The Moessner Miracle. Why wasn't this discovered for over 2000 years?

Today's video is about a mathematical gem that was discovered 70 years ago. Although it's been around for quite a while and it's super cool and it's super accessible, hardly anybody knows about it.
00:00​ Intro
04:58 Chapter 1: Making our own proof
09:55​ Chapter 2: Some more amazing facts
13:11​ Chapter 3: Post's proof
23:36​ Supporters
If after watching this video you'd like to find out more about Moessner's result, the following PhD thesis features a very comprehensive bibliography: ebooks.au.dk/aul/catalog/book...
The proof by Karel Post that I matholologerise in the second half of this video is contained in this paper:
Karel A. Post. Moessnerian theorems. How to prove them by simple graph theoretical inspection. Elemente der Mathematik, (2):46-51, 1990.
Post also proves a couple of generalisations of Moessner's theorem. Another good write-up of the same proof can be found in Ross Honsberger"s 1991 book More Mathematical Morsels. Honsberger says about Moessner that "he was internationally known in the field of recreational mathematics for many spectacular results in arithmetic". Have to have a closer look at some point at what else exactly he did :)
Post's article can be accessed here: www.e-periodica.ch (search for "Moessnerian theorems"). Sadly most other articles about Moessner's theorem are located behind paywalls.
Here is another very pretty proof of the basic cubes result by Anthony Harradine and Anita Ponsaing using actual 3d cubical shells www.qedcat.com/misc/StrikeMeO...
It's well worth exploring further than what I get around to reporting in this video. If you do, you'll discover interesting connections with super-factorials, higher-dimensional counterparts of Pascal's triangle, and so on.
Challenge for the programmers among you: write a program that turns a sequence of highlighted integers into the corresponding Moessner sequence.
Today's music is "Just Jump" by Ian Post. If you are interested in the t-shirt google "math whisperer t-shirt". If you don't understand the math whisperer bit, you did not watch the video to the end :)
Enjoy!
Burkard
14. Sep. 2021: I just added Russian subtitles prepared by Michael Didenko. Thank you very much Michael.

[@jaredjbarnes]

I don’t think I can express how grateful I am to live in a time when I can be taught by one of the best math teachers in the world from my bedroom. Thank you so much Mathologer, we don’t take you for granted.

[@walterpoelzing9412]

I was just thinking something like that. Burkardt has his Ph.D. in math and should be given a world Ph.D. in teaching math to math minded people from all leading universities across the globe. When is he coming up with his world tour? Would love to meet this guy in person.

[@jaredjbarnes]

@@walterpoelzing9412 I’m loving this idea!

[@KSignalEingang]

@@walterpoelzing9412 I can see it now ... "This sunday at Staples Center - Mathologer, with opening acts A Perfect Circle & Add N to (X)"

[@sandmann6851]

Yes! So true. Wir lieben dich Mathologer!

[@joem8496]

all the years of advanced maths and nothing like this was ever found in a lecture or book that I recall... so grateful for Mathologer and team!

[@frederickm9823]

I tend to forget that Burkard is actually German (?), so when he read out the title from Moessner's thesis I was like: "Damn, his German is amazing." and felt silly a few moments later...

[@friedrichschumann740]

He is German, more precisely from somewhere in Unterfranken. He said so in one of his videos. I think it was close to Würzburg or Würzburg itself.

[@renerpho]

@@friedrichschumann740 Würzburg, according to Wikipedia.

[@punditgi]

@@friedrichschumann740 Selbstverstaendlich! 🙃

[@Mathologer]

@@friedrichschumann740 I grew up in Veitshöchheim :)

[@ericbischoff9444]

@@Mathologer Oh, that looks like a beautiful place. I just finished spending a few years in Nürnberg.

[@mariusweber4990]

As a German with a rather casual interest in mathematics, I just want to point out how nicely Moessner presented his idea in the original text. No overstatement, no attempt at making it seem more complex or grandiose than it is, and a straight to the point, easy to understand text. It's literally just "here is a generalisation that does not seem to have been noticed so far, have a look". Great text and an awesome video!

[@mlmimichaellucasmontereyin6765]

Mmmm... Now, if only we could resolve The Ivory Tower problem of modern maths and The Status Quo Silo paradox of matamaths & SM QM. Sigh... The momentum of the subliminal culture of intellectual cowardice is probably too mammoth to overcome before our mass-self-extiction by chronic-egocidal mass-stupdity.

[@shoam2103]

@@mlmimichaellucasmontereyin6765 SM? QM?

[@kuick6814]

@@eurotrash5610 SM is statistical mechanics I believe

[@rog2224]

@@kuick6814 Such a welcoming discipline. Who can forget Goodstein's 1975 comment in his text book "States of Matter" "“Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics." TBH, I don't recall it being that hard, but I was following well worn paths on the topic, not forging a new frontier.

[@oooBASTIooo]

@@shoam2103 This is how people write who want to look smart in front of an audience of laymen.. Also, it is usually a sign of deep misconceptions.

[@jasonnybergoog]

That moment when you realize the number line is just the 1d case of Moessner's Miracle...

[@Joffrerap]

technically, there are all the numbers to the power 1.

[@jasonnybergoog]

@@Joffrerap actually to power zero (string of 1's...)

[@vj_henke]

Step 1: Take the numberline. Step 2: Don't cancel anything. Result: You got the numberline. It's a miracle.

[@Joffrerap]

@@jasonnybergoog when you say the number line, you don't mean 1 2 3 4 5 ... ?

[@jasonnybergoog]

@@Joffrerap I do. (I didn't say it was complicated :))

[@TrandusNinja]

I love how much of induction in this video even tho the word "induction" is not mentioned

[@Mathologer]

... there are also a "few" binomial coefficients in this video :)

[@BartonChittenden]

@@Mathologer 2 or 4, here and there.

[@minirop]

Do this? Pascal's triangle Do that? Pascal's triangle What time is it? Pascal's triangle

[@straft5759]

Don’t know how to solve a difficult math problem? Look for any Pascal’s triangles

[@MrConverse]

What day of the week is is? What month is it? What’s for dinner?

[@codekillerz5392]

Aurora borealis? At this time of day? At this time of year? Localized entirely within your kitchen? May I see it?

[@BroArmyCommander]

@@codekillerz5392 Pascal's Triangle

[@Mrpoopy62639]

Wait, its all Pascal's Triangle?

[@BardaKWolfgangTheDrug]

Decades upon decades of maths: *agonal spasm over sheets of paper* Mathologer: so basically we use Pascal's triangle here and that's it

[@steffen5121]

"And from here on, it's Pascals Triangles all the way down." 👌

[@nenmaster5218]

@@steffen5121 The Learning never ends, so call it silly, but i do have the hobby of asking people if i an recommend them science-chanenl or just education-channel in general to them! Mind if i do?

[@steffen5121]

@@nenmaster5218 sure, shoot!

[@nenmaster5218]

@@steffen5121 Ok, how about: -Oversimplified. -Is ok to be smart. -Sci Man Dan. -Professor Dave Explains. -Some More News.

[@steffen5121]

@@nenmaster5218 Thanks. The first two, I already know. I'll check out the other three.

[@force10guy26]

"If you're a 5 dimensional creature, that's a no brainer... Just start by visualizing 5D shells of 5D cubes" 🤯👽🤓

[@leonfa259]

Even as a 3 d creature ist is easy, just use a 3d cube x a 2d square. If you want to visualize more than 3 d, just breake it down in into multiple visuals of maximally 3 d. It is very useful if you deal with the visualisation of high dimensional data like Business reports.

[@BruceNitroxpro]

@@leonfa259 , This is intuitively obvious to the casual observer.

[@milanstevic8424]

@@leonfa259 literally you imagine the entire volume to continuously "react" to some data in the 2nd diagram (that can also react to data in some 3rd diagram). for example, if you take a 2D diagram with 10x10 black/white pixels, where pixel intensity depicts some f(x,y) then you can integrate another variable (z) non-spatially, if you imagine this flat image to change over time. now you can stack all such animation slices on top of each other, and create a 3D diagram. it works very intuitively the other way around. in a case where 3D is projected to 2D, it is simple to tell what is going on: 2D image is cross-sectioning the 3D volume. so adding a 4th dimension to existing 3 should be as easy as adding one imaginary influence. if you start with a 3D diagram with 10x10x10 black/white pixels, integrating another variable (w) non-spatially, would require imagining this volume to be animated like a cloud. again, if you could visualize 4D space in a spatial manner, you could stack all such animation slices on top of each other, but the simplest way would be to imagine 2 independent 10x10 flat boards, where f(x,y) is continually influenced by f(z,w), seemingly depending on some unseen (extradimensional and spooky) circumstances. additionally you can imagine that each full f(x,y) slice lives in a domain ruled by every unique f(z,w), inside a hierarchy. now it becomes easy to attach another one f(a,b) ruling over every f(z,w). obviously this is the same as having f(x,y,z) cubes being ruled by f(w,a,b), or playing a 3D game f(x,y,z,w), but you also get to choose the nature of the adventure on a 2D graph f(a,b). this is exactly what "animation" means btw: coming to life.

[@nenmaster5218]

The Learning never ends, so call it silly, but i do have the hobby of asking people if i an recommend them science-chanenl or just education-channel in general to them! Mind if i do?

[@henridelagardere264]

It seems a miracle indeed that anyone would do a half-hour video on the Moessner Miracle, until you read the name _Mathologer._

[@PeepzaHazMyNoze]

I searched for Mathologer the other day and noticed he had missed some months of uploading, watched some videos, only to see another upload today. Lovely!

[@MrConverse]

Search again. Maybe we’ll get another video tomorrow!

[@IFearlessINinja]

For the record, this is also a path to discovering derivatives and calculus. For many reasons it's a shame this wasn't stared at harder a while back. I wonder what we're just a few steps from discovering

[@alexandertownsend3291]

I am not seeing the connection. Could you explain it? It sounds cool.

[@gebruikerarjan]

Mmm sounds like another mathologer video= all the secrets of pascals triangel!

[@yxx_chris_xxy]

YouFearlessYouNinja: YouRight. Think of the numbers in different places in the triangles as deltas.

[@X_Baron]

Pascal's triangle has a connection to the prime numbers, but I feel like there's more to it.

[@BooBaddyBig]

I haven't proved it, but I'm pretty sure that the method of finite differences is in there if you look at it right. ANY sequence of a polynomial of order n comes out of n numbers if you repeatedly add them.

[@PapaFlammy69]

A new upload, great stuff

[@playerscience]

Oh hi flammy !!!

[@zeb1820]

Yes, but how can stuff that is less than three be great?

[@MaJetiGizzle]

Of course I’m seeing you here… Now just 3blue1brown and my KZbin math list will be complete.

[@h4z4rd28]

Hello from hilbi boi

[@benjabby]

A verified comment that adds absolutely nothing, great stuff!

[@gabrieldee345don5]

I am so regretful about not having a math teacher like you. Suddenly, after a life believing I am not good with maths, I started to understand stuff I never believed I would. :-)

[@petersontaylor2000]

The most unbelievable and beyond reason thing about this channel is that it does not have (yet) 1M subscribers! All the other assertions are left as an exercise for the reader...

[@JohnDlugosz]

He should do a collaboration with Colin Furze, who has >11 million subscribers. That should get him a few million cross-over views, and hopefully some percentage subscribe.

[@bhatkrishnakishor]

Just consult "the spiffing brit" and have your channel grow exponentially.

[@afterthesmash]

Some of us who adore math are not sheep who mash the subscribe button. Somehow, without ever a single use of the "subscribe" button, I'm rarely at a loss for excellent content. FWIW, I also do not subscribe to the afterlife.

[@MrSigmaSharp]

In university I always struggled with this number theory problems and their proofs. Mathologer should be a movement for teachers all over the world

[@djsmeguk]

This looks highly related to the discrete derivative and discrete integral which is cool stuff..

[@Mathologer]

You are absolutely right there :)

[@Bibibosh]

Idk

[@toniokettner4821]

isn't the discrete integral just counting squares?

[@mino99m14]

How is this related to a discrete derivative?

[@toniokettner4821]

@@mino99m14 if you flip pascal's triangle around, reading from left to roght you get sequences of numbers, where every row below another is the difference of the terms in the sequence above, thus discrete derivative

[@rafbambam]

Hi, I did find another way to "build" youre cubes. Instead of adding the hexagons, you can add 1+ (1+2+3), than add 7 + (3+4+5), and 19+ (5+6+7), ... If you look in the video at 8:39 you see the second cube. To go to the next cube you need to add one side, so 3 pieces, lets say on the left, then you add 4 pieces to the right side (3 pieces for the original cube and one piece for the just added row), and finaly you add the bottom with 5 pieces to get back to the cube. Now if you look at the numbers you add they are left and right from the yellow marked number in the top row (2 and 4 sandwidch the 3; 5 and 7 the 6,...) And these yellow numbers are diviceble by 3 (by construction) So you get a sequens of (3n-1)+(3n+1) and that is the same as 2(3n). We could write 2(3n) also as 3(2n). This we can convert in (2n-1)+(2n)+(2n+1). And this is the secuence you are adding to youre cube. (Note "n" is the base number from youre last cube, and the counting number of youre tripple (where 3 is the 1st; 6 the 2nd;9 the 3th;12 the 4th,..)) Notice that by every step you go up you start with the same number you stopt with the last time. (2n+1)=(2n+2-1)=(2(n+1)-1) So the cube you end up with has a sidelenght of 2n+1. And you start the next step just adding that side to youre cube I hope you can see how i find this a bit easyer to construct the cube. Also this is in a way the same you construct youre squares (if you start with "skipping" just one number at the top) if you notice that every odd number is the sum of its position + its position -1. (5 is the 3th odd number and it is 3+2; 9 is the 5th odd number and it is 5+4; ...). Also here you get that the last digit you added the last time you start to add the next time. (but you only get 2 numbers). Note; this methode does not work in the fourth or higher dimensions. Why? Well I'm not a mathematitian. So I don't know. And I have trouble imagining in four dimentions. Hope this is helpfull. Sorry for my bad English, I speak Dutch. Greetings from Belgium.

[@mlmimichaellucasmontereyin6765]

Bravo! Brilliant, I was thinking that there should be an easier, more geometrically logical way to show the intrinsic logic and potentials. BR, may I have your permission to use your result as additional proof of my proof of post-modern number theory, metamaths, RH, etc.?

[@rafbambam]

@@mlmimichaellucasmontereyin6765 wow, now I'm i m ressed, yes you May use it. Greetings.

[@rafbambam]

After a good night sleep I realised that I used the word "Cube" where I should have Said "cubeshell", becourse we are creating a Shell that goes over the previous Cube.

[@mlmimichaellucasmontereyin6765]

@@rafbambam BR - Kudos again & thanks! Will send you a link to an updated draft of my paper (proofs, etc.).

[@williamrutherford553]

I think this video shows why math education channels like Mathologer are so important; a simple, easy to explain proof didn't exist, so you made it! Maybe it isn't the most rigorous, but being forced into the visual medium instead of a dry, mathematical paper makes things WAY more understandable.

[@moonman2183]

Pure mathematics at its finest Thanks for revealing this bag of gems

[@Nikolas_Davis]

1:27, now that's what I call "completing the square!"

[@Mathologer]

:)

[@thegoldengood4725]

@@Mathologer :)

[@kohterg3713]

Mathematics and its magic it's terrifyingly beautiful!

[@nenmaster5218]

The Learning never ends, so call it silly, but i do have the hobby of asking people if i an recommend them science-chanenl or just education-channel in general to them! Mind if i do?

[@Sleeperguy23]

As a straight C, occasional B, math student I’m in awe of this channel. I discovered this channel just yesterday and I’ve already devoured several hours worth of math I otherwise would not have cared about. Idk if you are a teacher or were a teacher at one point but your presentation, explanations and enthusiasm is something I wish I had when I was taking math.

[@mattfisher7641]

Thank you! Your collection of KZbin videos is the modern day equivalent of Euclid's "Elements"...so clear, so precise, so pretty.

[@maitland1007]

Brilliant video! I'm so glad you're back and glad you have gotten through the stressful times. It always makes my day when I see one if your videos appear.

[@marcoantonio7648]

I'm a simple man, I see mathologer video, i make popcorn and watch it!

[@timburdack7366]

Wow, this is so amazing and beautiful! That's why I love maths and especially number theory, thank you for that video!

[@raphberry]

I watch and rewatch all your videos almost nightly, before going to bed, they’re all so beautiful. The music at the end of this one made me emotional enough to post today. Thank you for the wonderful work you do.

[@cauchym9883]

I wonder if Neil Sloane might be interested in including this Moessner-Transform and reverse transform into his database of integer sequences.

[@Mathologer]

That would be a good idea :)

[@briansammond7801]

Would it cause Sloane's gap to be filled?

[@steffen5121]

@@briansammond7801 🤞🙈😄

[@jannegrey593]

Yay! Another video - thank you. Can we get that one on Galois theory? Or did KZbin messed up again and didn't notify me and you already published it? Regardless - I love your videos!

[@Mathologer]

Galois, is the big one later this year :)

[@mohammedal-haddad2652]

@@Mathologer looking forward.

[@shoam2103]

@@Mathologer wow awesome!

[@Nightriser271828]

@@Mathologer fabulous! I did some reading on Galois theory for school and would love to see where you go with it.

[@mlmimichaellucasmontereyin6765]

@@Mathologer Dear Sir, I think you will see that my paper on post-modern metamaths metatheory, number theory, RH, etc., enables new group theory, etc.

[@mohammedal-haddad2652]

Once you mentioned the higher dimensions I expected the Pascal triangle will pop up somehow in this videos.

[@Mathologer]

When you dig a bit deeper you actually come across the higher-dimensional counterparts of Pascal's triangle (among other things :)

[@johnlinley4375]

I cannot tell you how much I enjoyed this lecture. It reminded me of several other Maths, and that in itself is one of its delights. Thank you.

[@infinitumneo840]

I'm always glad to see your videos. I've noticed an underlying geometry to many complex mathematical proofs and even physics itself. This illustrated that idea very nicely. It's good to see your videos again.

[@adityaruplaha]

Oh man the timing of this video couldn't have been any better. Thank you so much!

[@truesoundwave]

I'd love to see you do a video on dual quaternion skinning. It'd be combining topology, N-D transformations and general matrix/linear algebra. Alot of bad resources on it not really well explaining it's power.

[@Mathologer]

Have to admit that I have never heard of dual quaternion skinning. Will check it out :)

[@truesoundwave]

@@Mathologer Computer Applied usage of Matrix Multiplication! I could just request Dual Quaternions in general. Dual Quaternion Skinning(DQS) is a rendering algorithm mixing a good bit of...everything. Mixes Linear Algebra , Toplogy, Complex Analysis, Physics/Volumes seamlessly to produce real-life like animation replication. Take like a 3d model Bicep Flexing while bending it's elbow. Under a linear system, it doesn't look...right. Under a DQS system, it looks like when Mathologer flexes his giant biceps in real life. Not many good sources on it however at all yet if you ask animators which they would prefer if they have the choice, most would reasonably say DQS.

[@leif1075]

@@Mathologer How come you never replied to my message on your other video?

[@omp199]

@@truesoundwave A biceps muscle is not more than one "bicep"! It's not a plural. There is no such thing as a "bicep".

[@ericbischoff9444]

@@Mathologer or, to go more general, geometric algebra (basically, Clifford algebras). I think that could be "mathologerized" as it could be illustrated with small diagrams and, if the hard core abstraction is left behind, be put to the level of a general audience.

[@jabbarkhan2031]

I'm a High Schooler, didn't understand some parts😁. But I enjoyed the video so much🥰🥰 . Whenever I watch these Mathologar videos everytime I learn a new thing beyond High School syllabus. I just love this channel, he teaches in a very creative,fun and funny way.

[@hamidswift]

Dont worry, i didnt understand a thing.

[@sentienttoast1319]

I like how for this video I sat for 23 mins straight with full attention while I can't do the same thing for maths in school

[@sillymel]

Man, Post's proof really was an amazing proof. That was one of those proofs that just makes me go "Wow!"

[@rogerhudson2814]

One of the best, clearest, bits of number theory. Students have it so much easier than I did 40 years ago. Great video.

[@agreenwood3478]

I'm so very grateful for the time, work, and knowledge that goes into these teachings. Much love to the community of curious minds that coalesce around great men/women of science. It's a pleasure to be here with you all.

[@JohnDoe-py4ty]

How can it be that I'm not subscribed to this guy already? I love his pacing and enthusiasm for the subject - looking forward to more content like this!

[@Joffrerap]

15:35 very nice way to teach binomial coefficient

[@joshuarosen6242]

I recently watched an interesting video featuring Roger Penrose in which he expressed the view, which I share, that mathematics is discovered, not invented. When I see patterns like this it further strengthens my view that maths fits together too beautifully and with too many hidden but elegant patterns to be invented. This was very interesting and no, I hadn't heard of Moessner. Thank you for another informative and well-explained video. The internet isn't just stupid cat videos.

[@waylonbarrett3456]

I believe we confuse the language. It seems what we do is that we INVENT ways to describe patterns that we have DISCOVERED in nature.

[@joshuarosen6242]

@@waylonbarrett3456 While that would be a reasonable interpretation if we were talking about physics, we aren't. There is no nature in mathematics.

[@waylonbarrett3456]

@@joshuarosen6242 I disagree. Mathematics is explicitly natural. Before a mind existed, there were probably consistencies and patterns in natural phenomena. Only later would those patterns interface in the particular ways which we call perception and cognition and consciousness. When we abstract upon these patterns, our natural brains exhibit new natural patterns. Are you suggesting at some point it is supernatural?

[@joshuarosen6242]

@@waylonbarrett3456 If it were supernatural, it would be invented (presumably by some deity). I explicitly stated that I don't believe maths to be invented and so it cannot be supernatural. I'm not clear what it is that I wrote that you disagree with. I interpret your description of maths as being natural to mean it isn't invented, which is what I believe.

[@omrizemer6323]

This is an absolutely amazing video. I needed a few minutes to come back to my senses after watching this

[@tychophotiou6962]

Wow! That is absolutely fantastic! I am a maths teacher. Every time I watch one of your videos my respect for you increases even more. You are most definitely one of the greatest mathematicians of the 21st century!!!

[@johnchessant3012]

13:00 Taking a = 1 and b = c = ... = 2, we see that the output sequence for {n^2} must be {n! * (n-1)!}. Neat!

[@vj_henke]

But what if we take Input = {n^k} for any k > 2 ?! If you post the answer before me, you get a cookie.

[@jamiebradshaw4222]

@@vj_henke Not the person to whom you were responding, but I don't think you get anything nice. For k=3 the output sequence becomes n * ( (n-1)! (n-2)! ... 2! )^6 and I can't see any nice way of expressing it for k=4. If we relabel a=a_1, b=a_2, c=a_3, ... then we have a_1 = 1, a_2 = 2^k - 2, and a_n = n^k - 2*(n-1)^k + (n-2)^k for n > 2, which is a polynomial of order k-2, so I can't see any simple way of expressing the output sequence except just writing the whole product explicitly.

[@Mathologer]

Veeery good :)

[@Mathologer]

@@vj_henke Veeery tricky (but doable :)

[@Mathologer]

@@jamiebradshaw4222 Yes, nothing terribly nice.

[@Your2ndPlanB]

I'm so impressed by the way you make difficult mathematics easy to understand. Thanks for inspiring me to be a better teacher :)

[@DaveRS575]

One of my greatest challenges in mathematics was that I am primarily a "visual learner". None of my teachers were capable or willing to assist me with this issue. (1972-1984). My ability to visualize patterns and ultimately assign values to these pattern integers (reversing the process so to speak) ultimately helped me to understand how to develop compression and encryption algorithms and their respective keys, ultimately leading me into a career as a computer systems engineer, global network architect and designer regardless of my inability to comprehend the terminology used in common teaching methods used those days. I've recently discovered a renewed interest, even passion for physics and learning, through people like yourself who understand the importance of talking the time to teach both the concepts and the visual at a reasonable pace and in a manner that is not only clear and concise, but that you are brave enough to do so in a humble, fun, way without conceit or unnecessary arrogance so many people teaching in the field seem to have. I look forward to learning from you and your many videos here. It's no small wonder you have 100's of thousands of subscribers and millions of views and fans. Keep up the great work you've been doing. I've been looking for a good resource for my 8 year old grandson who is following me (and even exceeding me!) on my "return to mathmatics" journey! Thank you again.

[@Jukka70]

Welcome back, so great to have you back. I can relate to being busy. Am working at three location teach math at Adult education. 7 students alone in one location. So I really appreciate that you still have time for all of us. :). It not only amazes me the things you find to talk about, but to be able to find ways to explain things including things that are new proofs.

[@MattiaDonega]

Wow!! It's always a great pleasure to see your animated proofs! Challenge myself as a teacher to improve my skills and imagination for my lessons. Thank you a lot!!

[@Jack_Callcott_AU]

Herr Mathologer, Vielen Dank. This is one of the most incredible maths videos I have seen!

[@prgnify]

23:25 and you just made my day with your awesome explanation, walk-through and visualisation. Such a privilege to watch you, thanks for sharing

[@Etobio]

You are very engaging to your audience, and coupled with the excellent visuals this video was made very easy and pleasant to watch. Thank you!

[@aviralkumarbarnwal83]

One of the best teacher of mathematics.❤️❤️❤️❤️❤️ u sirji

[@benYaakov]

I can't believe for your effort in video Please never stop making such beautiful videos .

[@CODENAMEDERPY]

This is amazing. Thank you for making these videos for the world to see. They make it a better place.

[@LordDestrus]

As a person relatively unskilled in math, I am in love with stuff like this. I am subscribed and angry that I only just found your channel! Cannot wait to binge your content!

[@gdclemo]

Pascal's triangle is inaccurately named as it only has two sides!

[@toniokettner4821]

holy shit you're right

[@sebastianjost]

It's just an infinitely large triangle

[@toniokettner4821]

@@sebastianjost a triangle has 3 edges

[@BroArmyCommander]

I guess you could call them* Pascal's triangles* and make it a series :)

[@Kettwiesel25]

@@sebastianjost It might as well be an infinite square. You can only see two sides of it, since it is infinite. So how do you know it's a triangle?

[@rustybolts8953]

"Pretty obvious this pattern will continue forever." Fascinating how infinity is so intrinsically numerical...

[@aurielklasovsky1435]

That was so much fun! Pascals triangle rocks so good :) I think this is my new favorite video of yours. Please keep creating beautiful geometric profs, you the man!

[@enricoognibene1582]

Welcome back! Long awaited, but it was absolutely worth it! Great video

[@jasonmorello1374]

After a fashion, I think you have shown the nature of how slide rules work. Logarithmic scaling of two sets, and when compounded, brings to value in magnitude, similar to a pantograph.

[@punditgi]

Welcome back! We missed you, mein Herr! Hertzlich willkommen!

[@ronaldronald8819]

Pascals triangle always intrigued me. I looked at your explanation with amazement and delight. Thank you so much, an other secret revealed.

[@kennytran4]

It's always a joy to see these videos come out -- I remember reading your articles from the days of QEDcat on The Age.

[@Mathologer]

That's great :) Marty and I are just working on a second collection of these articles. Will be published later on this year. QEDcat is still alive !

[@Nikolas_Davis]

If the L-shaped part was called a "gnomon", I propose the term "kutos" (κύτος), for the 3D version. As in, the hull of a ship :-)

[@pierreabbat6157]

Με αρεσκει.

[@backwards3454]

Αρεσκομαι επι της νεας αυτης προτασεος

[@benvanrensburg4261]

Every time I see something like this, I wish (again) that I had nothing else to do than to learn Mathematics.

[@Ni999]

I'll probably never be able to say which of your videos is my favorite but if I had to try, this one would have to be on the list.

[@mateusildefonso6021]

You've proven this miracle with dynamic programming. You are a genius!

[@timburdack7366]

Uhh, a new Mathologer video! I'm soo excited!

[@jacksonstarky8288]

I stumbled on something similar to this in my last year of high school, 31 years ago... but my analysis of the nth powers resulted in a reduction to n! with some additional terms that I couldn't figure out what to do with at the time. I put my expanded tables into a spreadsheet that I still have... and I'm going to have to have another look at my data with this video in mind.

[@CL2K]

Wonderful. Love seeing how Pascal's triangle and the binomial formula can help us understand the sequence of integer powers.

[@charamomochi7629]

This was so cool to watch, thank you so much for this and I will definitely stick around for another.

[@tejarex]

Animated proofs should become routine in classrooms.

[@ankitnautiyal2568]

Yes so true, till what time we will be introduced to Maths in the same old bookish paragraphs. More so in maths before going to rigrous proof the pattern should be able to visualize by the learner and then think in an abstract way and generalize.

[@R.Daneel]

So very true. Particularly introducing math in school. Involve the visual cortex to get a few shots of endorphins when a complex concept suddenly becomes obvious. Great introduction to far more conceptual math subjects. So often on KZbin I'm left wondering "why didn't they just show us that in school?" Case in point: kzbin.info/www/bejne/lZaZqGh9qdCWfas (QI Pythagoras Demo)

[@InverseHackermann]

13:00 Spoilers below! Using a=1, b=c=d=...=2, the highlighted sequence is 1, 2+2, 3+2+4, 4+2+4+6, ... In each term of the sequence, we can split up the first term of the sum into k 1s and spread them out over the rest of the terms of the sum, giving 1, 1+(1+2), 1+(1+2)+(1+4), 1+(1+2)+(1+4)+(1+6), ... Each term of the sequence is the sum of the first k odd numbers, which are the squares. That means that the sequence generated by the squares is 1, 2*1^2, 3*2^2*1^2, 4*3^2*2^2*1^2, ... In general, the kth term is k(k-1)!^2=k!(k-1)!

[@Mathologer]

Veeery nice :)

[@danielprovder]

My inductive guess tells me the cubes yields (k)!(k-1)!(k-2)!

[@allozovsky]

@@danielprovder Seems more like n³ ↦ n×[(n-1)!⋅(n-2)!⋅...2!⋅1!)]⁶, as follows from the general formula.

[@allozovsky]

@@danielprovder But there is an input sequence that yields n!·(n-1)!·(n-2)! We may present it as n!·(n-1)!·(n-2)! = n¹·(n-1)²·[(n-2)!]³, which gives us a = 1, b = 2, c = d = e = ... = 3, so the input sequence would be n + 2·(n-1) + 3·[(n-2) + (n-3) + ... + 2 + 1] = (3n² - 3n + 2)/2 - it's only that the expression n!·(n-1)!·(n-2)! itself is not properly defined for n = 1, since (n-2)! = (1-2)! = (-1)! is hard to calculate :)

[@3Max]

We see a formula with the triangle(!) numbers, and the fifth element is exactly the number of seconds in a day... @Mathologer, coincidence? I think not!

[@nabilfares555]

Beautiful! Thank you so much for this beautiful treat.

[@Xubono]

Awesome, as usual. I have been missing you here. Welcome back! Not quite as much fun as in the Maths dept at The University of Adelaide, but VERY informative. I thought I enjoyed Mathematics in the past. Your videos open up an infinite assortment of extra dimensions of mathematical wonder. The first class video presentations and detailed explanations are unrivalled. Thank you Burkard. Best Wishes, David

[@Mathologer]

It feels good to be missed :)

[@Jaylooker]

It’s interesting to consider this prospective of the logarithmic product rule in the context of primes, the prime number theorem, harmonic series, and Reimann Zeta function.

[@mlmimichaellucasmontereyin6765]

Yes, indeed! See my paper @ michaellucasmonterey.com/metamaths/

[@timburdack7366]

13:00 My guess: a^2 = a×a, so I think the output sequence will be a^a.

[@Mathologer]

Hmm, good guess, ... but, no :)

[@eliyasne9695]

I gueesed it to be the double factorial of successive odd numbers, because: n²=1+3+5+•••+(2n-1) (2n-1)!!=1·3·5·•••·(2n-1)

[@kingkartabyo6206]

@@Mathologer 13:00 It’s pretty neat, as the crucial hint for this is 12:34. Although, how shifting a term forward affects the results is not that clear. So, n!(n-1)! should be the answer. And of course, made my day with your return!

[@allozovsky]

@@eliyasne9695 What if we reverse the task and try to find the initial sequence that generates the double factorial sequence - what would that sequence look like?

[@eliyasne9695]

@@allozovsky when I think about it, looking at the result at 11:50 it becomes obvious that we need a=1,b=0,c=1,d=0, and so on in an alternating pattern. So the sequence leading to the double factorial is: 1, 2, 3+1, 4+2, 5+3+1, 6+4+2, 7+5+3+1 ••• Pretty cool!

[@Piffsnow]

This channel always amazes me, even though I know it's good stuff. I'm a maths teacher and I learn so much here! Thanks!

[@5c0ttyd]

My first Mathologer video. Certainly won't be the last! Subscribed 👍 Very well explained, beautiful content, and a quirky likeable German dude with an infectious laugh. What more do you want?!

[@Nikolas_Davis]

Roger Penrose also presents the "cube-hegaxon" visual proof in his 1994 book " *Shadows of the Mind: A Search for the Missing Science of Consciousness* ". It is in chapter 2, section 2.4 (" How do we decide that some computations do not stop?")

[@Mathologer]

Yes, that's the standard "the hexagonal numbers add to cubes proof". That's what I adapted to come up with the visual proof for Moessner's miracle in the skip three case that I present in this video :)

[@mlmimichaellucasmontereyin6765]

Thanks!

[@Zavstar]

Pascals triangle seems to be everywhere

[@brad682]

This is the first time I’ve encountered Mossner’s Miracle, and it made my day too! Thanks for presenting a heady maths topic in such a way that a layperson can understand! That was really cool!!

[@nenmaster5218]

The Learning never ends, so call it silly, but i do have the hobby of asking people if i an recommend them science-chanenl or just education-channel in general to them! Mind if i do?

[@HxTurtle]

that must just about be the most mind bending video I ever saw. I know almost all 3 blue 1 brown videos, but what's so beautiful about this video is, that it always stays simple enough so that one can follow. I mean, there's no way to come up with all those correlations on your own, but once you get them presented this way, you actually see/understand the connections. that whole video is just about the edge of what I'm still able to follow without ever going over it .. that's why I like it so much.

[@cooleslaw]

Nice, never heard of him, but glad to learn about him now.

[@kshitijdalal4420]

Loved this Video! Always wondered whether there was any research on the addition of odd numbers to get squares. I knew how to get the cubes, but was pleasantly surprised to know that the pattern continues! Made my day, (the week actually). Loved the fact that some of the approaches in the video share a big similarity to the ones in your power sum video, grids and various others (And pascal's triangle!). Also, how you can use geometry to not only visualize, but also solve what is, essentially, a number theory and algebra problem. Gently makes you appreciate how interconnected everything in mathematics is. I hope you can keep making these videos, and that they reach an even larger audience, so everyone can fully enjoy and learn advanced mathematics. PS : Is your geometric proof the first one? PPS : I think, the output sequence might be n!*(n-1)! ? if a =1 and b, c, ... =2?, Since then each term would become n + 2(n(n-1)/2) = n^2.

[@Mathologer]

"I think, the output sequence might be n!*(n-1)! ? if a =1 and b, c, ... =2?" That's it :)

[@ericlester3056]

I love the embedded geometry of integers! Hexagons, triangles, squares from simple patterns.

[@ArnabNature]

I was waiting when you will upload a video. Love Mathologer ❤

[@jmcusack]

I can hear the ancient Greeks yelling, "Mathematics is just geometry, man!"

[@anhhoanginh4763]

i can see one acient Greek was drowned by the others because of some (a/b)

[@smurfyday]

That's a misunderstanding, worsened by the Common Core. There are whole worlds of math better explained without geometry.

[@robinbuehler]

Great video! A few years back I wrote my bachelor's thesis about homomorphic encryption. I would love to see how you explain how it works 🤠 I'm sure the different perspective will amaze me.

[@dantemeriere5890]

Just want to thank you for this incredibly interesting content. I love this channel.

[@DarkCloud7]

This was awesome to watch. I felt my eyes shining.

[@TedToal_TedToal]

Always something new and cool! And in the back of my head I’m wondering about whether there might be a lot of other applications here to turning a multidimensional problem into a 2D problem? Can we use this to help us visualize multiple dimensions?

[@jansacha772]

An hour ago I wondered when Mathologer would uploud a new video. I started to search answers in the whole internet an now he uploads !!! Coincidence?

[@BytebroUK]

I really really like that visual explanation. Thank you!

[@kennethlim8165]

Great video!! I really appreciate your breakdown and your visual representation is very helpful to my understanding. THANK YOU!!!

[@nerouchih3529]

Some of these and others similar you can find in 1st volume of Knuth's "The Art of Computer Programming" with invitation to prove ones

[@sirnukesalot24]

Sum, product, exponent, factorial... how does this pattern keep growing? Is the next step a factorial exponent? Where does product summation fall within this pattern? What do we do with the levels that haven't been named, let alone conceptualized yet? What can we do with them once we've figured that out?

[@EarthAltar]

That's when you build a warp drive engine for interstellar travel.

[@tsawy6]

There are generalisations. One cool one is the Ackermann function, which takes two natural numbers a and b, and if * is the b^th hyper operation returns a*a. It grows exceptionally massive exceptionally quickly.

[@balsoft01]

What you're referring to are hyperoperators. A really cool topic. And all of the hyperoperators have names, IIRC it's -ation, where is the "level" of the hyperoperator. Addition (a.k.a mononation) is the first hyperoperator, multiplication (diation) is the second, exponentiation (triation) is the third, and after exponentiation comes tetration. The hyperoperators are written as their "level" inside square brackets, so a + b is a[1]b, a * b is a[2]b, and a^b is a[3]b. The definition of hyperoperators is kinda hard, but the basic idea is that a[0]b = 1 + b and a[n]b = a[n - 1](a [n] (b - 1)), with a[n]0 being a for n = 1 (addition), 0 for n = 2 (multiplication), and 1 for n >= 3 (exponentiation and higher).

[@6612770]

@sirnukesalot24 I get the feeling that the solution to the Riemann Hypothesis is up there somewhere...

[@mlmimichaellucasmontereyin6765]

@@EarthAltar ...unless you then realize that you no longer need expensive machines to get you across the very dangerous vastness of universal stuff at or beyond the speeds of light [sic]. The speed of mind is infinitely faster (and safer, and cheaper, and Greener).

[@davidandrade5572]

Your content never ceases to impress me 🤯🤯🤯 makes regret changing from pure math to applied math...

[@alifepenneddown]

Mind = Blown! 🔥 What an amazing video for the topic!!!