Rethinking the real line
ฝัง
- เผยแพร่เมื่อ 16 ส.ค. 2023
- We take a geometric approach to rational numbers, to rethink how to organize the real line. Along the way, we visualize Diophantine approximation and continued fractions. And your favourite number, pi.
Much of the mathematics here is based on the following article:
Series, C. The geometry of markoff numbers. The Mathematical Intelligencer 7, 20-29 (1985). doi.org/10.1007/BF03025802
A big thanks to the Summer of Math Exposition competition for the motivation to make this happen, and a big thanks to my audience for forgiving my video-editing non-skills.
Some of the software used in creating this: Sage Mathematics Software, Manim, VPython, p5.js, Krita, Audacity, Kdenlive.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Music used in the video:
Walk Through the Park -- TrackTribe
George Street Shuffle -- Kevin MacLeod
Quarter Mix -- Freedom Trail Studio
Love Struck -- E's Jammy Jams
George Street Shuffle by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/...
Source: incompetech.com/music/royalty-...
Artist: incompetech.com/ - วิทยาศาสตร์และเทคโนโลยี
![[MPD직캠] 베이비몬스터 직캠 8K 'LIKE THAT' (BABYMONSTER FanCam) | @MCOUNTDOWN_2024.6.13](http://i.ytimg.com/vi/8qa-nm9bCLs/mqdefault.jpg)
The protective geometry view of the rationals reminds me of the gaps I'd see while driving past a vineyard.
Ok so that bit of projective geometry going from the 2D grid to the 3D representation blew my mind. What a fascinating video!
Everywhere I go with visual representations for math, I ended up seeing infinitely repeating fractals
I've watched a lot of SoME3 videos, and this is one of my absolute favourites. I can really feel the "It would be cool if I could animate this idea" mindset present throughout the video, and I love that you decided to share them. I can tell you had fun making this.
You've just transformed the way I think about numbers forever
As a mathematician, I am impressed and inspired by the beautiful and understandable visualization of some pretty deep mathematical concepts. Great work.
I remember the first time I started 'getting' continued fractions so fondly. It really does feel like breaking free from the trap of decimal expantion, which fails to elegantly represent even simple ratios like 1/3 or 1/7 (let alone simple irrationals!!!)
Seeing that vertical line representing pi coming down through the representation of the Fairey sequence brought to mind Dedekind cuts. Only being vaguely aware of both of these topics, it makes me wonder how they are related.
you can think of the process of moving from the top and iteratively piercing the arcs as building the Dedekind cut iteratively. Say, each time you pierce a bubble you add all rationals outside the bubble to the left of it to one set of the cut, and all the rationals outside the bubble to the right of it to the other set of the cut. In the transfinite limit you'll have the Dedekind cut. This kinda shows how much more data there is in a Dedekind cut than is needed to construct the reals, as just one of the two sequences given by the pierced bubbles would've already sufficed to build the rationals as sequences of on average quadratically fast convergence, but the Dedekind cut sets have a lot more junk in them.
I have played with and pondered continued fractions so many times and never seen this connection. I'm 7/22 ashamed and 1.618 delighted by this revelation!
Most real numbers are not computable, most are just the random L and R sequences
Exactly! And this is such a powerful way to understand that, which is totally new to me. Plug that into your Turing machine 😂
That is a cool way to think of it.
Maybe they don’t want to be found
Sorry, but what is L or R?
@@dsudaniel3003 Left or Right
A new classic here! I've had this video in my Downloads for some time.
Great visuals! Really enjoyed this explanation.
WOW!! I started anticipating the hyperbolic half plane - Farey sequence connection about a minute or two before you said it directly, so much so that I wrote it in my notes, and I *squealed* with excitement that it was on the right track!! I cannot WAIT to watch your next video on it, and I hope the Minkowski question mark function comes up in the connection as well! In particular, the R/L notation you used also reminds me of the modular group with the T generator... exciting!!
Sure ya' did, we can all write comments after watching a video through!
I totally thought she was saying "fairy" expansion the whole time 💀💀
@@erickugel1376 it feels that way, I'll be honest. The connection between primes and SL2Z is absolutely magical, the fairy product might be a bit more fitting
Thanks, nice introduction to Stern-Brocot type structures. Among bases, unary is of course the most basic. We can start constrution of number system (and lot else) from a chiral pair of symbols, and relational operators < 'increases' and > 'decreases' do fine. From these we get two basic palindromic seeds, outwards < > and inwards > . Square roots have repeating periods, which is nice.
Standard representations of continued fractions don't necessarily coincide exactly with Stern-Brocot paths, as here we have inverse paths as NOT-operations, with first bits on the second row interpreted e.g. as positive and negative. Eg. the φ-paths of Fibonacci fractions look like this:
LL
RR >>
LL and RR are inverse NOT-operations, and sow are LR and RL. BTW a nice surprice was that the Fibonacci words associated with LL and RR have character count of Lucas numbers, and likewise LR and RL Fibonacci numbers.
BTW Base 10 is not totally arbitrary (2 hands, 5 fingers in each, 10 fingers together),. Transforming standard continued fraction representations of sqrt(n^2+1) to the path information, the periods look like this:
sqrt(2): ><
sqrt(5): >>>>>
"A real number that cannot be described in finitely many English words"
*boom Berry's paradox*
Thank you very much for the visualizations!
Absolutely wonderful video! This is really cool and I think definitely deserves strong recognition in the #SoME3 comp. Thank you for making this!
This has made me very happy. Fabulous
A real pleasure. Thank you!
Very lovely video! Incidentally, if you take your diagram from 4:38 and shift each circle vertically so that they're tangent to the x-axis, you get what are known as "Ford circles". Astonishingly, they end up exactly tangent to _each other_ as well!
Didn't think I'd see one of my old professors on TH-cam. Nice video.
I didn't expect to watch this whole video but I did. Congrats kAN!
i am SO happy ur video won!!!!! this was so so sooooo good
Wow. That was amazing. Thank you for sharing
So thrilled you won!!! I've watched most of your videos and I really appreciated your way of explaining! Good job!
whatttttt this is the most exciting math video that ive seen!!!
Really fun video and great music choice! Thank you!
Oh man, the cliffhanger! Can't wait for that video :)
Amazing content. What a great motivation for continued fractions. This has to be my favourite #SoME3 entry.
Congratulations on being one of the winners! This was such an interesting video!
I think this is a WILDLY helpful video. Awesome job.😎
Your videos always give incredible insight, and this one was a joy to watch as well!
But one thing I can recommend regarding the animation is to make things fade in instead of just having them pop up (like with the zooming at 2:14 or with the spheres disappearing when the camera moves through them). Of course nothing important and can be ignored when it would take too long to implement, just something I think about as someone who plays around with procedural animations.
I've noticed that my favorite visualisations also inspire a metaphysical terror in me. These were very good on that account!
Absolutely awesome. Definitely my favorite some3 video so far.
So cool! I remember reading about continued fractions but this was a beautiful explanation of them!
A-M-A-Z-I-N-G video! Thank you so so much for the animated insights into the real numbers! I worked with continued fractions at my analysis course but I never imagined it this way.
And your channel name is awesome too!
This is SO BEAUTIFUL
It's incredible how far the 1/i^2 relationship stretches! It also describes an extraordinary range of natural processes, from harmonics to physical structures to pink noise. This in turn feels like a very natural method of approximating a position, and I can't wait to rethink some ideas with it in mind. Great video!
Thanks so much for this fascinating conference ! I loved already continuous fractions but you gave me further reasons to keep on...
Congratulations on winning #SoME3 !
Wonderful video.
I especially enjoyed the way you took us from 2d plane down to a first person view of the number line.
It's now got me thinking how this concept would extended to complex numbers...
You ma'am, are a genius!
definitely me favorite video of #SoME3 so far
Awesome video, you do a great job of showing interesting stuff while still keeping it basic and approachable. Keep it up :D
Neat! Continued fractions remain mysterious to me, but this is a great geometric connection.
I've heard about all these things separately, but never together, and with these visuals! Great work :D
What a great and well-explained video!
Really nice presentation ✌🏼🐻❄️
Thank you so very much for giving the Reals some Voice.
Outstanding! Subscribed!
Incredible video. Thank you!
Okay, but C. Series as a mathematicians name is just great.
There are quite a few interesting ways to use this! For example, you can make a line drawing algorithm out of this, that expands the slope of the line as a continued fraction, and draws the line recursively from this; because pi is about 355/113, the line (in pixels) is the same in the first chunk of 355 pixels as in the second 355 pixels, and this goes on for a while.
Beautifully presented and produced! Your visuals are very impressive -- if you could bring this sort of visualization to bear on the connection between Pell's equation and continued fractions, that would be stunning I'm sure.
What a lovely and interesting video, thanks!
Pls keep going on the hyperbolic geometry suff. I addicted to your video now!
This is awesome!
The Sylvester-Galai Theorem in Euclidean Geometry describes the existence of Irrational numbers! Whoa!
7:25 once you dropped down into the origin my brain immediately made the connection between the inverse square law & what was being talked about previously
This is fantastic, thank you!
Wow! great explanation and visualization - thanks!
Wow great video. I like the rethinking of the most fundamental concepts like that and the visualizations
amazing work!
This was brilliant! Really enjoyed it.
Awesome video!
great work!
An eyes openning video.
I love your room
I never thought of continued fractions as binaries.
beautiful and thought provoking, thank you so much!
Well presented. I've come across these representations but never truly understood them until now 🙏
Great video. Thanks!
This reminds me of a video by Numberphile years ago, IIRC, about how Phi (1.618...) is the "closest miss" of all those bubbles!
EDIT: I did not then understand the connection of such a projection to its significance within the Real Number "line". Thank you for filling in some gaps in my understanding. :)
Indeed, Phi corresponds to the sequence RLRLRLR...
I wrote a comment to the video with sum phi-observations included.
Opposite, actually. It's is the furthest miss. It is the hardest to approximate.
@@farklegriffen2624 The continued fraction / Stern-Brocot paths of φ is not approximate, it's periodic and thus exact. The basic path representation is LRLRLR etc, which looks much nicer with chiral symbol notation:
etc.
There's lot more to say, but let's leave that for another discussion.
@@santerisatama5409 Seems you misunderstood the comment you're responding to. I think they mean φ is the hardest real to approximate with rationals, which was the point of that numberphile video indeed.
Great work. I world like to watch more on continous fractions.
What is the argument that is "how the reals want to be organized"? It is beautiful and helpful (especially when using it in different contexts), but why would it be considered more natural?
She stated this very quickly at the end, but the argument is that if you assume that the rational numbers are your “starting point” for the real numbers (the basic things that you build real numbers out of) then this specific sequence of rationals is the best way you can describe the real numbers. It is “natural” in that, when looking at rational approximations, this sequence is the one that goes the fastest while also always existing.
If we’d chosen a different starting point then we would have gotten a different result. For example, if you start with the finite decimals (that is, decimal expansions which eventually terminate) then the infinite decimal expansion is the best sequence to go by.
For a slightly less arbitrary example, the copy of the real line which exists in the Surreal numbers is created out of dyadic fractions, that is, numbers which are equal to a whole number divided by a power of two. With this starting point, the binary (base-2) expansion would be best.
@@TheBasikShow Thanks.
Nicely done! 😊
Really cool!
Thank you for this video. I had never seen the motivation for the mediant spelled out clearly like you did using the 2D plane.
This was amazing.
this makes so much more sense than anything school ever tried to do with maths.
I love youm om! :)
Wonderful video!
Holy smoke didn't know there is such deep connection between the reals, projective geometry and complex plane.
This was great!
Great video
I've been working on this for a couple of years now. Mapping rationals onto a grid and intuiting irrationals as missing all the coordinates in the plane. Ugh, they have to start teaching this way in school!
Wow awesome video! Wld love a series on hyperbolic geometry and continued fractions
Congratulations on winning the contest; it was well-deserved! Is there some way to use this approach to understand the irrationality measures of a number?
If you're designing a calculator, you can use the Farey Sequence to algorithmically calculate the rational fraction for any decimal number. When I was designing my own calculator, I essentially did a binary search using the underlying concept.
This was a great video. This explained an interesting link between the p-adic numbers and the reals. In proofs, the reals are often denoted as being inf-adic, which sounded strange and mysterious before. But the business of choosing sub-bubbles to generate the continued fraction rep is highly reminiscent of the base p rep in p-adic number systems! Given that it makes it seem that the continued fraction rep is the “correct” way to write a real number in some sense. Your belaboring of the large number early in pi’s expansion is what clicked this into place for me. Thank you for dispelling my confusion. Anyone interested in details should read a bit of Gouvea. It is an excellent introduction to p-adics.
I wish this came up in my feed sooner!
really cool video! I liked it a lot
Yes it is Beautiful Mathematics and of course everyone will want to collate the labelling with what they think they know all-ready, ie in resonance-recognition.
Great video. Very briefly I thought this was going to veer into p-adic numbers.
amazing! i want to see the follow up!
Amazing!
This also sort of explains why the golden ratio φ is like the “most irrational number” - its continued fraction ‘address’ consists of only 1s - so all the rational approximations are similarly bad! Since the coefficients are always natural numbers, 1 is the worst possible!
Edit: fixed it’s instead of its.
I wonder if it would be possible to share this video with someone who’d never studied math beyond arithmetic and they would understand why some people find mathematics so beautiful.
My God this is such a perfect video
Not me googling "fairy subdivision of the real line" thinking it was another piece of colourful maths terminology (like the Friendly Giant!)
I wonder what the rules of arithmetic would look like for this system. Like you said it is most likely not going to take over base-10, but I am interested in seeing what progress we have made in this field.
Simply ... Wow! :-)