What's the Geometry of Numbers? - Minkowski's Theorem  

I'm currently writing up some lecture notes where I use Minkowski's Theorem to prove Lagranges 4 Square Theorem, so I was particularly interested when I saw your video. I really like your presentation. You are rigorous and the visuals are pretty nice. I also like that you not just show the Theorem but also apply it to a well known problem. Great job!

A very nice topic that I've not heard of before. In the proof at 3:24, you don't actually state why the set needs to be bounded and open. I thought about it, and actually this condition is just ensuring the set is measurable. So a more rigorous version of this proof would be: Let M be a measurable subset of R² with measure greater than 1. Consider the countable collection M' of measurable sets obtained by intersecting M with the integer square lattice; we have that the sum of the measures of each set in M' equals the measure of M. Now translate each set in M' by the appropriate integer vector offset to the origin. Then we have a countable collection of measurable sets, so they have a measurable union; and that union is a subset of the unit square, so its measure is not greater than 1. Now suppose the translated M' are pairwise disjoint. Then their union's measure would equal the sum of their measures. But this contradicts that the measures of M' sum to a value greater than 1, so the assumption of disjointness must be false. Wikipedia says there is a stronger version of these theorems called Blichfeldt's theorem that actually gives a lower bound on the number of lattice points that have to be contained within a set with a given area.

This is nice! I wasn’t familiar with either the lemma, nor either of the theorems. Nitpick(s) below the readmore line At one point you accidentally say that x^2 + y^2 “is the square root of the distance from the origin”, when you meant either that the square root of x^2 + y^2 is the distance, or that x^2 + y^2 is the square of the distance.

Excellent video. I did my PhD dissertation dealing with a generalization of this concept.

Great video, really cool proof of the sum of squares theorem!

Absolutely amazing video! Subscribed.

Fantastic video. This was sweet - I only have a couple criticisms. One is pacing - I think somewhat obvious stuff was presented at the same pace as less obvious stuff; the claim at 4:04 was really not obvious enough imo to be stated at a conversational pace and then move along to the next thing, it is worth at least drawing an example for why it's true. Also, at 4:30 - why should I imagine it?? You have the visualization tool, not me! :) that was a great opportunity to show the squares overlap with one another and highlight the intersection, and make a continuous analogue to the pigeonhole principle. The primary other criticism I have is not unique to you; it's true of most math presentations. In fact, it may not necessarily even be applicable depending on what your intent of the video is. If this is meant to be a cool visual reference of the proof of an interesting result, then I think it has accomplished that job well. If, on the other hand, this is meant to be an instructional video/tool, I think the criticism applies. The common presentation of mathematical information in textbooks, classes, and most parts of this video goes something like theorem->proof->examples->questions that the theorem answers, but most mathematical discovery often works like examples->questions about those examples->conjecture->proof. When presented in the second manner, the motivation becomes a lot more clear and the viewers can often be much more engaged (because now they are participants - they have asked the same questions about the examples, and they want to know the answer!). The video starts out strong in this regard, and draws various shapes on the plane, asking questions about the non-origin integer lattice points contained in the shape. But then, the restriction to symmetric convex bodies comes out of nowhere! Imo it makes more sense to "play the game" of how big can I make my shape without containing integer points, but then the viewer will probably arrive at the conclusion that non-convex shapes can get arbitrarily large, at which point it then makes a lot more sense to introduce the convexity restriction. Similar "playing around with examples" demonstrations can be done with the other conditions. At 3:30, it is so not obvious to me as a viewer why I would want to prove this in order to answer the larger theorem. Imo it is not hard to conjecture that the area constant in Minkowski's theorem is 4 if you play around with examples, trying to make your shape as large as possible without containing lattice points. At that point, trying to use the properties of convexity to force a lattice point to exist can guide you to the idea of a difference of two points being a lattice point, which then motivates the lemma at hand. The part of the video after Minkowski's theorem is awesome but suffers from a similar issue to the one I mentioned above, a lot of information is thrown at you without a lot of motivation as to how one could come up with that, and so it's fine if you pause the video and grok it yourself, but imo making a separate video with more drawn out motivation would be clearer as an instructional tool. All in all, only my perspective so I could definitely be wrong about some of the stuff I've written, but this was great!

Very nice. Can't wait to see the follow ups.

Please please make a new video on this topic! Looking forward to the sequel :)

Thank you, please post more videos like this!

This video goes really well with the other #some2 video "Why do we care about functions?" by Chillaxiom.

This was really cool! Thanks :)

Very cool! Like the ending problems :)

Pretty nice video, even more so for a first video. Animations and scripts are great (excluding that mishap at 8:20 when explaining the pythagorean formula)

Amazing Video man

Simple solution to 11:10. Dirichlet theorem: for every real number c there are infinite rational approximations m/n that satisfy |c-m/n| < 1/n^2. Let the line be y=cx. Consider the point (n,m). Then distance(point, line)

Nice!

Awesome video! Does symmetric in this case mean 180 deg rotation about the origin symmetry? Note, that after 180 deg rotation about the origin, the point (x, y) becomes (-x, -y)

Every time you open a box, and you insert a lemma or a theorem in it, you may want to add a figure(s) in the box

10:00 I may have missed something, but why does p divided a^2 + b^2? Also, where did the theorem "there exists u: p | u^2 + 1..." come from? Is it true for any prime p? Or only the ones of the form 4k+1?

At 8:55 did you mean u^2 = -1 (mod p) instead of u = -1?

8:55 instead of u = -1 (mod p) it should be u^2 = -1 (mod p)

At 8:55 you mean u^2=-1(mod p), not u=-1(mod p)

Watched

Based

Just one feedback: Your explanation of the main theorem at 4:04 was very confusing and took me a while to understand. It's partly because the animation doesn't go along with your words, so I didn't fully understand what you meant. For example, when you say "there are two points with the same position relative to the squares", actually show those points! And when you say "imagine taking the square tiles and putting them on top of each other", show it in the animation! Not everyone has a visual mind.

The audio quality is _very difficult_ to get past. Please clean it up _a little._

Love maths, but I dunno why I wasn't able to concentrate and understand what this problem does minkowski theorem solve ( which didn't make me invested in the video ) after 4:35 it gets a little better like in this video (ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-Q95PB_Dcjoo.html) I got to know why and what minkowski's theroem is .

Well, since you can find a rational number arbitrarily close to any real number, and the cartesian plane is infinite, no, you cannot choose a line through the origin which does not pass arbitrarily close to an integer point.

i just hate how a triangle and a pentagon are not symmetrical. why.

I don't think naming points X and Y are a good idea in Minkowski space

Easy answer to the question at 11:19 y=x +½

A bit dry (and your mic is not amazing), but there is potential here

10:40 - I have a hunch it may be 1.0149416 for 3 dimensions; the Gieseking manifold’s volume. - en.m.wikipedia.org/wiki/Gieseking_manifold - Cheers 🍻 ^.^

🕉️Hindu temples🕌⛩ use this type of equation💻 from many millions of years ago🔃 All the 🕉️hindu following👌 Chaturbuj chakradhar shri vishnu ji 4 mean chaturbhuj Ananta bhagwaan means infinity God Ananat infinity are very much used in 🕉️hindu culture and also used in getting married💍 Akshay tritiya is like never ending quantity Ananta chaturdashi are like infinity four 4⃣ side or something like that direction disha which are infinity in 🕉️every respect So indian🇮🇳 use this type of behavior from many millions of years

0:38 “Each point has 2 real number coordinates” If you are expecting this to be new information to the viewer, you should label the coordinates of your example points, so those viewers can have an example of what this phrase means. Consider cutting out or muting the “um”s in your narration. I usually cut the volume to 0 whenever I take a breath just to make things nicer to listen to. 2:45 “A square is convex […] this is not.” This point would be clearer if you added a midpoint to one side of the square and shifted that point towards the origin to make your concave shape example. If I misunderstand what convex means, making your 2 example shapes very similar makes it unlikely my misunderstood definition will work with the given example. Additionally, there are much more accessible definitions of convex than the one you used. “No interior angles are greater than 180 degrees” is the first I thought of. The “n points in M” style of phrasing isn’t one I understood until 3B1B used it. I know it can be tedious to add, but labeled example points would help the symmetry example, and the video in general, though your explanation works without them. 3:33 I just learned what ⊆ and Z mean 2 months ago while working on my own video, and I don’t know what the backwards E symbol means to this day. Adding a symbol explanation box somewhere would be useful if you’re targeting non-math majors. Just ⊆=“is a proper subset of”, Z=”all integers” should be enough. 4:08 2 example points would make this much clearer. 4:48 You talk about placing the tiles on top of each other, then don’t do it. Again, I know this would be tedious. If the library you’re using can’t to it, you could have rendered out that frame, cut out the tiles, and manually edited the tiles moving over each other in your video editor. 5:08 I don’t know what “S & C body” means. 5:30 Is this proof meant to work with any selected point Y, because I tried drawing this out in Blender and can’t get it to work? If not, the proof is difficult to understand without an explanation of how those points were chosen... unless the grounds for choosing them is that they satisfy the proof, in which case this become one of those “black magic proofs” where we just skip to having an answer and retroactively prove it was right. 7:13 It would help if you drew the parallelogram having the area of the determinant to the screen. I think I was only able to follow this segment since I already knew what a determinant did. 7:46 These examples for primes transformed into 4k+1 form was really good, but it’s way too fast. It also may have been clearer if you animated “5” into “5=2^2+1” instead of “5” into “2^2+1” so I can see the input and output at the same time. 8:30 Thank you for drawing the distance calculation to screen. I had a hard time following the final proof. There are a lot of pieces that go into making the proof make sense and not all of them are on screen. It would help if you write down each rule as you establish it, like labeling the radius as sqrt(2p). Despite all my issues, I was able to follow the video, and I can tell you were running out of time; I was too.