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

You've just transformed the way I think about numbers forever

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.

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!!!)

As a mathematician, I am impressed and inspired by the beautiful and understandable visualization of some pretty deep mathematical concepts. Great work.

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.

@tubebrocoli

9 ай бұрын

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.

"A real number that cannot be described in finitely many English words" *boom Berry's paradox*

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!

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!!

@X22GJP

6 ай бұрын

Sure ya' did, we can all write comments after watching a video through!

@erickugel1376

3 ай бұрын

I totally thought she was saying "fairy" expansion the whole time 💀💀

@lexinwonderland5741

2 ай бұрын

@@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

Most real numbers are not computable, most are just the random L and R sequences

@johnpeterson3386

10 ай бұрын

Exactly! And this is such a powerful way to understand that, which is totally new to me. Plug that into your Turing machine 😂

@lydianlights

10 ай бұрын

That is a cool way to think of it.

@cougar2013

9 ай бұрын

Maybe they don’t want to be found

@dsudaniel3003

8 ай бұрын

Sorry, but what is L or R?

@johnpeterson3386

8 ай бұрын

@@dsudaniel3003 Left or Right

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 '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): >>>>>

Great visuals! Really enjoyed this explanation.

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!

A new classic here! I've had this video in my Downloads for some time.

Didn't think I'd see one of my old professors on KZread. Nice video.

Neat! Continued fractions remain mysterious to me, but this is a great geometric connection.

Absolutely wonderful video! This is really cool and I think definitely deserves strong recognition in the #SoME3 comp. Thank you for making this!

Oh man, the cliffhanger! Can't wait for that video :)

Congratulations on being one of the winners! This was such an interesting video!

So thrilled you won!!! I've watched most of your videos and I really appreciated your way of explaining! Good job!

Amazing content. What a great motivation for continued fractions. This has to be my favourite #SoME3 entry.

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!

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.

Awesome video, you do a great job of showing interesting stuff while still keeping it basic and approachable. Keep it up :D

So cool! I remember reading about continued fractions but this was a beautiful explanation of them!

Absolutely awesome. Definitely my favorite some3 video so far.

Thank you very much for the visualizations!

Thanks so much for this fascinating conference ! I loved already continuous fractions but you gave me further reasons to keep on...

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...

I think this is a WILDLY helpful video. Awesome job.😎

I didn't expect to watch this whole video but I did. Congrats kAN!

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!

Really fun video and great music choice! Thank you!

A real pleasure. Thank you!

Wow. That was amazing. Thank you for sharing

i am SO happy ur video won!!!!! this was so so sooooo good

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. :)

@columbus8myhw

9 ай бұрын

Indeed, Phi corresponds to the sequence RLRLRLR...

@santerisatama5409

9 ай бұрын

I wrote a comment to the video with sum phi-observations included.

@farklegriffen2624

9 ай бұрын

Opposite, actually. It's is the furthest miss. It is the hardest to approximate.

@santerisatama5409

9 ай бұрын

@@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.

@ghislainbugnicourt3709

8 ай бұрын

@@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.

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.

whatttttt this is the most exciting math video that ive seen!!!

This has made me very happy. Fabulous

Incredible video. Thank you!

I've noticed that my favorite visualisations also inspire a metaphysical terror in me. These were very good on that account!

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!

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

Wow great video. I like the rethinking of the most fundamental concepts like that and the visualizations

Wow! great explanation and visualization - thanks!

Congratulations on winning #SoME3 !

What a lovely and interesting video, thanks!

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.

This is SO BEAUTIFUL

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?

@TheBasikShow

10 ай бұрын

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.

@ThingOfSome

10 ай бұрын

@@TheBasikShow Thanks.

Okay, but C. Series as a mathematicians name is just great.

beautiful and thought provoking, thank you so much!

This is fantastic, thank you!

This was brilliant! Really enjoyed it.

Pls keep going on the hyperbolic geometry suff. I addicted to your video now!

Well presented. I've come across these representations but never truly understood them until now 🙏

Thank you for this video. I had never seen the motivation for the mediant spelled out clearly like you did using the 2D plane.

definitely me favorite video of #SoME3 so far

Great work. I world like to watch more on continous fractions.

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.

Really nice presentation ✌🏼🐻‍❄️

An eyes openning video. I love your room

Great video. Thanks!

Wow awesome video! Wld love a series on hyperbolic geometry and continued fractions

Outstanding! Subscribed!

This is awesome! The Sylvester-Galai Theorem in Euclidean Geometry describes the existence of Irrational numbers! Whoa!

I've explored some of this myself, though I like multiplying 2×2 matrices containing only ones and zeroes. It allows continued fractions to be calculated associatively: don't have to start at the "deepest" part first, you can add more accuracy by calculating each successive term successively.

You ma'am, are a genius!

Thank you so very much for giving the Reals some Voice.

I remember finding some of these patterns and finding others (that unknown to me were already found long long before) I had focused in on the fractions between 1 and 2. I was looking for the best ratios for musical harmonies, 1/1 being unison and 2/1 being the Octave. I arbitrarily decided that how close that fraction in x,y to the 0,0 was it's strength of harmony.

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.

amazing work!

this makes so much more sense than anything school ever tried to do with maths.

My dumbass read this as “ranking all real numbers” like there would be a tier list of infinite length

Holy smoke didn't know there is such deep connection between the reals, projective geometry and complex plane.

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!

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?

I wish this came up in my feed sooner!

I never thought of continued fractions as binaries.

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.

Nicely done! 😊

great work!

really cool video! I liked it a lot

Awesome video!

Wonderful video!

Great video. Very briefly I thought this was going to veer into p-adic numbers.

Really cool!

Continued fractions can be quite useful. The continued fraction of many special functions converge faster and in a larger part of the complex plane than the Taylor series for the same function.

amazing! i want to see the follow up!

My God this is such a perfect video

🎯 Key Takeaways for quick navigation: 00:01 📏 *Introduction to real numbers and their representation.* 03:10 🧮 *Decimal expansion, limitations, and notable approximations.* 04:57 🎈 *Visualizing rational numbers, Reuleaux theorem, and their relationship.* 07:25 🌐 *Rational numbers explained through projective geometry.* 09:35 🛣️ *Fairy subdivision, its comparison to decimals, and pi's continued fraction.* 12:57 🥧 *Discovering Pi's true nature via its continued fraction expansion.* Made with HARPA AI

Awesome work, as a mathematician I came up with a pretty similar idea while trying to describe the real numbers without using any particular basis because it felt too boring and mostly going the wrong way into understanding the real nature of real numbers. What would be interesting next would be to work on the algebra behind the LR-notation. Are some calculations on algebraic numbers simplified?

Love the concept, visuals, and video as a whole. at around 4:45 you mention that according to Dirichlet's Theorem, the rationals are only covered by finitely many discs, but that irrationals are covered by infinitely many. Intuitively, this seems related to the way that zooming in on pi required passing through infinitely many lines that define it's Fairy location. I did find the former point vexing and have had to ponder the image for some time before really seeing this connection, and only vaguely at that. Would love to hear anything you have to say on the relationship between the number of discs covering a point and it's value. Thanks for the video.

@Robinsonero

8 ай бұрын

Oh, it seems obvious now. The number of discs covering a given point is exactly it's denominator. Lovely.

I had always felt that reals are pretty arbitrary so I had been seeking something more 'organic' (mostly looking at surreals). This video convinced me that's not necessarily the case! I wonder if we can use this to neatly speak about reals in constructive settings - like many proof assistants.

This was amazing.