Mathologer - 2020-10-17
Looks like I just cannot do short videos anymore. Another long one :) In fact, a new record in terms of the slideshow: 547 slides! This video is about one or my all-time favourite theorems in math(s): Euler's amazing pentagonal number theorem, it's unexpected connection to a prime number detector, the crazy infinite refinement of the Fibonacci growth rule into a growth rule for the partition numbers, etc. All math(s) mega star material, featuring guest appearances by Ramanujan, Hardy and Rademacher, and the "first substantial" American theorem by Fabian Franklin. 00:00 Intro 02:39 Chapter 1: Warmup 05:29 Chapter 2: Partition numbers can be deceiving 16:19 Chapter 3: Euler's twisted machine 20:19 Chapter 4: Triangular, square and pentagonal numbers 24:35 Chapter 5: The Ramanujan-Hardy-Rademacher formula 29:27 Chapter 6: Euler's pentagonal number theorem (proof part 1) 42:00 Chapter 7: Euler's machine (proof part 2) 50:00 Credits Here are some links and other references if you interested in digging deeper. This is the paper by Bjorn Poonen and Michael Rubenstein about the 1 2 4 8 16 30 sequence: http://www-math.mit.edu/~poonen/papers/ngon.pdf The nicest introduction to integer partitions I know of is this book by George E. Andrews and Kimmo Eriksson - Integer Partitions (2004, Cambridge University Press) The generating function free visual proofs in the last two chapters of this vides were inspired by the chapter on the pentagonal number theorem in this book and the set of exercises following it. Some very nice online write-ups featuring the usual generating function magic: Dick Koch (uni Oregon) https://tinyurl.com/yxe3nch3 James Tanton (MAA) https://tinyurl.com/y5xj2dmb A timeline of Euler's discovery of all the maths that I touch upon in this video: https://imgur.com/a/Ko3mnDi Check out the translation of one of Euler's papers (about the "modified" machine): https://tinyurl.com/y5wlmtgb Euler's paper talks about the "modified machine" as does Tanton in the last part of his write-up. Another nice insight about the tweaked machine: a positive integer is called “perfect” if all its factors sum except for the largest factor sum to the number (6, 28, 496, ...). This means that we can also use the tweaked machine as a perfect number detector :) Enjoy! Burkard Today's bug report: I got the formula for the number of regions slightly wrong in the video. It needs to be adjusted by +n. In their paper Poonen and Rubenstein count the number of regions that a regular n-gon is divided into by their diagonals. So this formula misses out on the n regions that have a circle segment as one of their boundaries. The two pieces of music that I've used in this video are 'Tis the season and First time experience by Nate Blaze, both from the free YouTube audio library. As I said in the video, today's t-shirt is brand new. I put it in the t-shirt shop. Also happy for you to print your own if that works out cheaper for you: https://imgur.com/a/ry6dwJy All the best, burkard Two ways to support Mathologer Mathologer Patreon: https://www.patreon.com/mathologer Mathologer PayPal: paypal.me/mathologer (see the Patreon page for details)
I am so blown away.. I was never a big math guy though I did use a lot of geometry and right angle trig in my construction life.. But now here in my old age (68) I see the amazement of math laid out before me. The wonder that a few of my fiends had talked about but I could not see.. Oh to take this knowledge back 50 years and do it all over again... What fun it would have been.. Thank you my friend for giving me a taste of the fun and joy my old friends had in their day.. They are gone now but I remember.. thank you!
@tinkmarshino very wise
@tinkmarshino your words are too hopeful, now it seems hard for human race to even get past 2050
@Aarav We have to many distractions my friend.. A simple life is an honest life...
@Aarav we be around for a long time to come.
The older you get, the wiser you become , with ever new understanding you realize how much you have to learn or how little you know! Lucky for me I only seek to learn 1 thing, life (everything) lol. Not just everything but simplified and connected, teachable through concepts and applicable in my reality! That all lol. Luckily for me I didn't have to look far for the answers
🦸🦸♂️🦸♂️⚠️⚠️⚠️😇😇😇
As a 50 year old man, I may have done better in maths if I had teachers like you! Thank you for your simplification of complex maths. :-)
You are welcome :)
aint that the truth brother.....
"We do real math, which means we prove things" *squashes pentagon into a house shape"
29:51 squashes house into a cube and a half (along the diagonal)
Yeah this got me for a bit, but:
The nth triangular number is (n(n+1))/2
and the nth pentagonal number is (3n^2 - n)/2.
This can be written as (3/2)n^2 - n/2
-> n^2 + n^2/2 - n/2
-> n^2 + (n^2 - n)/2
-> n^2 + (n(n-1))/2
Let m = n-1 so we have n^2 + (m(m+1))/2
So the nth pentagonal number is the nth square number plus the (n-1)th triangular number.
Er- perhaps he was just envelope-ing it? We are discussing patterns after all!
The geometry doesn't change though. It still has five sides and the same number of dots and you can see visually that the number of points increases in the same way when lengthening the sides of the pentagon.
12:49 "When is this pattern hunt going to end? Soon, promise!"
40 minutes left in the video
40 minutes is soon, depends on the context
That soon was on a scale of the rabbit hole of mathematics which ironically enough seems to be a diverging infinite sum of new ideas and findings :p
To be fair we did find a pattern, we only continued after we found it
Oh good someone else noticed that too... I'm not going crazy... haha hehe haha... 1,2,3,4,5,... what's next 6? no! 42!! aaaahhhhh!
@ReznoV Vazileski ⁹
Hi Mathologer, could you please explain why the perimeter of the ellipsis has no finite formula?
Best part of this was realizing how I can use the logic to solve 3 of my yet unsolved ProjectEuler problems! Awesome video!
Project Euler? What is that?
It's a very popular (and difficult) library/ competitive coding platform. I haven't got to the point that this is useful yet, but some problems are just crazy.
Love the video, Partition numbers are what got me so interested in OEIS. I was hoping you were going to go into A008284 which is kind of a transformation of Pascal's triangle but spits out the partition numbers.
I find the process and revelations fascinating. But the engineer half of my brain asks, "Is there a real world application for this (yet)?
Every video is so damn interesting and explained incredibly well. Words can't explain how thankful I am to have found a channel like yours.
This is an awesome video! I didn't know this version of the pentagonal number theorem, and it's a lot more intuitive than multiplying out lots of generating functions. Really enjoyed every minute of it.
Pure magic Burkhard. I went though this video in detail with my gr. 9 students this week. Curriculum be damned..! It’s so fun to see them light up when understanding. I hope they appreciate that very intense math concepts are made accessible to math neophytes thanks to your phenomenal animations and eloquence. Very Much appreciated by me at the very least.
I made it to the very end! Love the visual proof as always. My first guess at the second "what comes next" is 1, 3, 8, 21, 55, 144, just the Fib numbers.
Oh yah.. every other Fib!.. heh, I guessed 55 as well -- but way less elegantly than yours. I was thinking 2[p(n)+p(n-1)]-[p(n-2)] ... double the sum of the last 2 numbers and subtract the 3rd last number. Works out to the same thing -- but honestly I cant figure out why it's the same... I gotta ponder that for a bit
This is literally magic, the video kept getting more and more interesting (and complicated) and I more and more amazed
D:
D colon
That would explain why, when I'm guessing he's saying "mathematician", I keep hearing "mathemagician"
314 likes , pirfect!
Dear Mathologer, please explain the connection between partition numbers and Bell numbers in an equally, if not more, pleasing and accesible way! Bell's recursive formula is such a mind-boggler!
I made it to the very end! And I actually followed everything you presented, cuz by the time you got to the p(n)(O-E) setup, I bursted aloud: "Some are zero, and the others will be pentagonal exceptions alternating between 1 and -1!!!" I felt sheepishly proud, but really, it was only obvious because the previous 47 minutes were presented so masterfully by you!
That's great :)
This has blown my mind. This is now my favorite Mathologer video, as I can actually follow along with it to the end.
I like when this guy laughs, he sounds like he really loves what he does and gives good vibes
The hardest "What comes next?" is the year 2020
Absolutely true
@Uncompetative touching to see how optimistic we all were in 2020 to think that was the worst, don't you think?
in 2022 the hardest "what comes next" is russia :D
*2022 but yeah 2020 was the correct answer until 2021, then 2022, whats next lol maybe nothing is next
I made it to the very end. Can't say I fully understand Euler's Pentagonal Formula, but I'm happy to know it exists and that you have visually given me enough to feel I've discovered a new facet of the universe today. Thank you!
Due to covid, I finally managed to have enough time to watch a whole mathologer video :)
Omg I can't imagine me and my friend tried to deduce a closed formula for circle regions, unbeknowst to it is utterly complicated :D, Though it was funny (we actually derived couple of formulas related to the problem) but we failed :D
i always wanted to dig into partitions but never got around to it. Thank you for outlying it and making it so easy to follow! Euler used to be my favourite as well, that dude was amazing.
Good job Mathologer, keep it up
I made it to the very end! I'm glad I did and learned lots and lots and lots of interesting number theory. Thanks Mathologer team!
Lovely mathematics being expounded here! This is perhaps my favorite Mathologer video yet.
19:59 I would never have recognised that pattern in my entire life ,I wonder how does Euler does this ?
Great video! While looking at partition numbers (without repetitions) myself, I noted another nice pattern that that if p(n) is the nth partition number, then k<n appears as a summand in p(n-k) partitions of n.
16:08 - Challenge Accepted:
Firstly, by 666th partition number, do you count the first 1 (from 0) as the first?
If so: 11393868451739000294452939
If 666th is the one associated with 666 then: 11956824258286445517629485
returns in 1 second. Looked a lot cleaner when I thought I could just use ints, but then the numbers got v. big. v quickly.
What do those numbers look like translated to diagram or cymatically?
A fun fact I found on Wikipedia: Jordan Ellenberg cameos in the film "Gifted" (about a math prodigy) as a professor lecturing on the partition function and Ramanujan's congruences. Apparently this gives him a Bacon number of 2.
Thanks a lot for your great content! 👍😁
One question though: What software do you use for the animations?
The problem at the end is extremely interesting. Changing the sum to product is called "norm of a partition" (Sills-Schneider 2019). There are very few papers on this very subject.
Thus, the sum of norms is quite intriguing to ponder upon.
The first time i was studying partition numbers I could only find proofs involving generating functions, which I am not particularly fond of. How and where did you find these brilliant combinatorial arguments? Or are these generating functions in disguise but mathologerized?
Those "complete the sequence" questions are my pet peave. The thing is, any number can continue any sequence, and there will be a formula (a polynomial; actually, infinitely many polynomials) to produce the resulting new sequence. That type of question is routinely used in school tests and intelligence tests, but what it really tests for is a kind of learned bias toward small integers.
@Cristian M I guess that's true, in general only upper bounds of Kolmogorov complexity can be provided.
@Stephen Holland how do you get from 2 to 5?
@Code:A Channel 2 is WBBM, channel 5 is WMAQ, channel 7 is WLS, channel 9 is WGN, and channel 11 is WTTW. These are Chicago area television channel assignments. I’m joking about these numeric sequence questions. The answer varies depending on you knowledge base.
@Tetraedri_ If you only know the first n terms of the sequence, then it must have some description already. Otherwise the original comment applies and there is not a unique answer. Waffling on about finding a shorter programmatic description of an undeclared algorithm has nothing to do with the point being made by WhyCantI. Print "123, 748, 869" may be hard to improve upon but is irrelevant unless you like mathematical mental masturbation. Usually in school they mean the lowest order polynomial with integer coefficients where the sequence is generated by plugging in 1, 2, 3 etc. The teacher never explicitly says this in some form, and quite often also does not give enough terms (numbers of the sequence) for the solution they want to be determinable and unique. I remember the first time I ran into this in a quiz to determine how smart the PhD candidates were for a job. I could not follow the example questions and answers at all and had to ask how they were computed. When explained that they were fitting polynomials of the second degree, they then in the real test asked a series of increasingly complex series. I was using discrete differentiation to find the polynomial coefficents and then backtracking to get the answer. Then they jumped to degree 3 and then 4, etc. (generalizing their own rules) I answered them all easily because my technique was iterative and I just had to compute to the next polynomial degree. After the test they accused me of cheating because no-one had answered them all before. So, yes, I do think "find the next number in the sequence" is mostly BS material.
@h00d b0ii My go to when I need a sequence of numbers with a property. Saves me a lot of legwork, sometimes.
I've come back to this channel after a long time and while watching this video I felt my long lost love for mathematics more and more. Thank you for this.
Amazing Video, as usual. I love how you delve into the details of the proof. Some further motivation on "why partitions are interesting" would be welcome at the beginning of the video, especially for viewers (like myself) whose training is more in applied mathematic, physics, engineering, and computer science. In spite of not understanding the significance of partitions, I followed your reasoning with delight. "If the journey is enjoyable, the destination may be less important"
probably the best and most mindblowing maths video ive ever watched
Really excited to try to read about it on my own
thx for the links in the description :D
I instantly thought of that solution for the sums of positive integers. Very similar to the power set formula and concept.
Dear Mathologer,
Seeing your video this morning has brightened my day so incredibly much. Your videos allow me to transcend my body(have pain) and live in a world of pure mathematics. Please never stop. - Your fan and student.
This video was so simple to understand. Thank you, Mathologer, I like the way you provide such great incites into the heart of the problem.😀
Edit: After again clicking on this video for the second time, I now truly HAVE MADE IT TO THE END. (Did not notice before)
Imagining if such machines exist for primes or other seemingly incalculable sequences is making my mind go wild. It seems the hardest part is finding the machine that works.
Wow, so many connections between the subjects! This is one of your best!
You simply turn math into magic. That's why I truly want to learn ♥️🥺
I just watched the first 5 minutes and have to really compliment the way you present you material. It's inspiring how you structure it in a way that makes it engaging. The "tricking" shows how important it is to really check what's going and that's what math is all about :)
:)
When you mentioned the opening and closing of the gaps, my mind immediately went to binary with their two states of on or off.
@Mathologer, Serious question: How/where/when did you learn about this? I personally hold two degrees in mathematics and nothing I ever studied came remotely close to this sort of material. I am a high school calculus teacher and I love anything unique or niche in mathematics that I can put in my back pocket to show to my students if/when the time arises. But SO much of your content is so unique that I begin to wonder about the differences in mathematics degrees' content in different parts of the world. I wonder if you are a true master of math history and have explored very very deeply the nooks and crannies of the great mathematicians who came before.
Aside from all this, thank you so much for your content. It is such an oasis, especially in the current times in which we live.
❤️
I admire your "ease" of presentation on all videos and topics (and the great visualizations)...
You’re so much fun and it’s so fun to see you have fun with your presentations!
I love you Mathologer. Really. Few other channels dare to dive into such a level of details.
And even when it gets too complicated for a video, we at least get the main intuition.
Love it!
The chapters on this video are incredibly helpful. Sometimes you need to rewatch just a small segment before moving on.
I made it to the very end!
Amazing work, I didn't even know this was 50 minutes long.
Tubluer - 2020-10-17
Mathologer: What does a partition have to do with a pentagon (aside from beginning with "p").
Me: blinding flash of insight They both end in "n"!!!
Elias Cotton - 2021-01-04
@Pez pentagon and partition are both physical objects..
ffggddss - 2021-01-05
@Pez Yes, that's included in the match/search strings we've been batting back & forth here...
Fred
NotWearingPants - 2021-02-14
If the similarity is in terms of both matching the same regex, then we might as well just /(partition|pentagon)/
Poquihuf - 2021-12-14
Yay 1000th like
Eduard Rutetskyy - 2022-03-17
OMG ARE YOU SOME ILLUMINATI GENERAL OR SOMETHING????? xD