I like how the final graph, which was described as the Alps, was actually used in the animation following it. The quiet enthusiasm of Neil Sloane is very enjoyable.
While playing around with the OEIS myself, I actually discovered that the Stern-Brocot Sequence makes genuinely quite good music when you use the "listen" feature.
4:49 the way he whispers *"Graaaph"* ... With such a calming and peaceful voice. I imagine myself finding inner peace in a beautiful natural environment somewhere in Nepal at a valley on a small island in a lake, with only the sounds of birds and the silent water... And as I am about to fall asleep, this old man comes by and whispers "Graph" into my ear as I fall deeply asleep
Honor to the amazing Neil Sloane, to Douglas Richard Hofstadter for his contribution to mankind represented by "Gödel, Escher, Bach: An Eternal Golden Braid," and to Brady Haran for letting me know such pure geniuses! From my heart thank you all for making my life even more beautiful and enjoyable! Brady go ahead!
The last one looks strikingly similar to Sierpinski triangle, which can be obtained by bit-wise AND-ing the x and y coordinates and marking null results on the plane... (a & b)==0 is essentially telling "a and b have no overlapping bits" you explained verbally in the last part.
This episode is pure delight! I love the graphs, espeically the "alps" at the end. We need more of the amazing and interesting sequences to fill our heads with morning smiles and wonder.
Q Sequence: If a(n) ~ n/2, for all n < k, then: a(k-1) ~ k/2 a(k - a(k-1)) ~ k/4 and a(k-2) ~ k/2 a(k - a(k-2)) ~ k/4 , so a(k) = a(k - a(k-1)) + a(k - a(k-2)) ~ k/4 + k/4 = k/2 So, if the "ribbon" has slope 1/2 for the first few terms, it follows that it will for every term. So, if it has slope 1/2 for the first few terms, it will never die!
Neil sloane is my new favourite numberphile appearance :) I can listen to him for hours! Not that I understand 90% of what he says, but I still like to listen :)
So sad the series ended! Really liked the concept and really liked the mathematitian. It's so cool that there are so many sequences showing fractal behavoir when n is large enough!
1:49 Actually looks like the border of a pine tree forest, with the little pines growing from the seeds of the larger pines in the interior as time goes by. An expanding forest.
I love the way his desk is (dis?)organised In particular that the books are labelled on the pages rather than the spine and that there is a Mac sitting precariously on a stack of books where one is labelled "UNIX"
At 7:45, the snow of the alps look like bent Sierpinski triangles. If you tweak the Sigrist series, could you possibly get an array of Sierpinski triangles?
Please please please Grady, do some more graphs. These are the most interesting and satisfying series(:P) to watch. All the other content is great, but the graphs of these sets are extremely satisfying to see :D
The angle of the ribbon graph at 5:47 minutes in this video is 26.56505118 degrees. This angle is half of the 3, 4 corner angle that the 3,4,5 Pythagorean triangle (47th problem of Euclid) forms. Set the vertical units on this graph to the same scale as horizontal axis and you will see it. This is a "divine" angle and can also be found in the Freemason Compass and the Chinese Ying Yang duality circle. It's too bad he said that he wasn't going to share the equation for the proven ribbon graph.
the unknown sequence almost definitely will go on to infinity, because every term is smaller than n, for lack of a better phrase. theyre all assembled from >n, and all the number >n are by definition smaller than n, and they themselves are assembled from smaller numbers. Like he said, you dont get to N, or even close, theyre all close to half n. No set of numbers you add together can be greater than their own sum. If you add all the integers you might accomplish a supertask and reach some type of infinity, but you can always just add one to it, so no matter how many terms you add together, they dont ever become larger than infinity. the ratio to how far back you go for the two things you add remains the same, it is bounded by upper limits, because as it shows, you eventually reach a point where you dont go any further back, or forwards, before ballooning out again. Its form is dictated by its ratio, and its ratio is bounded, its irrational, but it is the sum of rational numbers. It cant EVER become larger than itself. No sum is larger than itself. As the numbers become larger, you add smaller and smaller numbers together, you might go further back, but then it becomes a smaller number to add together, and those small numbers guarantee that the NEXT term will be smaller than n, so, the sums might get bigger to a certain point, but then you go so far back, you find the small numbers again, and only add small numbers together. There is a reason it looks cyclic, because you can only grow the number of hops back you go so far, because eventually that means you add together SMALL numbers, and then dont go very far back. In fact, i think i can prove it. every term is n+1 greater along than the last one. But the smallest increment you can add is 1, so you cant ever add an increment greater than 1 to the step n. If you go back so far that you reach the first or second term, you just add 1. but then the NEXT term, is n+1, meaning even if you go n terms back, you find 1 again. It MIGHT get stuck in a loop of only adding 1 to itself forever, but it wont ever get into undefined, because it cant grow larger than 1 step at a time.
4:46 "Pretty obviously it's never gonna happen (...) It hasn't happened, and it's not gonna happen" Isn't that quite a strange thing to claim without a proof? Maths have surprised before, on conjectures that seemed "obvious".
Olof Andersson, agreed, especially with the first episode’s ‘fly straight, damn it!’ graph kept in mind - behaviour can abruptly change - sure, it *probably* won’t happen, but it could
@@jamesimmo True, but this graph is sort of self regulating. It grows at N/2, but to escape you need to have offset of larger than N. For that to happen, you'd have to pick two numbers that have average value larger than N/2. And if you have a neighbourhood where you have large offsets, then those large offsets make you pick numbers from near the start of the sequence where the numbers are small, which makes future offsets smalller. So even though it's not known to be impossible, the nature of the sequence makes it extremely unlikely to escape far from the median N/2 values and the further you go you have more of the sequence to land on, so hitting those two deadly numbers that cause you to get offset >N becomes less and less likely. So even though it's not proven, I'd say it's pretty strong conjecture.
Watch the full Amazing Graphs Trilogy (plus an extra bit): ruclips.net/p/PLt5AfwLFPxWLkoPqhxvuA8183hh1rBnG
It says "Invalid parameters" when I click that link
@@esotericVideos Same with me. I think there's something wrong with the link.
@Numberphile The link is broken. It says "Invalid Parameters" and shows an error page
this man is the best. love the sequences
Let me guess, you forget to change the status to public. Then you have no problem yourself.
Petition for a "Graph of the Week"
YES!!
All in for it!
Where do I sign
SIGNED
Yes
I wouldn't mind if it will become a new regular series.
MajkG MajkG PLEASE THIS!!!
Neil would have to add it to the OEIS.
Pun?
Love.
Literally said “ohhhh” out loud at that last one, that was certainly my favorite
fractal patterns are (almost) always pretty
??
@@Triantalex what’s wrong
I like how the final graph, which was described as the Alps, was actually used in the animation following it. The quiet enthusiasm of Neil Sloane is very enjoyable.
That was my favorite too
Keep ‘em coming because these graphs are funny. Excellent work by the way.
Inject this directly into my veins.
Inhale the graphs, exhale the graphs
:)
??
This Graph series is turning out to be one of the best series on Numberphile.
While playing around with the OEIS myself, I actually discovered that the Stern-Brocot Sequence makes genuinely quite good music when you use the "listen" feature.
I have a dream. I submit an interesting sequence to the OEIS. Neil comments this sequence of mine in a Numberphile video. That's it.
That’s a grand slam.
We need a graphophile channel dedicated to uploading cool graph every week
Make Sequencephile, starring Neil Sloane please, or keep them coming here. Either way, thank you amazing appreciated.
This guy is amazing, the graph stuff is like mathematical ASMR.
I love these. It feels like productive ASMR.
4:49 the way he whispers *"Graaaph"* ... With such a calming and peaceful voice. I imagine myself finding inner peace in a beautiful natural environment somewhere in Nepal at a valley on a small island in a lake, with only the sounds of birds and the silent water... And as I am about to fall asleep, this old man comes by and whispers "Graph" into my ear as I fall deeply asleep
😅😅😅
Please keep this series up!
"Adam Savage" in the Patreon supporters. Wonder if it's the best known one.
Indeed it is.
I never met the dude.
Creator of Minecraft Markus Persson Notch is a 3blue1brown supporter
@@philipabelanet5476 do you know?
@@wasdwasdedsf He's mentioned being a fan of math & science channels.
Prof Sloane's graphs make my day.
He is NOT a professor, certainly not anymore
@@yvesnyfelerph.d.8297 ye lol he's 80 years old
@@leo17921 And doesn't look a day over 60.
1:35 This kind of reminds me of the Pythagorean triple tree, constructed out of the Fibonacci sequence. Mathologer made a video about that one.
7:53 Do the flags which have numbers, are they actually in order of the sequence they just talked about? Nice easter egg, Numberphile ;D
Honor to the amazing Neil Sloane, to Douglas Richard Hofstadter for his contribution to mankind represented by "Gödel, Escher, Bach: An Eternal Golden Braid," and to Brady Haran for letting me know such pure geniuses! From my heart thank you all for making my life even more beautiful and enjoyable! Brady go ahead!
Thos mans excitement for numbers is so infectious. He could talk about toilets and make it fascinating.
I cannot explain why but I have loved this series - thanks!
7:52 is a Sierpinsky triangle thingy (like most binary recursive series seems like)
These three videos have been some of the best on Numberphile. I'd absolutely love to see more
I love this series so much, keep it up!
I see what you did there lol
The one is amazing. I love how a number sequence can translate into nature so well.
The last one looks strikingly similar to Sierpinski triangle, which can be obtained by bit-wise AND-ing the x and y coordinates and marking null results on the plane...
(a & b)==0 is essentially telling "a and b have no overlapping bits" you explained verbally in the last part.
You cannot possibly understand how happy I am to have this video I was so sad when I watched the second and this wasn't out yet
I was looking for more content about graphs and sequences by this guy. His voice is so pleasant to listen to. More please!
The Alps graph has a bit of Sierpinski triangle kinda look to it
Yeah. A bit spooky.
Neil might be my favorite of them all. He gets so excited and I like that
I can't believe equations are better at drawing than me
to be fair, a lot of shapes and their combinations are just equations or series thereof
Neil Sloane's OEIS graphs and strange sequences are my favorite videos.
2:32 "well... I guess..." said all bashful... I love it.
This episode is pure delight! I love the graphs, espeically the "alps" at the end. We need more of the amazing and interesting sequences to fill our heads with morning smiles and wonder.
Wow, I love so many Numberphile presenters, but Neil Sloane has really rocketed toward the top among my favorites. He has an artist's soul.
Q Sequence: If a(n) ~ n/2, for all n < k, then:
a(k-1) ~ k/2
a(k - a(k-1)) ~ k/4
and
a(k-2) ~ k/2
a(k - a(k-2)) ~ k/4
, so
a(k) = a(k - a(k-1)) + a(k - a(k-2)) ~ k/4 + k/4 = k/2
So, if the "ribbon" has slope 1/2 for the first few terms, it follows that it will for every term.
So, if it has slope 1/2 for the first few terms, it will never die!
These videos inspired me to recreate all these awesome graphs with Python. So much fun!
Great to hear.
Please don't stop making this series I love it so much
5:30 that reminds me of note frequencies when talking about relations between them, for example an octave is 1/2 or 1:2, etc.
Neil sloane is my new favourite numberphile appearance :) I can listen to him for hours! Not that I understand 90% of what he says, but I still like to listen :)
4:24
I'd like to rename this list to "counting badly" (or slowly)
YEAH
MORE GRAPHS
Brady: So this is the third and final episode of
NOOOOOOOO
So sad the series ended! Really liked the concept and really liked the mathematitian. It's so cool that there are so many sequences showing fractal behavoir when n is large enough!
I need more of this! Just shows you how beautiful math is.
Imagine being in a dark room and hearing this guy whispering his favorite sequences
In the last graph i saw Sierpinski triangles sweeping to the right (where the "mountain shadows" are). How interesting! 7:44
This might be my favorite playlist on RUclips
At 1:52 looks exactly like Sagrada Familia, take a look!
These Neil Sloane videos are the best
I genuinely enjoyed the last two videos so much that I got excited seeing this notification
1:49 Actually looks like the border of a pine tree forest, with the little pines growing from the seeds of the larger pines in the interior as time goes by. An expanding forest.
I love the way his desk is (dis?)organised
In particular that the books are labelled on the pages rather than the spine and that there is a Mac sitting precariously on a stack of books where one is labelled "UNIX"
This sequence of videos on the graphs has been really amazing!
At 7:45, the snow of the alps look like bent Sierpinski triangles. If you tweak the Sigrist series, could you possibly get an array of Sierpinski triangles?
More Neil, please. He’s always interesting and he’s a very nice person!
I would watch these graph videos forever
YES!! My favorite video series on RUclips continues!
I really, really, really like to listen Mr. Neil Sloane! More pleaaaase!
Please make more! I love Neil Sloan and the videos are so interesting
Love the editing
And the way he says graph
Yeah!!!! This series gets me HYPE!!!
Fascinating. Dunno why but it feels like we're exploring the fabric of the universe or something by analysing how these sequence behave.
Cool!
0:01 Stern's Sequence
1:56 Hofstadter's Q Sequence
5:40 The Chaotic Cousin
6:08 Remy Sigrist
That Hofstadter and his level-crossing feedback loops!
Here's a thing: Hofstadter's series is non-primitive recursive but functional computable.
"ill put down a 0, no-ones is going to object to that"
The subtle sarcasm is strong with this one.
We need sooo many more of these! Great stuff. Wonderful. Amazing.
this is my favourite channel to watch at 3am
Please please please Grady, do some more graphs. These are the most interesting and satisfying series(:P) to watch. All the other content is great, but the graphs of these sets are extremely satisfying to see :D
PLEASE don't end this series!!!
So interesting! And Neil Sloane makes my day every time he’s Numberphile. So - could we have some more, please?
Thank you for these amazing joyful videos!
The angle of the ribbon graph at 5:47 minutes in this video is 26.56505118 degrees. This angle is half of the 3, 4 corner angle that the 3,4,5 Pythagorean triangle (47th problem of Euclid) forms. Set the vertical units on this graph to the same scale as horizontal axis and you will see it. This is a "divine" angle and can also be found in the Freemason Compass and the Chinese Ying Yang duality circle. It's too bad he said that he wasn't going to share the equation for the proven ribbon graph.
Neil seems like a real mellow fellow on the surface but lurking just below is a real excitement about sequences.
the unknown sequence almost definitely will go on to infinity, because every term is smaller than n, for lack of a better phrase. theyre all assembled from >n, and all the number >n are by definition smaller than n, and they themselves are assembled from smaller numbers. Like he said, you dont get to N, or even close, theyre all close to half n.
No set of numbers you add together can be greater than their own sum. If you add all the integers you might accomplish a supertask and reach some type of infinity, but you can always just add one to it, so no matter how many terms you add together, they dont ever become larger than infinity.
the ratio to how far back you go for the two things you add remains the same, it is bounded by upper limits, because as it shows, you eventually reach a point where you dont go any further back, or forwards, before ballooning out again. Its form is dictated by its ratio, and its ratio is bounded, its irrational, but it is the sum of rational numbers. It cant EVER become larger than itself. No sum is larger than itself.
As the numbers become larger, you add smaller and smaller numbers together, you might go further back, but then it becomes a smaller number to add together, and those small numbers guarantee that the NEXT term will be smaller than n, so, the sums might get bigger to a certain point, but then you go so far back, you find the small numbers again, and only add small numbers together.
There is a reason it looks cyclic, because you can only grow the number of hops back you go so far, because eventually that means you add together SMALL numbers, and then dont go very far back.
In fact, i think i can prove it. every term is n+1 greater along than the last one. But the smallest increment you can add is 1, so you cant ever add an increment greater than 1 to the step n. If you go back so far that you reach the first or second term, you just add 1. but then the NEXT term, is n+1, meaning even if you go n terms back, you find 1 again. It MIGHT get stuck in a loop of only adding 1 to itself forever, but it wont ever get into undefined, because it cant grow larger than 1 step at a time.
I notice the "Remy Sigrist" sequence looks very similar to the "Balanced Ternary" sequence from a previous video.
Reminds me of Zelda!
Really liked this series - not as difficult as other topics but very satisfying.
Hopefully there will be more someday
5:40 is a fractal, with the second left quarter half the size of the right half and so on
Neil Sloane is such an inspiration.
"third and final of the trilogy" 😭
More please. These are the best parts of math imo
I could watch these all day!
This series could replace every future upload on Numberphile and I wouldn't even be that mad
I actually audibly gasped I was so excited to see this!
4:46 "Pretty obviously it's never gonna happen (...) It hasn't happened, and it's not gonna happen"
Isn't that quite a strange thing to claim without a proof? Maths have surprised before, on conjectures that seemed "obvious".
Olof Andersson, agreed, especially with the first episode’s ‘fly straight, damn it!’ graph kept in mind - behaviour can abruptly change - sure, it *probably* won’t happen, but it could
@@jamesimmo True, but this graph is sort of self regulating. It grows at N/2, but to escape you need to have offset of larger than N. For that to happen, you'd have to pick two numbers that have average value larger than N/2. And if you have a neighbourhood where you have large offsets, then those large offsets make you pick numbers from near the start of the sequence where the numbers are small, which makes future offsets smalller. So even though it's not known to be impossible, the nature of the sequence makes it extremely unlikely to escape far from the median N/2 values and the further you go you have more of the sequence to land on, so hitting those two deadly numbers that cause you to get offset >N becomes less and less likely. So even though it's not proven, I'd say it's pretty strong conjecture.
Favorite videos of recent times! Thanks :D
I absolutely love these videos. More please!!!!
Aha, 7:44 the ALPS, I wanna climb it. & 7:55 funny skating👌👌👌
Keep going
Everytime I watch the graph videos I get reminded that Maths can be beautiful at times
4:50 fibonachi exists in negative values, and so we wouldn't have any problems if we needed the -3rd number of the fibonnachi sequence, for example
This is like educational ASMR. This man has invented a way to stimulate all the hemispheres of the brain at the same time.
We need more than just a trilogy.
We’ll see what we can do.
My favorite graph is simply plotting how many iterations a number x goes through the Collatz Conjecture to reach 1
These videos are phenomenal, please do more of these!
This is professor Farnsworth from futurama come alive in a slightly less crazy form
I see amazing graphs, I thumbs up. Six seasons and a movie please!
I love this serie... And the series that are presented!
Sloane is amazing as always
Creates that type of fascinating mentality for maths