343867 and Tetrahedral Numbers - Numberphile
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- Опубліковано 20 тра 2024
- Featuring James Grime. Bug Byte puzzle from Jane Street at bit.ly/janestreet-bugbyte and programs at bit.ly/janestreet-programs (episode sponsor) --- More links & stuff in full description below ↓↓↓
Dr James Grime discussing triangular numbers, cubes, pentagonal numbers, hexagonal numbers, tetrahedral numbers and Pollock's Conjecture.
James Grime: www.singingbanana.com
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Thanks to viewers for helping find the 343867 sums, including Arne, Alex, Sam, Felipe, Pablo, Ewoud and Michael. - Наука та технологія
Bug Byte puzzle from Jane Street at bit.ly/janestreet-bugbyte and programs at bit.ly/janestreet-programs (episode sponsor)
120 is also a triangular number that I am using in Pyramid Chess, a pyramid of 120 hexagons.
seems more like a bean dish puzzle!
can anyone explain this differently?
"There exists a non-self-intersecting path starting from this node where N is the sum of the weights of the edges on that path. Multiple numbers indicate multiple paths that may overlap." Not quite catching how it relates to the numbers in the graph
@@ChrisTian-uw9tq You can follow any path and choose when to stop. The edge weights you pass (not the nodes) need to sum to the number (or one of the numbers) of the dark green node.
@@Artaxo Then how is the pre-populated 31 meant to have its following edge filled to sum to 31 if max number allowed is 24?
The stop motion is georgious. Appreciate the effort
By our man Pete 👍🏻
@@numberphile Thanks Pete! :)
@@numberphile Thanks Pete! :)
Pete ftw
Wait you didn’t just put an overhead camera on top of James’ paper and let him slowly move all the dots around then edited out the hands?
Whoever animated this episode, you earned your paycheck.
Reminded me of an episode of Gumby.
A big applause for all the stop motion inserts and the clay balls and the discs! Wow ❤ I adore the clay Bollocks run Pollocks 😅
I wonder if it's actually stop motion or if it was just made to look like stop motion (like the Lego movie).
@@sergio_henriqueall real, moving little things around and taking photos
I've come to the comments section to write how happy I am to see Dr James Grime again on Numberphile and how much he's been missed, but I see everyone's done the same thing already!
Very happy to see Dr Grime back on numberphile!
For reference Lagrange actually proved any number is the sum of four squares. Which is why it is usually called Lagrange's four-square theorem.
I loved that game when I was in grade school
6:34 The Fermat-Haran Conjecture 😀
Gaus and Euler, the people who took a look at mathematics and went "that s***'s boring, but I can fix it."
The latter being the guy who gave us the base of the natural logarithm and the formula: e^πi + 1 = 0.
@@JamesDavy2009
Honestly the two were so important that listing any one thing they did as an example feels like it can only ever understate their contribution.
Even that formula is just one example of an expression that drops out of what is an entire mathematical framework that Euler pretty much constructed from scratch, and that entire framework is just scratching the surface of his contributions to mathematics.
@@JamesDavy2009 Many things in math are named after the second person who discovered them, because the first person was always Euler.
8:34 “I said «Pollock’s», you’ve heard me quite distinctly.”
😂
The only mathematician owned by a dog
Sounds like ☝️
Was he trying to make sure that people weren't mishearing him as saying "bollocks"?
@@shruggzdastr8-facedclown I think so. It took my a while to realize that since it isn’t used as profanity (or really at all) in my dialect of English.
Pollock's conjecture is bollocks.
Or, alternatively,, Pollock's conjecture is the dog's bollocks.
James is really Mr. Numberphile =D
I haven't watched the video yet, but I'm very excited about the combination of Numberphile, James Grime, and a specific large number.
A perfect storm
@@numberphile Superior highly perfect storm 😉
EYPHKA! Delightful historical coincidence that you can still write this Greek word with Latin characters
ЕВРИКА
ΕΥΡΗΚΑ is not EYPHKA 🙂
@@jlljlj6991 I see what you did there
@@jlljlj6991 oh don't go splitting hairs
@@jlljlj6991I can’t tell the difference
I always remember triangular and tetrahedral numbers because of the song 12 Days of Christmas. If you interpret the lyrics as listing all gifts up to that point (including previous days), then the running total of gifts is the first twelve triangular numbers. If instead you interpret it as listing the gifts for only that day (i.e. the gifts from all previous days are given again, leading to, e.g., 12 partridges in 12 pear trees) the running total of gifts is the first 12 tetrahedral numbers.
I think having that damn song stuck in my head in class was the reason I worked out the tetrahedral formula.
The animation / stop-motion is looking smooth as heck
Always waiting James' videos❤
Same 😅❤
That joke about Fermat's margins is so granular, and I'm 100% here for it
Parker squares are the 21st-century version of Fermat margins.
Fun fact: 2024's the only tetrahedral year all our lives~ And there's a book all about triangles coming out later this year! Seems like a triangle-y type of year~
True, but 2024 is also the only year we'll live through that is also a dodecahedral number and the first one since 1330. Every (3n + 1)th triangular number is the nth dodecahedral number.
That IS a fun fact!
For anyone curious 1771 was the last one and 2300 will be the next!
2024 is also the only dodecahedral number year we'll live through.
GRIIIIIIME
I MISSED YOU, MAN
welcome back, singingbanana!
Fr
This guy makes maths ALOT more fun than when I was in school.
What wonderful video! As usual, perfect presentation by Mr. Grime and a generally very interesting topic 🤗
Thanks so much. 🙏
Glad you enjoyed it! Cheers.
I watch your videos, I don't understand anything about numbers, but I like your enthusiasm and your healthy joy, greetings from Chile
I love maths! James adores it.
Good to see James Grimes again!🌞
My favourite banana! 🍌
I love that Gauss uses the same asterisk I his writings that I overuse today.
✺✺✺ I switched to the Sixteen pointed asterisk 😄 ✺✺✺
Classic Numberphile with the OG presenter! ❤❤❤
I'm happy to see James again, being guest in other channels. Hopefully he'll upload new video in his own. 🤞🏼
Glad to see James Grime again!
Great to have you back on Numberphile, James, and thanks for the video! And congrats on the ring 😉
2:54 'try and go even further' sounds a lot like 'triangle even further' lol. was that intentional?
How did I miss a James Grime vid!
First things first: Click like!
Now let’s watch what this video is about…
James' closing comments are spot on. I was in high school (late 1980s for me; my brain is very middle-aged now) when I found the pattern of adding consecutive odd numbers to generate the square numbers, and then I figured out that the Nth level difference between consecutive N-dimensional numbers was N! (N factorial)... it's easiest to see this with the square/odd numbers, in which adding 2! starting at 1 generates the odd numbers. I found some hiccups in the first few iterations at each new power, but in general the pattern normalized at N^N.
Why why WHY is this so fascinating? It should be complicated, abstract and boring but it's interesting as heck and I don't know why
I like Brady's proof by pronouncement.🎉
Nice!
I love (that) stop motion!
I love James' little speech at the end of this
Amazing video as always, I’m glad with the stop-motion, can’t imagine how much work it took to make
i've used triangular numbers to verify if a group of unique integers (in any order) was a gapless sequence or not. i was goofing around with some very basic arithmetic and i kept getting results that were oddly familiar. they turned out to be triangular numbers! around this time i had just been introduced to triangular numbers from numberphile!
my specific use case was to determine if a set of years had gaps in it. turned out that there were much easier ways for me to do this programmatically with code, but i'm still proud of having such an epiphany as a non-mathematician.
i have a working demo and explanation that i can link to, but i don't want this comment to go to spam jail!
basically, the formula is this: `(max(set) * length(set)) - sum(set) = T(length(set) - 1)` where `T(n) = (n * (n + 1)) / 2`. `length` is the amount of entries in the `set` of unique integers.
It is a cool find and definitely works assuming the integers are unique, however, if you know the maximum you probably also know the minimum and thus max(set) - min(set) = length(set) - 1 is likely easier to check.
@@benjaminpedersen9548 lol of course i was overthinking it! it's funny because i did think of something like this but i must've forgotten to -1 from the length before i derailed and went on this magical journey. also, i almost immediately found another way to do this leveraging the native features of the programming language i was using. i ended up not using my original idea at all. but i won't let that take away the epiphany i got from this "discovery", however useless it may be. 🤣
thank you for the simplification!
Numberphile is extra special with Dr. James.
Numberphile's vid editor is probably my favorite person in the world that I don't know
I think it makes sense to me that it doesn't require more than n n-gonal numbers. Here's my hand wavy intuition/psuedo-proof:
Lemma: any sequence of n-gonal numbers starts as "1, n, ...". This is almost by definition: you start with 1, then add as many red checkers as it takes to make n sides. Of course, that's n checkers total. So the second number in the sequence is n.
So now let's just keep adding checkers (start with 1, then 2, etc.) to see how to arrange them into at most n n-gonal numbers. If we add 1 checker, it might take 1 more n-gonal number. If we add 2, it might take 2 more n-gonal numbers (a 1 and another separate 1). Once we get to adding n more checkers, then it only needs 1 more n-gonal number, because those extra n checkers can be arranged into 1 "pile" (the lemma). So this shows that every n new checkers we add, it sort of collapses back down to one extra pile.
Of course, that alone doesn't necessarily mean the "collapsing" keeps it under n piles *forever*, but it's some sort of intuition. I wonder how close this is to the real proof, if at all
Wow, I just noticed that for the square numbers you used square waves and so on. Pretty nice touch!!
I noticed that but i forgot what they were called :)
Ive never been this early to a numberphile
me neither
Welcome to the party
same
today I was reminded about figurate numbers and went to read more about them. and now you release a video :D love this coincidence!
The animations are great, but the synth effects i liked even more. Reminded me of those VHSes math teachers might put on in the 90s showing weird math ideas.
It’s incredible that to this day, every episode gets its own special animation to make visualize the lesson in a delightful way. Stop motion!! Brady you animate so well!
What a beautiful video. Thank you.
I really appreciate the precision with saying (every time) that any POSITIVE WHOLE number :)
My favorite tetrahedral number fact: The numbers along the finite diagonals of the multiplication table sum to the tetrahedral numbers. 1, 2+2, 3+4+3,4+6+6+4, ...
thank you so much for your kindness and information
Your method and solution are so intresting!! Wish you the best 🙂
Loving the sound effects!
Has a very 70s animation vibe (or thereabouts) ✨
This video, more than any other, reminded me of a Sesame Street episode brought to us by the number 343867.
I don't know exactly why but this is the most beautiful fundamental proof I've stumbled upon in Mathematics thus far. Thanks so much for making this video!
Oh Cauchy, always ruining my life by being a better mathematician than I could ever dream of aspiring to be
Great vid!
James is the best
Great animation. The kind of thing that hooks the kids.
The Katamari speaking sound effects are perfect
4:19 Funny: The first way that occurred to me was one you didn't mention. Seeing as 4|28, I divided it by 4, getting 7=4+1+1+1, then enlarged, getting 28=16+4+4+4.
Gosh i love this channel !
And we love people who love the channel :)
That stop motion was pretty sweet!
The difference between the same (in order like the 5th pentagonal and the 5th hexagonal) pentagonal and hexagonal number is a triangular number and then the difference between the next pentagonal and hexagonal numbers is the next triangular number
Same goes for square and pentagonal
Triangular and square etc
Very interesting
James is back!!!
I love the episodes with connection to Geometry.
I love how we can shine a light on an arbitrary number like 343,867 with this channel
Also always great seeing James in a video!
Love that they still used the brown paper
Just when we thought we had seen everything, Numberphile comes up with yet another 👍
I have encountered a lot of content on this channel where people have checked a conjecture up to a very large number but with no proof,
i think it would be rather more useful to learn about all of the anomalies unproven conjectures which even after checking it up to very high numbers
would eventually show something unexpected.
Knowing about all of the anomalous unexpectancies would give one a good head start approaching any new theories.
Cool. I solved the bug byte puzzle. Took me about 2 hours, but it was fun.
Did you use a computer?
11:33 this is exactly why I started working on the Collatz conjecture. I knew I'd learn a lot by thinking about the numbers and how they connect, and I was right.
🤯
this was an excellent one
Lovely video that reminds us all why we love math!
Also, please come to Toronto when you can, there's pretty fun math happening here :)
Now do it for all 4-dimensional pyramid-numbers 😄
hyper numbers
The stacking sound effect is adorable
Getting a new Singingbanana and a new Engineerguy video in one day, nay, within an hour of each other is *crazy*
Give that man a wider margin!
As the number of people mentioning they are happy to see Dr Grime back approaches TREE(3), I'll just add my contribution!
Given that you seem to need at most 5 tetrahedral numbers to construct any number and at most nine cubes, it would seem that one would in general need at most n+1 3D-numbers to construct any number, where n is the number of vertexes the 3D-number has.
I’m a simple man, I see james, I click 🙌🏻😹
0:40 "Do you know who else loves triangles? Matt Parker, because they're not squares"
"Every triangle is a love triangle if you love triangles"
- Pythagoras...probably
They're Parker squares . . . only off by one vertex.
I imagine the way to prove the conjectures is through the jumps between singles. So for example with the triangle ones the jump from 1 to 3 is 2, 3 to 6 is 3, 6 to 10 is 4, 5 the next, 6 the next, you get the picture. Presumably the numbers between will only refer the the Ngonals that came before.
there's nothing like James Grime in a Numberphile video
It might not be the main point of the video, but, I am really enjoying the sounds in the animations. Assuming these where created by the animator, this is really classy sound design, very buchla-esque / synthi etc. It really adds to a great video; always good to see James Grime. Like +1
5:20 also that was my conjecture :)
1. So if we name triangle-, square-, pentagonal- etc numbers as "plane" numbers;
2. And we have proof that we can write any whole number as sum of 1n of n-numbers for "plane" numbers;
3. Also we can name tetrahedral-, cube-, dodecahedral- etc numbers as "volume" numbers;
Could there be relation between shape of plane and quantity of planes to describe how many "volume" numbers we need for different shape of volumes?
Or something further beyond: relation between quantity of planes and volumes, and shape of these planes and volumes for description of "hyperspace" numbers?
My favorite tetrahedral number is 4060. It is the 28th, and it is exactly 10 times bigger than the 28th triangular number which is 406. And 28 itself is a triangular number
Also, the digits of all these numbers each add up to 10
That stop go animation must have taken ages to do. Good job Brady and his elves
Great animation
He's back!!!!!
Classic numberphile!
Yesterday on CTC, Simon added triangle numbers together. Is it planned or coincidence that you uploaded this video about tetrahedral numbers today?
Using 1 seems like a cheat
underrated
nice physical animations :D
Cracking the cryptic mentioned a few days ago the concept of tetrahedron numbers. Nice coincidence 😃
James is great. :)
Fascinating, cool sponsor too :P
I was wondering if "Any number can be written as N N-gonal numbers" is optimal? IE, for all N, do there exist numbers (for that N) such that *require* N N-gonal numbers? Or are there some N for which you can do better than N N-gonal numbers?
What a coincidence, one of yesterday's videos on friends of the channel Cracking The Cryptic involved a puzzle where the solution path touched on tetrahedral numbers!
And the triangular number for nine appears in almost every video lol.
@@jimi02468 shh that's a secret
For the longest time I thought Grimey was the host of Numberphile as he was in so many videos, until I saw Brady!
I see Dr Grim, I upvote.