The Slightly Spooky Recamán Sequence - Numberphile
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- Опубліковано 13 чер 2018
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The Recamán Sequence on the OEIS: oeis.org/A005132
Full length video of Tiffany Arment coloring the patern: • Recamán Sequence (40 m...
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"Can you show me an interesting sequence?"
"Sure, I can Recamán-d you one"
I love this so much, probably more than I should
irock123432 1
That was brilliant, you should be proud. 😂
100% Recamán-dable
Praised be the punhnster
Thanks I hate it
the 99735th term of the recaman sequence is 19
@ateb3: It's surprising that it takes that long to return to such a low number.
ateb3 Alex said that it is assumed that the sequence will go through every number. So I wanted to see for myself if the sequence contains all numbers between 0 and 100 inclusive. After the 404th term there are only three missing 19, 61, 76. When I found that it took until term 99735 to hit one of the missing numbers I thought it would be worth mentioning. If the missed numbers are smaller than the jump size then the only way for the sequence to hit these numbers is if the current number on the number line is just a little bigger than the jump size. Having a small gap between current number and jump size occurs several times on the way to 19. It isn't obvious at all when exactly one of the missing numbers will finally be included if at all. The 181654th term of the sequence is 61 and the 181644th term of the sequence is 76.
I was wondering if there were any unusual hold outs. I wish they talked more about the meaning of the sequence, even if it is only abstract at this point.
ateb3 If you're unamazed by something, you don't have to say it out loud, and if you want to, at least don't say it in such a rude manner.
Thanks for the more detailed follow up, fire fist. What IS surprising is how quickly the last two integers are cleaned up ... only 10 iterations apart, after nearly 200,000 iterations.
This is Earth Radio. And now, here's... human music.
Hmm. Human music. I like it!
Is that a reference to something? Like was that a joke in Futurama?
Rick and Morty, I think. Futurama's a close guess.
Yes, it's from the fourth episode of the first series of "Rick and Morty". The episode is titled, "M. Night Shaym-Aliens!"
Rick and morty when Jerry's in the interstellar daycare for Jerry's from every universe.
Schönberg: "In our tone-row, we use all 12 notes in an octave before repeating a note"
Recamán: "Hold my beer"
Now there's a topic: what's the relationship between this series and the 12-tone row?
worldnotworld neither has a tonal center and both are by their nature chromatic, so the sound is somewhat similar to our ears - our ears are attuned to hearing tonal music so a lack of tonality becomes a defining characteristic for these kinds of sequences. Both sequences also have their inner logics which are nevertheless difficult to predict as youre listening, i suppose
Slightly Spooky Sequences
Send shivers down your spine
Wow
geez I love that song
Semicircles shock your soul
And seal your number line
@@_fedmar_ underrated comment
@@maximumdosage Why thank you
Seems like a decent idea for a numberphile t-shirt design...
Or earrings! I'm already thinking about how I could make some for myself.
I was thinking tattoo.
@@dragoncurveenthusiast 3d printing?
Dragon Curve Enthusiast you could do a logarithmic version of this so you could fit more on an earring.
You can make that using quilling paper,
only if you are skilled.
"What do you want to be when you grow up"
"I want to be a colourist that features on Numberphile"
The guy who invented the Recaman sequence is my math teacher
I met him in real person. He is a wonderful professor. :3
What’s his name?
@@ferrismesser Bernardo Recamán
bonxbonx r/whooosh
LoDefGaming Well, yeah, but the name of inventor of the Recaman sequence is quite obvious because of the "Recaman" part
He connected the points with Parker semi circles
Glad this meme is still alive.
Whole thing looks kinda like a Parker sine wave
Google says, "No results found for 'Parker semi-circles'." :(
A "Parker ____" is a solution to a difficult task that is declared accomplished by ignoring one of the rules for accomplishing the task.
In other words, a half-assed effort that provokes unwarranted satisfaction ... and spawns memes.
Oh! That is funny. Thank you for explaining that. :)
The only reason is sounds "Spooky" is because of the arbitrary choice of using the chromatic scale. If you used a different scale, say a pentatonic, then it would sound completely different :P
A lot of the character is from the limitation of 72 notes, the 73rd being the first again.
it would be interesting if you did like a 12 note limitation, making a sort of 12-tone row but with some infrequent repetition
BunniBuu I have absolutely no idea what just came out of your text box
Music theory jargon, don't worry about it :P
I think Alex was a bit confused about the musical implementation. He seemed to bounce between a regular (major) scale, with 7 notes per octave, and a chromatic scale, with 12 notes per octave. And when he showed sound examples, all of them used the chromatic scale.
If you use a standard piano for these, the major scale, starting with 0=A₀ (lowest note on the piano), will end at B₇=50, because the next note in the A major scale is C₈#, 1 semitone above the top note on the piano. (BTW, middle C is C₄ and each numbered octave starts with C and goes up to B.)
If you use a chromatic scale,, starting with 0=A₀ , the highest you can go will be C₈=87.
Incidentally, if you do this on a Bösendorfer Imperial Grande, you have 97 keys to use, starting with C₀ and ending with the same C₈ as before.
Then using a major scale, it would be a C scale, with C₀=0 and C₈=56.
Using a chromatic scale, you'd go from C₀=0 to C₈=96.
I'm mystified where he got 72 from; it doesn't follow from anything he said. Unless you accept his contention that you're limited to 6 octaves; but you're not.
Fred
This made me think of Langton’s ant, how two simple rules on a grid at first creates what looks like chaos but always at some point it creates a diagonal highway that stretches on for ever.
It ends at 91 in the book
First correct answer I saw... DM me your address if you want a book! :)
Sebastian Cor gg
No I was too late :'(
I sent you a message I think, could you check?
I was second, sad life
Imagine if they chose keys in a musical key instead of just the notes. It would probably sound like some kind of crazy house music.
Every one of these "spirals" has a number at the center. Do these numerical values have any particular importance?
enolastraight these numbers at the center of said “spirals” are the actual numbers in the sequence if you were to represent this sequence using the actual numerical values of each term. The spirals are simply a visual aid to demonstrate the jump from each term. It just so happens that the spirals form a nice visual which is why that is how it is shown here, but the center points for the spirals are the actual terms in the sequence.
@@masongiacchetti5478 ugh no, most numbers are on the outskirts of the spirals, where the spirals intersect with the number line. He means the sequence of numbers *literally* at the centre of each spiral i.e. the centre of each back-forth oscillation
Only odd jumps have a center. How do we handle the cases on even jumps? Do we even consider half circles to be sufficient in defining one of your numbers or would be look for the very special cases of, say, a number in the center of two consecutive odd jumps that form an (almost) closed circle? I'm interested in your question as well. Also, if we can find a clean definition, we have another integer sequence to be considered for the OEIS, which would be lovely.
@@felixmerz6229 Allow non-integer centres!
I happened to be re-watching this, and only just now noticed the Fig Newtons + Leibniz cookies on the shelf. That's wonderful.
Brady it's taken the better part of two years, but you have convinced me to seriously consider subscribing the Brilliant. ha ha. Thanks for the marvelous content over the years mate.
They’ve got some great stuff on there. Fiendish and fun. And they are a great supporter of Numberphile too.
I'd never heard of this sequence before, it's pretty neat! I like the audio version a lot.
(This might be clear to some, but in case anyone was wondering, “twiddle” is because we identified some octaves. The difference in notes would get bigger and bigger forever if we allowed arbitrary pitches, but when the notes were several octaves and a half step apart, it was played as a half step apart)
I used the same rules and wrote a program to do it. After checking my math, I did end up going forwards to hit the same number twice.
You know that's Gallifreyan.
Ruben 😂😂
Ruben new head cannon: the language of gallifreyan is actually derived from this or a very similar but more elegant sequence.
I love this.
Exactly what I thought, bro.
Ruben it says, “ *S P O O K Y S C A R Y S K E L E T O N S* “
What isn't explicitly mentioned here is that even though you can't jump backwards to a number that has been used, you CAN jump forwards to a number that has been used. This seems like a very convenient rule put in place just to make a nice pattern, else it would end pretty quickly (on the 24th jump, I believe). Whether all numbers are accounted for is still waiting to be proven, but the pattern will go on forever.
I was wondering about a terminating sequence when you couldn't go backward AND you couldn't go forward.
Nice "legal" interpretation... If it's not expressly prohibited, then it is allowed. 😊
What if we mod 12 or mod 7 or mod 5 and then play it?
Might sounds nicer
Abdul Muhaimin sound*
Thanks, I now have my new CROP CIRCLE pattern ! 👍👍👍👏👏👏
Amazing how such simple rules can produce complex patterns
I am glad to say that Bernardo Recaman is my teacher right now! Awasome person he is. Proud of him and everything he has achieved
This is by far the most interesting UA-cam channel I'm subscribed to
Que bueno es escuchar de nuevo sobre matematicas colombianas en este canal!!!! De verdad que me emociona y llena de orgullo 🇨🇴🇨🇴🇨🇴
Just when it thought I would never need Spanish again after finishing the last Spanish GCSE exam earlier today...
Then this video...
¿Pero qué estás diciendo?, si el español es un idioma tan fácil. Saludos.
Jeisson Sáchica I only understand half of that...
Same. 😆
"But what are you saying, if Spanish is such an easy language? Greetings"
Something along those lines
The audio version sounds like something J.S. Bach would have written if he was a 20th century atonal composer.
All that Contrary motion 😍
I'd call it 'snakes and ladders'
Look up Conlon Nancarrow player piano studies here on YT. Lots of atonal piano pieces that are impossible for humans to play at tempo. Incredible stuff.
imagine that sequence in a microtonal scale
+toofast4ya And now I really want this to happen.
It appears very similar to a lense flare. As a matter of opinion, I cannot think of anything but lense flare when I see it.
Thats interesting, thanks. Wonder if it has some kind of antenna application as well.
Hmmm.....
*lens
@@scotthammond3230
We Are Now Experimenting With Said Antenna Design On The "Grand Sakura" Array!
this video was amazing. reminds me of why i first started following number-file. Short, sweet, spectacular.
It actually doesn't take any screen shots to solve where it ended. While following the numbers at the beginning of the video, the last iteration was subtracting 46. Also, because the 38 - 65th iterations are in the same circle, all you have to do is add up the number of iterations left and add to that value. So at the 45th iteration (adding 45), we ended at 81. We need to add one for every time we pass the 47th, 49th, 51st, 53rd, 55th, 57th, 59th, 61st, 63rd, and 65th iteration. And because the 66th iteration is off the page, this was the perfect place to stop. But adding 10 iterations to the number would leave the 65th iteration (adding 65) to be at 91 and cause the 66th iteration to not be possible going backwards, since we used 25 as the 17th number in the squence (18th if you count 0), and cause the 66th iteration to jump forward, off the page, to 157.
If you can't go forward you go back but what if you can't go back or forward? I'm assuming this never happens but has this been proven?
@@bepamungkas Thank you.
I was wondering the same thing!
What if you can't go back AND you can't go forward?!?! The sequence terminates.
This was never alluded to nor hinted at. I *assume* (trust) that this study has gone on long enough to dismiss this possibility.
Am I wrong???
@@josephmelnick3446 Looking at it again I think you can always go forward but I'm not positive.
@@josephmelnick3446 The sequence is not a permutation of the integers, i.e. its definition allows for any integer to appear in it multiple times. If you tried to go backwards and would hit a number that's already in the sequence, you need to go forward instead. However, if by going forward you hit an already existing number, that's allowed. E.g. 42 is both the 21st and 25th number in the sequence.
How have I never heard of this?! This is amazing! 😍
Having followed the sequence up to its first 25000 iterations beginning at 0, the only numbers less than 1000 not accounted for within the sequence are the following: 19, 61, 76, 133, 223, 366, 828, 829, 830, 831, 832, 834, 835, and 879. It is hard to imagine at this point to see how the integers 19, 61, and 76 especially will be occupied by the sequence, but it does go back towards and after the 100,000th term. I very apparently got bored.
Visually, I would like to see the semicircles replaced with spirals so that at the intersections with the number line the radius of the spiral coming in is equal to the radius of the departing spiral.
Let’s see it in 3 dimension now
How would you put it in 3D?
it would be spheres
But you would only be able to see the outside sphere.
Well, not if you're a transdimensional being with 4d eyes
oh i thought you meant increase the number of axes for the semicircles to wrap around in... sort of like rotate 90deg per turn instead of 180deg?
6 octaves equals 48 semitones now? Tsk. Tricked by the name. That should be 72, surely?
Yes; 73 if you include both endpoints.
If you use a standard scale instead of a chromatic one, 6 octaves will have 6·7+1 = 43 notes, with both endpoints included.
But with an 88-key piano in front of you, why on Earth would you limit this to 6 octaves??
Fred
I mean, it's arbitrary either way.
Why 6 octaves? Why 12TET? Why chromatic? In the end, I figure that representing an integer sequence aurally is simply meant to give an impression of it's orderliness, and I'm sure that the representation they give it does that pretty well.
+ MiskyWilkshake: "I mean, it's arbitrary either way."
Well, no, not entirely. There *are* some ways to decide these things.
1. Human hearing is limited to 10 octaves (20 Hz - 20 kHz); many can hear only 9 or even 8. A standard acoustic piano has 7¼, so that's a reasonable limit to set.
Plus, pianos are pretty widely accessible to lots of people, so they can play around with it.
2. In the western world at least, everyone is pretty familiar with the semitone (= 1/12 octave) as the smallest practical musical interval. I think that some microtonal division could work for this, but you can't go too much smaller than a semitone, or most people won't be able to distinguish such slight variations in pitch.
3. All in all, what you want to aim for is the largest collection of different pitches that satisfy those two constraints.
Given these considerations, I'd say use a chromatic scale, running 8 octaves, from 27.5 Hz (A₀, the lowest note on a piano) to 7040 Hz (A₈); even though that's almost an octave above the highest note on a piano. That's 97 separate pitches. Or just limit it to the piano; 27.5 Hz (A₀) to 4190 Hz (C₈).
Fred
One of my favorites so far. Incredible content here.
At the end there, to me it looked like a sound profile or speaker illustration. Like how someone would illustrate sound waves emanating outward and in a specific bubbly region or whatever.
I was drunk on chocolate ice cream before i watched this video and understanding the sequence made me laugh a lot, I do not know why
Yay another colouring section by Tiff, love those!
Nice video, I really enjoyed it! I've always been fascinated by the effect that music has on psychology, and this ties math into it!
Most numberphile videos delight but this was exceptional. Wow!
You'd expect that small numbers would be completed rather quickly, but 4 is filled after 131 steps, and 19 is filled only after 99734. Brilliant! (no pun intended)
I don't think you know what a pun is lol
what's the pun?
It really does sound horrific. But the description of a clash between order and chaos makes perfect sense. That's exactly what makes something uncanny, it resembles something normal, but there is something aberrant about it.
If it was perfectly chaotic we'd hear white noise, if it was perfectly ordered we'd hear music. (pleasing music)
infintiyward I think the main reason it sounds so horror-like is that they used a chromatic scale. Had they used a diatonic one, it would have sounded much better.
@Tim H. True, that's part of it, but a chromatic scale alone doesn't have that much turmoil. Those aberrant notes would make pleasant intervals sound unsettling too.
Depends on how much input, if you allow a random amount of frequencies to play at random intervals I imagine you'd have something like white noise. How do you decide the range of that random amount?
TootTootMcbumbersnazzle
Yesssss yes. Of course, I must have forgotten to drink my coffee today.
Perfectly ordered would be boring. The great composers and artists knew when and which rules to break
Looks like the Gallifreyan writing in Dr Who. Also, he may have been talking about Kandinsky when he was talking about artists doing circles, in case you were wondering.
Including negatives and reversing the rule when negative just makes a spiral
I believe the number the sequence "ends" in this video is 91, the 66th number of the sequence with the previous number being 26 and the next being 157.
God of mathematics seems to be good at dwawing but seems to be poor at composing.
They are just playing the notes in sequence, In order to make music you have to introduce timing to the mix. You can do pretty amazing stuff using like the digits of pi for the notes and the digits on e for the intervals between them
Olbaid Fractalium in the fundament of the music lying the physic of waves, the fundament of physics is mathematic. LOL
it only sounds bad because you are used to listening to music that sounds like 123412341234123412341234 123412341234123412341234 etc
Olbaid Fractalium mathematically spaced chords sound gross too
Idk I kinda liked it!
From Wiki: Conjecture - Neil Sloane has conjectured that every number eventually appears, but it has not been proved. Even though 10^15 terms have been calculated (in 2018), the number 852,655 has not appeared on the list.
This makes the number line feel like an orbit looked at straight on that includes the observer as a value on the line with zero being antipodal to the observer and all the other circles being orbits you would have to walk into to approach the center.
You should listen to Ligeti's etudes and musica ricercata: they are based on similar concepts
Think I found my new tattoo.
I'd love to hear what the sequence sounds like on a certain scale, like C major or something.
The spookiest part is how you have your expensive computer perched on a book overhanging the edge of the table. Let's calculate the odds that you're going to knock that off accidentally at some point.
Reminds me of certain crop circles.
Yes, that's what I thought of as well
It sounds like music you would hear in cartoons from the 60's like Tom & Jerry.
So amazing. Fascinating pattern, but that's a gorgeous pattern!!!
LOL. If you're a musician, that progression made total sense at every point and in both directions.
The first number to be repeated is 42. O_o
The 42th number is 79, which is the fourth number to be repeated.
The 79th number is 153 and the 5th number to be repeated.
The pattern does not persist afterwards. And I had to start counting at 1 (not 0).
what do you mean with a number being repeated?
42th?
Can the sequence ever go back twice in a row?
Yes it can, it goes 63, 41, 18 an n={22,23}
If youre interested about brilliant's rod problem, here's my take:
The time it takes the rod to complete 1 rotation is 1 divided by its frequency so 1/200 minutes
The time inbetween flashes is also 1 divided by the frequency so 1/201 minutes
So the flash comes in before the rod will have completed 1 resolution,
-At this point we can firstly say that the Rod will definitely not Appear Stationary. For that to be the case the Frequencies would have to match
Now that the Rod is spinning clockwise and the Flash comes in shortly before the Rod completes 1 resolution, it will have completed the fraction of 200/201 of 1 whole rotation
-this means it will appear to have moved 1-200/201 of one resolution in the opposite direction
And thus it will APPEAR to rotate counterclockwise.
In the end it never appears moving really, but the snapshots taken show the rod being just a tiny bit off of 1 whole resolution so you would more favourably preceive it to move abit counterclockwise and not almost fully clockwise, with each snapshot
just 0.5 views? this is clearly one of the best numberphile clips..
Is there actually a proof that it's infinite? It definitely doesn't seem obvious that there can't be a point when you can't go backwards but also can't go forwards.
The number line is infinite, so even if you can't go back, there will always be another number higher up the number line that you can go to.
+Flamarius Also, I believe repeats are avoided only when subtracting (not quite sure of that one).
Flamarius is right, 42 is repeated in the sequence as are many other numbers. There's no restrictions on forward movement, only backwards
Yes, at first I thought they were claiming that it might just hit every number exactly once, and I thought that would be amazing. Then I worked out part of the sequence for myself and found that 42 was repeated. At first I thought maybe I'd made a mistake, so I checked, and then I noticed that the video did actually show that there were repeats, and that was disappointing.
Still, it means that the first repeated number is actually 42. Forty-two! So that must be significant. :)
What a freaking mess >:(
So, apparently:
If you aren't allowed repeats when going forward, then it ends right before you hit 42 again.
If you are allowed repeats, it's infinite because you can always at least take the "add" branch.
prove that it doesn't terminate? like is there a situation where you go n steps back but n+1 steps forward is already occupied right
plus n+1 back from the n steps back is also occupied? plus the mirror version of all of it (i.e. +n forward works --> n+1 forward doesn't & -(n+1) backward doesn't.
Yes, at the 21st and 25th terms. They’re both 42.
I've seen the coloring book that has this! that's so cool, now i feel bad for not picking it up when I had the chance
Bernardo Recaman is a professor at Universidad de los Andes in Bogotá. He has been a mentor to me and i´m proud to know him. He is my friend and this sequence is a challenge because of the mysteries it holds. I'm glad there's a Numberphile video about his sequence and he may be happy about it too.
yo this is so cool I wanna hear 10 mins of that (maybe there r repeats thatd be cool)
Rigby Go to the website in the description :)
Specler X oh sick tysm
Honestly it looks like gallifreyan xD
-someone get the reference please-
Actually, circular gallifreyan, 'cause there are other gallifreyan alphabet ^^
Google got the reference for me
I was literally gonna comment that!
Doctor who?
That's awesome
I liked the music version, it reminds me of the songs I've played on my piano. The Renaman sequence looks and sounds musical.
This again reminds me of how values of atoms get increase along the periodic table: weird new systems go on for a while, get replaced by others, some long and others short...
I saw something similar in that other episode about "turtles and roses" on polygons.
It wasn't invented, it was discovered...numbers, patterns, iterations have existed since our universe was born...
yeah ok cool but you get what he means
Well that’s one view, but it’s not the only view.
Looks like a ripple in water
Would love to see two perpendicular lines that use the Recaman Sequence. Maybe have it going from one line to the other in strait lines.
Raise your hand if you also watch Numberphile and are completely lost as soon as they jump into unpacking their ideas. ✋ Yet I'm still subscribed...
I was waiting to hear why it's interesting but all I got was "it sounds weird on an arbitrary scale". Why is it mathematically interesting?
It probably isn't mathematically interesting in the sense of shedding light on "important" problems. But it is one of those peculiar mathematical objects that is very easy to define but not at all straightforward in its behaviour. There is also an obvious question to ask about it that is very easy to state but that nobody has so far been able to answer: Does every natural number appear in the sequence? Nobody knows! People think that every natural number will appear, and apparently it has been checked up to some high number, but nobody has proved it. This is reminiscent of the Collatz Conjecture.
Any chance of a link to a high res image of that 600 number graph?
Do you want one, I was going to create an app to draw it
It ends at 91. And I'd appreciate getting that book. (signed by you and Alex if possible ;) )
Here's what i did for anyone interested:
try and draw the spiral or whatever thats called upto 25;
look up the rest of the series from there. here it is:
..., 25, 43, 62, 42, 63, 41, 18, 42, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, ...
match ur spiral with the spiral in the picture, and u should notice that after 25 there's no coming to 26 for a long time. Keep that in mind for now. The next thing we do is trace backwards from the endpoint of the spiral, until we're next to 25. Common sense, its 26 thats next to 25, so the term before the endpoint must be 26. Now refer to the Recamán sequence, and we see that its 91 thats after 26, so the endpoint is 91.
Yeah you can probably try other methods to find that, but this made a lot of sense to me.
Another thing I tried was use a digital measurement app to find the endpoint; first I defined 1 unit as the space between 0 and 1 (the tiniest semi circle in the spiral), then I simply checked the distance from 0 to the endpoint, which turned out to be 94.3 something units. I then jumped in excitement that I solved it, then I noticed that 94 isn't a term in the Recamán sequence. Then I thought a little more about it, and finally have the solution (above).
One of the most beautiful Numberphile videos. I think Grey would like this. :3
damn it i got spooked!
I have the collatz conjecture for my ringtone almost a have of a year :P
Those biscuits at the back have to be the most subtle in-joke of all time.
Sequences as music! Awesome!
pretty sure the "human" quality is just a result of it sounding like a piano
No, they were referring to the multiple interlocking up-and-down-y line that you can hearm
Nope - the one where it is just ascending also sounds like a piano. They mean the little flurries and stuff that are not obviously a sequence. To me some of those bits sound like experimental or modern or whatever its called Jazz.
elevown oh, jazz
elevown like, hitting a few random keys at weird intervals type jazz
yup - I dunno enough about it to know even its proper name lol but I've heard it. Regular Jazz is fine, but the experimental/modern stuff - half of it Does just sound like randomly hitting keys lol.
I've just spent the past half hour listening to the digits of pi on the OEIS website.....
Sequence is endless, every positive integer is included.
The musical term for this is COUNTERPOINT. The melodies are going either towards eachother or they out-distance from eachother, whereby they sometimes cross eachother.
I already have his book: "Alex's adventures in numberland"
Then you might want the sequel, _Alex through the Looking Glass,_ as well.
I can't decide if that title is witty or ignorant. Carroll was a math teacher and wrote the book to be all about math and logic. Maybe Alex knows this and enjoys the pun anyway
Owen Keller Alex knows this, because everybody knows this.
8:37
Probably came from a jump which was a multiple of 72, nothing special.
All sorts of interesting stuff to investigate here: Like tracking the sequence of how many times you can go back without having to go forward; in your graphical approach, (Maybe start with cylindrical coordinates and project them onto a plane?); how many times does the line intersect itself; how would you determine which Fourier series that would sum to these numbers;... ? There's got to be a doctoral thesis in there somewhere, probably more than one.
Very interesting sequence, and quite beautiful on top of it!
3:02
There's a mistake in your master coloring 😉
(At the number 43)
I believe it stops at 103, and the highest number represented (farthest right point) is 129. The picture also appears to start at 1 rather than 0. Could be off quite a bit though, I just did it assuming equal spacing between numbers and measuring from the most top down image available (the paint filling part near the beginning).
bits of it alternate between sounding like the Viridian Forest theme and the Rocket Hideout theme from the original Pokemon games
Reminds me of the endless staircase in SM64...
(I know it's more a Shepard tone but still)
JekyllGaming99 Reminds me of 'The Devil's Staircase' by György Ligeti.
György Ligeti - L'escalier du diable
Ligeti - "The Devil's Staircase". My thought exactly.
The audio at the beginning is much better than the one in the video imo.
I recently purchased the "Patterns Of The Universe" coloring book. Small world!
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