In the first graph (base 3), it does look likes the triangular pattern i have seen seen somewhere else......maybe fractals.....can you make such similar pattern from similar kind of definition of (other)functions.....I would love to see a dedicated video on such functions giving rise to these pattern/fractals.
There is an UA-cam channel, called "Experimental mathematics", where Neil Sloane has some very interesting presentations on integer sequences. I highly recommend checking it out.
It's all the more ironic since he specifically mentioned fractals later for the forest-fire series (but not for the wisteria series which also looked like a fractal).
Cool thing i found (probably was already found but i did it anyway): You can do the first sequence with any base if you replace all non-zero even digits(d) with -1*d+1 It gives you the same fractals but with any length of 'arms' on a V shape.
@jj zun It kinda does. It will still produce a feactal shape. But with my method you get a nice V shape. With other rules you will have an asymmetrical (but still repeating as a fractal) shape.
1:13 It should be noted that the column labelled Balanced Ternary isn't actually Balanced Ternary. The Balanced Ternary (where T conventionally denotes -1) for decimal numbers is as follows: 1 is 1 (= 1*1) 2 is 1T (= 1*3 + 1*-1) 3 is 10 (= 1*3 + 0) 4 is 11 (= 1*3 + 1*1) 5 is 1TT (= 9-3-1) 6 is 1T0 (= 9-3+0) 7 is 1T1 (= 9-3+1) 8 is 10T (= 9 + 0*3 -1) etc Similar to using Roman numerals, you write 2 as "three less one". What is described in the video is a mapping in which numbers are written in ternary, partially "bit-flipped" with 2s turning into -1s, and then rewritten in decimal. It's a perfectly fine thing to consider and explore, but it isn't Balanced Ternary.
@@williamhautekiet9061 That's what I put! I wrote it incorrectly at first and then corrected it immediately afterwards to be as you say, 5: 1TT. Are you still seeing my original, incorrect version?
OH! Thanks a lot! As someone who use balanced ternary quite often, I was really sad to see he's not talking about the real balanced ternary here, without even saying that that's not the real one! Thanks for bringing some precision here
What strikes me is that the StarWars sequence is a way of seeing the Sierpinski Triangle *from below*. Whenever you start building a fractal, you start from low detail to high detail and there is no meaning to ''the highest possible detail'. Yet here the graph expands outwards, as if you started to draw a Sierpinski Triangle backwards!
The "Starwars"-graph looks actually like a modifyed version of that triangular fractal(1.divide a big triangle into four smaller equal shaped and sized triangles 2. divide each of the three corner triangles like step 1 3. repeat step 2 on all smaller and smaller corner-triangles of next bigger triangles).
What a coooool few graphs those all are! Please more videos with this man! He's amazing and seems so passionate about what he's explaining. I remember when I read about his work in the book Alex's Adventures in Numberland, SO cool!
So, to clarify during the forest fire sequence it was stated that they jump by 0 and then they jump by 3 but the difference in jumps between the two was 2, and 3 - 0 = 3 which is not 2 so there's a bit of a mistake there.
the forest fire one could also be defined as "no point is the average of two other points", where a point is the (x,y) pair. (i-j,f(i-j)) + (i+j,f(i+j)) cannot be allowed to equal (2i,2f(i)).
Over the past few weeks, I've actually been playing with the "Forest Fire" sequence and have come up with a few related sequences I might submit to OEIS. Instead of limiting to positive integers, you can specify that instead of picking the lowest number that doesn't create arithmetic progressions, you pick the lowest absolute value, with positive numbers before their inverses. The graph is pretty cool, more like sand dunes. You can also start with this sequence and define another sequence that's based on running partial sums. As expected, there's a strong positive bias, but there's quasi-periodic behavior, and it seems possible to me that there will be an infinite number of negative partial sums. I'm also playing with a different "greedy" algorithm. Instead of computing values of the sequence incrementally, if you have with some N that's the size of the sequence you want to compute, you can go all the way up to N filling in 1s whereever possible, and leaving blanks where it's not possible. Then, fill in the blanks with 2s whereever possible, etc. That sequence looks like a series of horizontal bands with gaps in between them. One question that can be asked is whether the gaps between bands extend to infinity. I haven't worked it out yet, but it seems like it shouldn't be difficult (and is perhaps trivial for an actual mathematician) to show that there can never be a 3 in the sequence. However, it seems like there's a gap between around 750 and 950 but there's no clear boundary. I suppose the same extension to non-positive integers could be applied to this algorithm too, but I haven't looked at that yet!
For the graph at 7:00 If you do it at a point n and find the n + 1 point, if you make a central symmetry with this new point as center and from a previous point, the result is a white space, and if you do it with all previous points with the new point as center then you get the central simetrie of the previous graph in white space.
After I watched this video, as I was playing with that good old generalisation technique, I came up with this - "Not average" sequence: 1. a(1)=1; a(2)=1 2. For every a(n) all possible sums of type s(n, i) = a(n - i) + a(n - 2*i) are calculated { i < [n / 2] }. 3. If s(n, i) is even, we have v(n, i, 1) = v(n, i, 2) = s(n, i) / 2 4. If s(n, i) is odd, we have v(n, i, 1) = (s(n, i) - 1) / 2 and v(n, i, 2) = (s(n, i) + 1) / 2 5. We seek the smallest natural number that do not appear within the arrays of numbers v(n, i, 1) and v(n, i, 2). This is the next member of the sequence, a(n). 6. Repeat points 2. - 5. for a(n+1). The sequence is like: 1, 1, 2, 3, 4, 1, 1, 4, 4, 5, 6, 7, 2, 1, 7, 1, 1, 2, 8, 7, 9, 7, 1, 6, 9, 9, 6, 9, 10, 3, 10, 11, 3, 11, 12, 1, 1, 13, 3, 2, 1, 13, 3, 13, 13, 10, 1, 1, 3, 1, 9, 11, 14, 3, 14, 13, 15, 15, 10, 15, 16, 1, 16, 4, 17, 4, 12, 11, 9, 17, 16, 15, 17, 17, 18, 15, 19, 18, 11, 5, 19, 19, 20, 21, 1, 4, 21, 20, 22, 22, 4, 22, 23, 24, 25, 1, 25, 2, 2, 26, 4, 16, 1, 1, 27, 24, 4, 3, 5, 22, 25, ... and at first glance has most erratic and peculiar behaviour. If you look at these numbers, scattered on the plane, a fractal structures do appear... which is not that strange, because the sequence writes onto itself. The big question is how to know (or prove) that these structures will go on to infinity or there is some kind of loop that dramatically will change the behaviour of the sequence (like the Langton's ant or that sequence showed to us here recently - I forgot its name). "Nobody knows", my favourite quote of Mr. Sloane, meaning "Let's find out", of course. I don't know, there must be some theory lurking in these questions. You know what beautiful mathematics appear from the thin air, when one explores the prime numbers, for example... and they belong to sequence that also writes onto itself, by definition. So, who knows what mathematical wonders await us in these sequences?!
It makes sense that any rules you use to generate numbers show some kind of regularity unless you pick something specifically that would not be regular, like first appearance of a given integer in the digits of pi. Even the primes show some nice regularity in Ulam spirals (I'm sure there is a numberphile video on that). Other regular rules give random (or pseudo random) numbers. The generalisation and categorisation of what types of rules generate which is surely a fascinating study!
About the smoke pattern, maybe you could find a new constant by comparing the sizes of the fractal parts, like in the bifurcation diagram :D these are interesting, keep them coming!
A sequence i found that looks pretty random but i don't know if there might be something interesting to it: *1.* Start with a 1. *2.* For every term the rule is a(n) = a(n-1). *3.* If a(n) is bigger than n, subtract n from a(n). The sequence itself is 1, 2, 1, 2, 4, 2, 4, 8, 7, 4, 8, 4, 8, 2, 4, 8, 16, 14 etc.
//Here is my processing sketch code for the last one. Runs in exponential time however. Not sure if there is a better algorithm import java.util.List; import java.util.ArrayList; int count=1; List values; void setup(){ background(0); size(1040,768);
Theoryphile (new thing I just made up where I try to delve a little deeper into the theory behind numberphile videos): The first sequence can be understood as a repeating pattern of operations running in order, recursively. Middle - up - down. If you split it into powers of three - the first 3 terms, the first 9 terms, the first 27 terms - you'll see this same middle-up-down shift pattern playing out on different scales. Each time we cross a number with a new power of three, the sequence zooms out and repeats the same pattern. We can also see two gradients at work, the maximal upper and lower boundary, which contains all the points with a slope of 1 and -1 respectively - and the rough "centers of each triangle", which go as 2/3 and -1/3 respectively. That the difference between the effective gradients is one is a nice sanity check that this curve reaches all integers - converting from a decimal integer X (which covers all positive integers), to a balanced ternary Y (which covers all integers, exploring the positive integers at roughly 2/3 the efficiency). Although the process is not random, we can think of the points as if they were uniformally distributed random numbers - choosing from the same width of distribution - I.e. the same difference in gradient between the median bounding slopes. Will update if I have further thoughts
Well hang on if you're going from positive one right the way down to Infinity you naturally would have gotten every integer both positive and negative which should be the same size but given the way it looks with the tertiary system it looks like it would be twice as big even though they should be the same size. Am I thinking about this right or what?
I think the fractal structure is intuitively obvious. If you think of generating this sequence as plotting its points on the (n,A(n)) plane, then at each step you reflect all the points generated so far about the last point you generated, and mark their images as verboten. Then you choose the lowest positive non-verboten point for n+1 and continue the procedure. If the cloud of points for 1 to N exhibits some large scale density structure, then that structure will be roughly imprinted, upside-down and in "photographic negative", on the structure between N and 2N.
It's amazing what you can find when you just start compiling numbers and shifting them into graphs. Of course, sometimes what you find is just a rather beautiful set of numbers...
Hands up if you went to a school where your teachers apparently went out of their way to make mathematics way less interesting than they clearly could have..
Watch the full Amazing Graphs Trilogy (plus an extra bit): ua-cam.com/play/PLt5AfwLFPxWLkoPqhxvuA8183hh1rBnG.html
! Invalid parameters.
Link is broken
wait what? i guess not.
In the first graph (base 3), it does look likes the triangular pattern i have seen seen somewhere else......maybe fractals.....can you make such similar pattern from similar kind of definition of (other)functions.....I would love to see a dedicated video on such functions giving rise to these pattern/fractals.
17 likes?
oh..18
So can we just have like fifty videos of these? Because wow they're satisfying.
Yeah, it might as well be a series. There’s probably a lot of them.
Can we just have more videos of Neil Sloane he has a lovely voice :)
I would like an infinite sequence of Neil Sloane videos
Yes please, more of these and more Neil 🙏🙏🙏
There is an UA-cam channel, called "Experimental mathematics", where Neil Sloane has some very interesting presentations on integer sequences. I highly recommend checking it out.
"Allright then, this is brilliant." With all the excitement of someone being told to clean their room.
Aroight 😂
Came down to find exactly this comment, lol
And yet, I paid more attention to this Brilliant promotion than any other one I've ever seen.
false.
_alright this is the sponsor today, let's get this stuff over with so i can get my money and all that, yep_
These two episodes of amazing graphs just blow my mind away to the outer space. Great videos!
The parallellograms right?
Why? Minus one minus one minus one etc etc... boring.
graphs are blow jobs!
Wait till you see the third one
I never want the Neil Sloane videos to stop, he is just the best.
He's a legend
10:25
"The third and final one is coming very soon"
"Pretty obviously, it's never gonna happen"
Quality editing.
...If you look at the grafffff
??.
"I can, therefore I must."
~Neil Sloane, 2019
"Are you sure?"
??.
Wait so is no one going to point out that that balanced ternary graph is a Sierpinski triangle???
ITS THE TRIFORCE
yes
See also A213541.
It's all the more ironic since he specifically mentioned fractals later for the forest-fire series (but not for the wisteria series which also looked like a fractal).
It's close but not quite (it's not a blacked out triangle in the center, more like a blacked out hexagon)
Amazing that the Sierpinski triangle emerges from graphing the balanced ternary sequence, while also covering all integers exactly once
This is like watching Bob Ross. Very calming and satisfying.
Cool thing i found (probably was already found but i did it anyway):
You can do the first sequence with any base if you replace all non-zero even digits(d) with
-1*d+1
It gives you the same fractals but with any length of 'arms' on a V shape.
@jj zun It kinda does. It will still produce a feactal shape. But with my method you get a nice V shape. With other rules you will have an asymmetrical (but still repeating as a fractal) shape.
You should go to OEIS and submit that sequence
@@matthewweitzner8956 which one? There's an infinite number of them (one for each value of d).
@@pafnutiytheartist Choose your favorites, and list any number of terms in a row
I can't believe Neil Sloane is 80 years old
More proof that exercising your brain keeps you sharp into old age
I always enjoy Numberphile, but WOW!! These videos are amazing. Looking forward to third in the series. 😀😀
Neil episodes are probably my favorite. The animations of Neil are a nice touch.
People to mathematicians: why do you do all this work that doesn't have any real applications?
Mathematicians: I can and therefore I must.
Because mathematics is the only area of human enquiry where one can prove anything.
@@frankupton5821 Actually I wouldn't give math that much credit, after all we have theorems like Gödel's incompleteness theorems
Mountain climber: "Because it's there"
We can't prove everything in maths, but we can prove (or disprove)most things, whereas in other subjects we cannot conclusively prove anything.
@@sardonyx001 actually, Gödel's incompleteness theorem say we can't prove EVERYTHING, not anything. We certainly can (and do) prove many statements.
These Neil vids sure aren't Sloane down!
👉😎👉
1:13 It should be noted that the column labelled Balanced Ternary isn't actually Balanced Ternary. The Balanced Ternary (where T conventionally denotes -1) for decimal numbers is as follows:
1 is 1 (= 1*1)
2 is 1T (= 1*3 + 1*-1)
3 is 10 (= 1*3 + 0)
4 is 11 (= 1*3 + 1*1)
5 is 1TT (= 9-3-1)
6 is 1T0 (= 9-3+0)
7 is 1T1 (= 9-3+1)
8 is 10T (= 9 + 0*3 -1)
etc
Similar to using Roman numerals, you write 2 as "three less one".
What is described in the video is a mapping in which numbers are written in ternary, partially "bit-flipped" with 2s turning into -1s, and then rewritten in decimal. It's a perfectly fine thing to consider and explore, but it isn't Balanced Ternary.
Shouldn't 5 be 1TT in balanced ternary? It makes more sense to me to only allow the symbols T, 0 and 1. Then
6 is 1T0
7 is 1T1
8 is 10T
etc.
@@williamhautekiet9061 That's what I put! I wrote it incorrectly at first and then corrected it immediately afterwards to be as you say, 5: 1TT. Are you still seeing my original, incorrect version?
@@scottclowe Now I see your correct version, I don't know why I couldn't see it at first :)
Thanks for that, I didn't get that part in the video at all.
OH! Thanks a lot! As someone who use balanced ternary quite often, I was really sad to see he's not talking about the real balanced ternary here, without even saying that that's not the real one!
Thanks for bringing some precision here
“it’s well defined, it’s unique, and it’s CRAZY” 6:40
I always like listening to the graphs on OEIS. There's some of the craziest horror/ boss music stuff you'll hear.
What strikes me is that the StarWars sequence is a way of seeing the Sierpinski Triangle *from below*. Whenever you start building a fractal, you start from low detail to high detail and there is no meaning to ''the highest possible detail'. Yet here the graph expands outwards, as if you started to draw a Sierpinski Triangle backwards!
The "Starwars"-graph looks actually like a modifyed version of that triangular fractal(1.divide a big triangle into four smaller equal shaped and sized triangles 2. divide each of the three corner triangles like step 1 3. repeat step 2 on all smaller and smaller corner-triangles of next bigger triangles).
That Star Wars-y graph at the beginning also resembles an array of Sierpinski triangles with one corner extremely stretched-out.
These graphs are so cool. Please make more of these! I might just hang one as a poster in the study room :D
The first thing I noticed with the Balanced Ternary Graph is how it looks like an infinite fractal of Triforces.
look up Sierpinski Triangle, I think you'll enjoy it
@@iveharzing Infinite Triforces within Triforces
go back to breaking brawl with mods
What a coooool few graphs those all are! Please more videos with this man! He's amazing and seems so passionate about what he's explaining. I remember when I read about his work in the book Alex's Adventures in Numberland, SO cool!
"It's well defined, it's unique, and it's crazy" im definately using this haha
I don't understand the math, but I love to listen to him explain things!! Very satisfying. And he usually looks so happy =)!!!
So, to clarify during the forest fire sequence it was stated that they jump by 0 and then they jump by 3 but the difference in jumps between the two was 2, and 3 - 0 = 3 which is not 2 so there's a bit of a mistake there.
I would love to see a continuing series of these. Please don't stop at 3! The graphs and the professor are amazing
0:01 Balanced Ternary
1:50 Wisteria
3:37 Forest Fire
7:42 Artificial Sequences
the forest fire one could also be defined as "no point is the average of two other points", where a point is the (x,y) pair. (i-j,f(i-j)) + (i+j,f(i+j)) cannot be allowed to equal (2i,2f(i)).
I have been wasting so much valuable homework time writing C# programs that print back Neil Sloane’s sequences lately.
Over the past few weeks, I've actually been playing with the "Forest Fire" sequence and have come up with a few related sequences I might submit to OEIS. Instead of limiting to positive integers, you can specify that instead of picking the lowest number that doesn't create arithmetic progressions, you pick the lowest absolute value, with positive numbers before their inverses. The graph is pretty cool, more like sand dunes. You can also start with this sequence and define another sequence that's based on running partial sums. As expected, there's a strong positive bias, but there's quasi-periodic behavior, and it seems possible to me that there will be an infinite number of negative partial sums.
I'm also playing with a different "greedy" algorithm. Instead of computing values of the sequence incrementally, if you have with some N that's the size of the sequence you want to compute, you can go all the way up to N filling in 1s whereever possible, and leaving blanks where it's not possible. Then, fill in the blanks with 2s whereever possible, etc. That sequence looks like a series of horizontal bands with gaps in between them. One question that can be asked is whether the gaps between bands extend to infinity. I haven't worked it out yet, but it seems like it shouldn't be difficult (and is perhaps trivial for an actual mathematician) to show that there can never be a 3 in the sequence. However, it seems like there's a gap between around 750 and 950 but there's no clear boundary.
I suppose the same extension to non-positive integers could be applied to this algorithm too, but I haven't looked at that yet!
My variation is that the minimum is how many terms back you saw the last term.
I want to see a million of these kinds of videos, it’s so interesting and it’s all stuff nobodies ever heard of
Could you please include links to OEIS for these sequences in the description? Thanks!
Can't get enough of these graphs!
"It's well defined, it's unique, and it's crazy." Yeah me too, Neil, me too.
My thoughts exactly. :P
Laughed hard at that point xD
false.
All of these graphs blow my mind!
LOVE the videos with Neil!
Thx for using my piece for A123456
these videos are some of the best in this channel
I love this type of video :) the patterns that arise are fascinating!
The moment that he reveals the grAph! so satisfying!
For the graph at 7:00
If you do it at a point n and find the n + 1 point, if you make a central symmetry with this new point as center and from a previous point, the result is a white space, and if you do it with all previous points with the new point as center then you get the central simetrie of the previous graph in white space.
After I watched this video, as I was playing with that good old generalisation technique, I came up with this - "Not average" sequence:
1. a(1)=1; a(2)=1
2. For every a(n) all possible sums of type s(n, i) = a(n - i) + a(n - 2*i) are calculated { i < [n / 2] }.
3. If s(n, i) is even, we have v(n, i, 1) = v(n, i, 2) = s(n, i) / 2
4. If s(n, i) is odd, we have v(n, i, 1) = (s(n, i) - 1) / 2 and v(n, i, 2) = (s(n, i) + 1) / 2
5. We seek the smallest natural number that do not appear within the arrays of numbers v(n, i, 1) and v(n, i, 2). This is the next member of the sequence, a(n).
6. Repeat points 2. - 5. for a(n+1).
The sequence is like: 1, 1, 2, 3, 4, 1, 1, 4, 4, 5, 6, 7, 2, 1, 7, 1, 1, 2, 8, 7, 9, 7, 1, 6, 9, 9, 6, 9, 10, 3, 10, 11, 3, 11, 12, 1, 1, 13, 3, 2, 1, 13, 3, 13, 13, 10, 1, 1, 3, 1, 9, 11, 14, 3, 14, 13, 15, 15, 10, 15, 16, 1, 16, 4, 17, 4, 12, 11, 9, 17, 16, 15, 17, 17, 18, 15, 19, 18, 11, 5, 19, 19, 20, 21, 1, 4, 21, 20, 22, 22, 4, 22, 23, 24, 25, 1, 25, 2, 2, 26, 4, 16, 1, 1, 27, 24, 4, 3, 5, 22, 25, ... and at first glance has most erratic and peculiar behaviour.
If you look at these numbers, scattered on the plane, a fractal structures do appear... which is not that strange, because the sequence writes onto itself. The big question is how to know (or prove) that these structures will go on to infinity or there is some kind of loop that dramatically will change the behaviour of the sequence (like the Langton's ant or that sequence showed to us here recently - I forgot its name). "Nobody knows", my favourite quote of Mr. Sloane, meaning "Let's find out", of course.
I don't know, there must be some theory lurking in these questions. You know what beautiful mathematics appear from the thin air, when one explores the prime numbers, for example... and they belong to sequence that also writes onto itself, by definition. So, who knows what mathematical wonders await us in these sequences?!
I could watch Professor Sloane make graphs all day. Art with numbers. These are great. More, please and thank you.
We need like 10 more parts
On this thumbnail Sloane is even more similar to Homer Simpson
It's the Man
the Myth
the Legend
ASMR Maths Teacher
RoverKnight it’s Scott Sterling
Never seen such a convincing brilliant advertisment and I have seen many of them ^^
this guy is a legend
Why does it look like a Sierpinski's triangle?
Looks more like what happens if you try to generate one with chaos game, but use something less than 1/2 for the rule.
@@bengineer8 specifically 1/3
@@japanada11 That is an amazing explanation, thanks :)
These are so cool. Can this be a series on here? I could look at these awesome graphs all day.
love this guy, we need more of him!!!
6:50
When Thanos snapped, this graph was made
It makes sense that any rules you use to generate numbers show some kind of regularity unless you pick something specifically that would not be regular, like first appearance of a given integer in the digits of pi. Even the primes show some nice regularity in Ulam spirals (I'm sure there is a numberphile video on that). Other regular rules give random (or pseudo random) numbers. The generalisation and categorisation of what types of rules generate which is surely a fascinating study!
About the smoke pattern, maybe you could find a new constant by comparing the sizes of the fractal parts, like in the bifurcation diagram :D these are interesting, keep them coming!
Let's all be honest: Neil Sloane is the absolute BEST we had in Numberphile!!!!
Utterly fascinating! Looking forward to the next video. Brilliant! :)
These are very satisfying. I would love more.
"five, is one-two, which becomes one, minus one. One, minus one means one three, minus one, so its two"
Quick maths?
2:52 Literally loled at how sick this is :')
this is amazing great guy. his passion for what he does is obvious
I love how the first frame of the video is the name of my favorite additive number system.
Aight Brady congrats on getting me hooked on this, now please give me more before I go mad
Could you do a video about Mandelbox? I'm very interested in fractals like this and want to know about the mathematical background about it.
Numberphile already has a couple of videos on Julia sets and the Mandelbrot set :)
@@blumousey I mean the Mandel*box*
Fantastic videos. Keep going with these please.
Make a series of this!
Master of sequences...
Hattsoff off to you NumBerpHile👍
BRILLIANT NUMBERPHILE...
A sequence i found that looks pretty random but i don't know if there might be something interesting to it:
*1.* Start with a 1.
*2.* For every term the rule is a(n) = a(n-1).
*3.* If a(n) is bigger than n, subtract n from a(n).
The sequence itself is 1, 2, 1, 2, 4, 2, 4, 8, 7, 4, 8, 4, 8, 2, 4, 8, 16, 14 etc.
I envy people that get paid to think about these things
1:28 Legend of Zelda!
The head sequence does have an equation. It is a massive sum of sine functions but it still has an equation
*graph makes sierpinski's triangle*
now what nerdy franchise does that remind me of?
neil: star wars
me: *facepalm*
Thank you! I had hoped I wasn't the only one.
What
@@TykoBrian7 Legend of Zelda.
Although it does look more like a massive fleet of star destroyers
Starcom's Shadow Force emblem.
//Here is my processing sketch code for the last one. Runs in exponential time however. Not sure if there is a better algorithm
import java.util.List;
import java.util.ArrayList;
int count=1;
List values;
void setup(){
background(0);
size(1040,768);
stroke(255,0,255);
values=new ArrayList();
}
void draw(){
boolean doContinue = true;
int t=0;
while(doContinue){
t++;
boolean foundError =false;
for(int i=1;i100000)noLoop();
}
“You won’t get a large drupe because the digits are all small”
10 seconds later
“Well yes they had a lot of big digits”😂
Theoryphile (new thing I just made up where I try to delve a little deeper into the theory behind numberphile videos):
The first sequence can be understood as a repeating pattern of operations running in order, recursively. Middle - up - down.
If you split it into powers of three - the first 3 terms, the first 9 terms, the first 27 terms - you'll see this same middle-up-down shift pattern playing out on different scales.
Each time we cross a number with a new power of three, the sequence zooms out and repeats the same pattern.
We can also see two gradients at work, the maximal upper and lower boundary, which contains all the points with a slope of 1 and -1 respectively - and the rough "centers of each triangle", which go as 2/3 and -1/3 respectively.
That the difference between the effective gradients is one is a nice sanity check that this curve reaches all integers - converting from a decimal integer X (which covers all positive integers), to a balanced ternary Y (which covers all integers, exploring the positive integers at roughly 2/3 the efficiency). Although the process is not random, we can think of the points as if they were uniformally distributed random numbers - choosing from the same width of distribution - I.e. the same difference in gradient between the median bounding slopes.
Will update if I have further thoughts
Does anyone know if the forest fire sequence has any alternate names? I can't find the sequence anywhere other than this video.
@8:33 "yeah right..." always so polite
Well hang on if you're going from positive one right the way down to Infinity you naturally would have gotten every integer both positive and negative which should be the same size but given the way it looks with the tertiary system it looks like it would be twice as big even though they should be the same size. Am I thinking about this right or what?
MORE GRAPHS AND MORE SLOANE... please.
Neil's videos are so damn interesting!!!
These are amazing! More graphs please.
The smoke waves look like murmurations.
I want you to know that I fall asleep to almost all of these videos.
I think the fractal structure is intuitively obvious. If you think of generating this sequence as plotting its points on the (n,A(n)) plane, then at each step you reflect all the points generated so far about the last point you generated, and mark their images as verboten. Then you choose the lowest positive non-verboten point for n+1 and continue the procedure. If the cloud of points for 1 to N exhibits some large scale density structure, then that structure will be roughly imprinted, upside-down and in "photographic negative", on the structure between N and 2N.
I love typing sequences into the OEIS search to see if they have it. I weirdly love when it doesn’t match anything in the OEIS,.
Lovin the crazy unknown properties of those graphs!
More... More... These are fantastic
This has been your most inspirational video✌️π Thank you so much~
This was a really fun video!
This man has a beautiful soul.
There should be an entire channel for this
Please keep making these videos
I'm addicted to this after two dose. Give me episode III
These video series are amazing, it's like being on LSD without taking LSD
It's amazing what you can find when you just start compiling numbers and shifting them into graphs. Of course, sometimes what you find is just a rather beautiful set of numbers...
Brady you have a gift for asking questions.
Hands up if you went to a school where your teachers apparently went out of their way to make mathematics way less interesting than they clearly could have..
When will you make part 3 public?