Easy to follow and concise enough without glossing over too many details.

[@Boe1771]

At 1:19 there's a small typo in the formula; 4-1-1-1 != 0. Off course the number D should be 2 as well. Great video!

[@dominiquelaurain6427]

Thanks Matthew. Good video (short,clean,motivated) and very well explained that not so famous theorem. Good luck for the SoME competition ;-) If you miss winning no worry, behave like Parker in magic squares....the joy of math is not the result but your style.

[@mooncowtube]

Beautiful video, with lovely slick animations that make the key points so clear. I hope you make more, and look forward already to seeing them!

[@JasonCantarella]

This is amazing, Matt! I'm linking it to the class page now. 😀 I'm glad this theorem stuck with you-- once I learned it, I could never forget it either!

[@isbestlizard]

What an awesome relationship. Up there with Eulers one about faces, verticies and edges!

[@admiralbananas]

There is actually a generalization of this theorem to polygons, making it even more similar to Euler's Characteristic! www.maths.ed.ac.uk/~v1ranick/papers/banchoff5.pdf

[@amoryketchum727]

You did a wonderdul job explaining this concept, and the animations turned out great! Proud of you 😊❤

[@SimonTyler0]

A neat result and a great exposition - if this is the first time you've made something like this then I can't wait to see more

[@thaliskatsikeas225]

underrated content keep it up!!

[@columbus8myhw]

Very nice! This area of topology is full of results that make you look at things you thought had no structure - aimless curling doodles - and showing you that there's a lot of structure there after all. The animations are very smooth, the exposition is clear, and I love the "a-ha" moment from swapping the direction of the arrow. My one comment is that it seems like the criterion of simple closed curves is a bit late in the exposition, but this is a minor thing.

[@admiralbananas]

Thanks so much for the feedback. I agree with the constructive criticism. For the next one I'll try to make the setup and criterion of the subject earlier and more lean.

[@nosuchthing8]

As a child I was working on something like this for fun. I never got anywhere of course, but it seems like enough info is there.

[@rugbybeef]

This was very well explained and an interesting idea. I like how you broke it down why and how different species of curve points (double points, inflections, bitangents) changed the count along the graph explaining how each derived its value. I'm not entirely sure what motivated the discovery of the theorem. I would have loved a little more exposition about the applicability of the theorem or context as to what the mathematician were pursuing when they described it.

[@admiralbananas]

I didn't find any notes from Fabricius-Bjerre on the writing of this paper. When I was taught this lecture as an undergrad, I was told that the creation of the crossing function was done in the pursuit of a completely different result. While studying the properties of the crossing function, which he thought may help him with a different conjecture, he stumbled upon the result in the video. It certainly makes for a nice "happy little accident" story but I have no way of backing up if it's true. I am also not aware of industry applications of this result. I think it's a nerd-sniping equation. Us math folk are probably going to find a novel equation like this interesting, regardless of real-world use 😉 There are some cool generalizations of this theorem though, and maybe those have applications! Check out the description of the video.

[@fallenflame8678]

Great video! I hope to see you make more videos like this in the future.

[@admiralbananas]

I'm already working on the next one!

[@enstucky]

Hey! Found your video on the SoME voting page. My process didn't allow me to write official feedback, but I liked your video and wanted to give you my thoughts anyway :) The only really bad thing I could say is that the hook might be a little hit or miss since it appeals only to an aesthetic sense of simplicity. For me it was a hit, though; I had fun with this video. But speaking of simplicity, as someone who's done (a small) amount of geometry in life, I know how much work you put into making the explanation as "airy" as it was. You have the technical disclaimer, but you simply use it to bracket off tedious objections, and then don't let them drag down the argument for the rest of the video. I also appreciated that you kept the video fairly text-light; the narration leans heavily on the excellent animations, which are well able to hold it up. All things considered, it's a pretty little argument that your fluid animations bring to life- nicely done!

[@admiralbananas]

Thanks a bunch for the detailed feedback. I'm working on another video currently and it's helpful to know what I should continue doing and what I can do better. :)

[@hellkr]

Parker Square t-shirt! Lovely! 😁

[@ChiralSymmetry]

Amazing. Thanks for the video! As a math enthusiast, one has to wonder if these results generalize to higher dimensions (surfaces, tangent-planes, etc.) And if so, does the generalization more or less get you to things like Euler's formula for polyhedra? Sort of, you can deform stuff and/or add features, but even so, a formula about the features remains invariant.

[@admiralbananas]

I also tend to think about how things can be made more general, but I'm not sure where you'd even start to generalize a theorem like this. What would be the 3-dimensional versions of double points, inflection points, and bitangents? Well, inflection points is not so hard to see, since they have the same definition on higher dimensions (signed curvature changing sign). But in 3D a curve that intersects itself almost certainly will do so in a line of intersection; hardly a "point". Also, 2D bitangent "planes" would be hard to come by in finite quantities. I think this is just one of those theorems that's stuck in 2D and we just have to appreciate it for what it is 😌

[@HamsterFurtif]

Very interesting video, thank you for contributing to the SoME!

[@KSignalEingang]

I really enjoyed this! Between your choice of topic and your explanation, it really hit the sweet spot of being interesting and non-obvious, but not too difficult to make sense of. One thing I'd have been curious to know is whether this theorem has any uses outside this very specific problem, whether practical applications or other more complex mathematical results.

[@admiralbananas]

For applications, I do not know of any. There are some quite interesting generalizations of this theorem though. If you're interested, I added some links in the description of the video. In particular, I find Macway's article to be very accessible.

[@ominollo]

Great video about an interesting topic 🙂

[@omargaber3122]

You can join ##SoME3 with this video, thank you

[@admiralbananas]

I'm not sure what you mean, do you mean the tags in the video?

[@apollo261]

Great video, about an amazing theme I didn't knew about!

[@admiralbananas]

I'm really glad you enjoyed it.

[@drder]

Keep up the good work! Also, love the shirt!

[@nartoomeon9378]

Hmm... I think we can move backwards. Every good tuple (T+, T- , D, I ) can define some "class" of good curves..... But we can use (T+A, T+B, T+C, T-A, T-B, T-C, D, I ) for more precision.)

[@jadeglaze3390]

Never made anything like this before?! Wow! Nice work!

[@admiralbananas]

Lots and lots of hard work, but it gets easier as you go and it's very rewarding. If you've thought about doing something like this before, give it a shot!

[@carlosharmes2378]

Nice way of indivision ❤

[@johnchessant3012]

This is awesome!

[@tyrjilvincef9507]

I can't believe I have never heard this fact before.

[@admiralbananas]

That's exactly what I thought when I learned about this as an undergrad. I've always been fairly engaged with main infotainment on KZbin, and it feels like a good "for fun" theorem and proof that I would've heard of before.

[@darndirtydogbob5205]

This should be called the teenage ninja turtle theorem

[@admiralbananas]

In my files, I named that curve "goggles"

[@user-pr6ed3ri2k]

4:38 I think the 3+ int thing makes there he 3 tangents that go thru the same ooint

[@Axacqk]

I'd like to see a deforming proof. First show that when you smoothly deform any eligible curve, the artifacts can only appear and disappear in ensembles, and these changes preserve the invariant. Then show that in addition to smooth deformations, the invariant is also preserved by an operation that excises a small loop and replaces it with a simple arc - that will take care of all possible winding numbers, because you can always smoothly create two loops and then surgically delete one to increment or decrement the winding number.

[@madhuragrawal5685]

Have you done any algebraic topology formally?

[@admiralbananas]

There's probably a way you can do this. That's a similar style of proof to the classic proof of Euler's Characteristic, and as it turns out, there is a generalization of the Fabricius-Bjerre theorem to polygons (Banchoff). I've certainly brainstormed ideas as to how you'd approach proving the Fabricius-Bjerre theorem, but ultimately this is the proof I was excited to show because it is a more visual and accessible version of the one in the original paper. If you come up with a deforming proof on your own do let me know, I'd be interested to read it.

[@Axacqk]

@@madhuragrawal5685 No I haven't, only watched YT videos, thank you or I'm sorry depending on why you're asking ;)

[@madhuragrawal5685]

@@Axacqk ah lmao, saw many big words so wanted to make sure. Just gives an idea of how standard an idea like this might be

[@choco_jack7016]

i see that parker square shirt

[@admiralbananas]

He's the mascot for giving things a try :)

[@berendkooiker3538]

I really enjoyed this video! I hope you stick to it :)

[@admiralbananas]

That's the plan!

[@shadamethyst1258]

Could you also prove that there is always an equal amount of type B and type C tangents, meaning that on average the bitangents have a +2 influence?

[@xaviergonzalez5828]

What a video! Thank you! New subscriber

[@xaxuser5033]

Excellent video

[@ZaqZiemba]

poggers theorem

[@beutyindetail]

love the t-shirt

[@tricksshotsfinity4415]

Nice vidéo

[@shohamsen8986]

Your argument around 5:48 is based on the fact that f_c(x)=0 for some x. This is what you are using to conclude that the total jumps is 0. How do you know that such a point always exists. Let's say your starting point has f_c=2. How would you find your point for f_c=0?

[@shohamsen8986]

I would assume that one need only study the convex hull of this shape and points on the convex hull which are not straight, cause curved sections usually share regions with the original curve.

[@admiralbananas]

It does not actually rely on the assumption that f_c(x) = 0 for some x. It relies on the fact that the *sum* of all the jumps is equal to zero. This has to be true because if you move x exactly one time around the curve, f_c(x) has to be the same value at the beginning and end of your journey (because it's the same point). This means that, in your entire trip around the curve, all the effects of the jumps cancel each other out. If they didn't, then you'd be able to move around the curve and end up with more crossings than you started with. This is why we can use the values in the table to build an equation that equals zero. A point that is on the convex hull of the shape will always have f_c(x) = 0, so if we wanted to assume such a point exists, we would be safe to do so.

[@admiralbananas]

​@@shohamsen8986as for a study of just the convex hull of a shape to prove the theorem, it would be hard to generalize to features that are in the center of the shape since we can add loops to the curve which are not necessarily perturbations of the convex hull. Maybe I'm wrong though and there is a simple connection. If you come up with a proof, do let me know!

[@luckyw4ss4bi]

At 1:19, the bottom equation shown is invalid

[@admiralbananas]

NOOOOOO and I thought I had caught everything 😭 I appreciate your engagement and attention to detail! The issue was the # of double points; should be 2. So equation should have been: 4-1-2-2/2 = 0

[@luckyw4ss4bi]

@@admiralbananas no worries! I knew it wasn't a big mistake, personally. These things always happen!

[@DerDieDasBoB]

Phee...and i thought i was cracy that 4-1-1-2/2 is not zero...

[@leif1075]

What do you mean the sum total of jumps has to equal zero..how does that make any sense..? Are you assigning a direction to the jumps somehow so when you sum them they get to zero? And why did yiu make it s function ofnthe crossings instead od something else? Like the tangent libes or something or it's not quite possible to graph that?

[@matteogauthier7750]

It means add all the jumps together. If there’s a jump from 1 to 3, that’s a value of 2. Jump from 5 to 4, -1. 4 to 5, 1. With this you can surely see how their sum must be equal to 0, because you end with the same number of points as you started with.

[@admiralbananas]

Matteogauthier is right. As for "why did we definite the function this way", it's hard to give a better motivation than "if you study it, we can prove the theorem". Fabricius-Bjerre discovered this theorem because he was studying this function for unrelated research, and he sorta stumbled upon the relationship in a happy accident. At least so I was told. If this is true, then that means that the details of the proof are what actually lead to the discovery of the proof, rather than the theorem being something that inspired us to create this "crossing function". To put it another way, you can think of the proof in this video to be a retracing through the steps of how the theorem was discovered in the first place, as opposed to the typical "here's a cool fact, here's what it looks like to apply some rigor to it". Hope this helps

[@leif1075]

@siradmiralbanana thanks but you would agree them it's not the first function anyone or most ppl would think of for this right. I would think k you have a function in terms of the tangent points and show it is constant always 1/2 right? Tha ks for sharing.

[@leif1075]

@matteogauthier7750 if tou add them as vectors then I see..I don't think he was clear about that in the video. Thank you.

[@guillaume5313]

You won't be a participant of SoME if you don't put the #SoME3

[@admiralbananas]

Thanks!

[@guillaume5313]

No problem, nice video !

[@MrBeiragua]

4-1-1-2/2 = 1, not 0

[@admiralbananas]

Yeah....it should have been 4-1-2-2/2=0 😬 Good catch!

[@wil8785]

[6:23] nice

[@admiralbananas]

😂

[@gamerhegel7780]

quite sus

[@زكريا_حسناوي]

محتوى الرياضيات ممتاز، ولكن الرجل الخنثوي مستفز للغاية

[@user-pr6ed3ri2k]

Importantdissunctionsltme