TH-cam has a lot of trash on it - and then it has things like this. I think this is a serious contender for the best STEM-related video I've ever seen.
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.
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): >>>>>
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.
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!
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.
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.
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!
Here's how to calculate the numbers used in the sequence of a continued fraction. You take the number, - the biggest whole number you can find, then flip the remainder as a fractional addition, and then repeat. ie: 3.14 - 3 = .14, 1/.14= 7... 1/... = 15... 1/...= 1... 1/... = etc where 3,7,15,1 would be the 'address' instead of 3,1,4,...(base 10 address) Loved the video, and now that i understand the process to generate continued fractions, I agree with you, it's actually VERY easy to calculate the continued fraction. It only requires simple subtraction and division, where as calcuating base 10 of a number also requires simple subtraction and division. Now getting the actual 'value' back from a continued fraction, is a lot more involved (every address you go, requires another division and addition (and the addition involves multiplication), where as with any normal base address it only requires 1 simple multiplication per address, so indexing in is much easier than indexing out, which means numberical bases are here to stay, like you said, but its still quite clever and I loved thinking through the arguments and watching the visualization. Edit: I did some more thinking about this, and actually you need the base 10 system to write out the continued fraction in the first place. So this is an index, of an index. If we wrote all the numbers in base 2, the continued fraction would look way different. Doesn't change much, but I did overlook that, this isn't a 'better' address divorced of the base 10 address, it's an even more specific, base 10 address.
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!!
@@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
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...
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!
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.
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. :)
@@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.
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!
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
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.
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.
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.
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.
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.
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!
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.
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.
I came across a rational approximation of Pi a few years ago, while playing with my phone's calculator app. I had been quite fond, still am, of 355/113, since it is easily memorizable, but still irrational. Now I'm no engineer, nor a math student. I might describe myself as an 'innocent bystander'; I do possess a bit of curiosity. So I rounded off Pi to 3.1416, which is as accurate as I'm likely ever to need, and Lo 'n' Behold, I had a rational approx for Pi. It can be expressed as 3927/1250, or 3+(177/1250). I await the inevitable call from the Nobel Prize committee.
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.
I wish I had a better mastery of continued fractions. I still feel like I don't know basic things about them. This video inspires me to finally dig in on them. Thank you!
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.
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?
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.
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.
"Why does it have this amazing 292 early on?" (13:38) Seems to me it must be "Because it can't be otherwise". No matter what crazy universe one finds themselves in, mathematical truths always apply.
I feel the urge to correct you here, but that can be a slippery slope so please keep in mind that I'm trying to expose my view on the subject instead of imposing it. Questions starting with "why" can be interpreted in various ways. You seem to have understood it like when people say "Why does the golden ratio have so many properties", implying "it must be magical ! / it must have been chosen by a god !". In that case "why" implies that a choice was made, with an intention. The usual math answer is "it's just a number", meaning that it doesn't need to be chosen to have interesting properties, they emerge from math definitions, and if we really wanted then any number (maybe not all transcendentals ?) can have interesting properties if we dig deep enough. I believe that was your intention when you said "it can't be otherwise" however that sounded dogmatic, and seems to shut down any attempt to study the number further. Instead, I think "why" here means "can I find a neat explanation for this property ?". It's not about finding a spiritual significance to the number, but rather looking for a reasoning that makes us go "oh yeah it makes sense". Maybe even "oh so this can be applied to something else...". And then about crazy universes, I don't know how you can claim anything about them since I presume you can't visit them either. I see what you mean about math being self-contained and independent of the world but that's not a truth, that's an opinion that can't be tested. Many people think instead that math is closely related to the world by the fact that it emerges from our brains and their sensory experience of the world. We could claim that until we got an idea of what's a circle, pi just didn't exist. Again that's just an opinion, a way to see things. In that context, mathematical truths don't always apply, because until their subject is defined, they are nonsensical. Or maybe your comment was an "anthropic" joke, because a "crazy universe one find themselves in" implies that "one" is a human, so the universe can only be ours until proven otherwise.
@@ghislainbugnicourt3709 I don't care if you find me dogmatic. I try to speak as unambiguous as I can and let people respond how they like. The "Why" was not mine, but the video author, and I simply added my answer. As for math in other universes, all mathematical truths must hold because mathematical truths transcend physics. I don't even know what it would mean for any universe to have different mathematical truths. It's not that I know so much about math or physics. It's because the question is a category error. But try to describe such a universe if you can.
So it seems we will need a method like this origin perspective when we get faster than light travel, to avoid stars. It would be interesting to apply this to a 3D star map to find the origin of the universe , if there is one.
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!
My dumbass read this as “ranking all real numbers” like there would be a tier list of infinite length
good luck finding a tier list with an uncountably infinite amount of tiers
by Cantor's diagonalization arguments such a tier list cannot exist
>
Everywhere I go with visual representations for math, I ended up seeing infinitely repeating fractals
TH-cam has a lot of trash on it - and then it has things like this. I think this is a serious contender for the best STEM-related video I've ever seen.
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.
"A real number that cannot be described in finitely many English words"
*boom Berry's paradox*
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): >>>>>
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.
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!
Great video. Very briefly I thought this was going to veer into p-adic numbers.
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.
Great visuals! Really enjoyed this explanation.
Holy smoke didn't know there is such deep connection between the reals, projective geometry and complex plane.
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.
whatttttt this is the most exciting math video that ive seen!!!
So thrilled you won!!! I've watched most of your videos and I really appreciated your way of explaining! Good job!
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!
Here's how to calculate the numbers used in the sequence of a continued fraction.
You take the number, - the biggest whole number you can find, then flip the remainder as a fractional addition, and then repeat.
ie: 3.14 - 3 = .14, 1/.14= 7... 1/... = 15... 1/...= 1... 1/... = etc
where 3,7,15,1 would be the 'address' instead of 3,1,4,...(base 10 address)
Loved the video, and now that i understand the process to generate continued fractions, I agree with you, it's actually VERY easy to calculate the continued fraction. It only requires simple subtraction and division, where as calcuating base 10 of a number also requires simple subtraction and division. Now getting the actual 'value' back from a continued fraction, is a lot more involved (every address you go, requires another division and addition (and the addition involves multiplication), where as with any normal base address it only requires 1 simple multiplication per address, so indexing in is much easier than indexing out, which means numberical bases are here to stay, like you said, but its still quite clever and I loved thinking through the arguments and watching the visualization.
Edit: I did some more thinking about this, and actually you need the base 10 system to write out the continued fraction in the first place. So this is an index, of an index. If we wrote all the numbers in base 2, the continued fraction would look way different. Doesn't change much, but I did overlook that, this isn't a 'better' address divorced of the base 10 address, it's an even more specific, base 10 address.
A real pleasure. Thank you!
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
i watched your video a few months ago and ive been thinking about it constantly, its changed the way i view number! super thanks!
I never thought of continued fractions as binaries.
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
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...
Didn't think I'd see one of my old professors on TH-cam. Nice video.
Congratulations on being one of the winners! This was such an interesting video!
A new classic here! I've had this video in my Downloads for some time.
I've noticed that my favorite visualisations also inspire a metaphysical terror in me. These were very good on that account!
Amazing content. What a great motivation for continued fractions. This has to be my favourite #SoME3 entry.
Thanks so much for this fascinating conference ! I loved already continuous fractions but you gave me further reasons to keep on...
Really fun video and great music choice! Thank you!
Thank you so very much for giving the Reals some Voice.
Oh man, the cliffhanger! Can't wait for that video :)
Absolutely wonderful video! This is really cool and I think definitely deserves strong recognition in the #SoME3 comp. Thank you for making this!
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!
Neat! Continued fractions remain mysterious to me, but this is a great geometric connection.
This is awesome!
The Sylvester-Galai Theorem in Euclidean Geometry describes the existence of Irrational numbers! Whoa!
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.
So cool! I remember reading about continued fractions but this was a beautiful explanation of them!
Pls keep going on the hyperbolic geometry suff. I addicted to your video now!
Thank you very much for the visualizations!
definitely me favorite video of #SoME3 so far
I think this is a WILDLY helpful video. Awesome job.😎
Absolutely awesome. Definitely my favorite some3 video so far.
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.
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!
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
i am SO happy ur video won!!!!! this was so so sooooo good
I didn't expect to watch this whole video but I did. Congrats kAN!
This has made me very happy. Fabulous
Great work. I world like to watch more on continous fractions.
Wow. That was amazing. Thank you for sharing
Well presented. I've come across these representations but never truly understood them until now 🙏
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.
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.
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.
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.
What a great and well-explained video!
You ma'am, are a genius!
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.
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!
What a lovely and interesting video, thanks!
Wow great video. I like the rethinking of the most fundamental concepts like that and the visualizations
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.
Awesome video, you do a great job of showing interesting stuff while still keeping it basic and approachable. Keep it up :D
Really nice presentation ✌🏼🐻❄️
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.
I've heard about all these things separately, but never together, and with these visuals! Great work :D
This was brilliant! Really enjoyed it.
Okay, but C. Series as a mathematicians name is just great.
Wow! great explanation and visualization - thanks!
Incredible video. Thank you!
Thank you for this video. I had never seen the motivation for the mediant spelled out clearly like you did using the 2D plane.
I came across a rational approximation of Pi a few years ago, while playing with my phone's calculator app. I had been quite fond, still am, of 355/113, since it is easily memorizable, but still irrational. Now I'm no engineer, nor a math student. I might describe myself as an 'innocent bystander'; I do possess a bit of curiosity. So I rounded off Pi to 3.1416, which is as accurate as I'm likely ever to need, and Lo 'n' Behold, I had a rational approx for Pi. It can be expressed as 3927/1250, or 3+(177/1250).
I await the inevitable call from the Nobel Prize committee.
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.
Outstanding! Subscribed!
I wish this came up in my feed sooner!
I wish I had a better mastery of continued fractions. I still feel like I don't know basic things about them. This video inspires me to finally dig in on them. Thank you!
Gosper arithmetic is worth checking out.
Wow awesome video! Wld love a series on hyperbolic geometry and continued fractions
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.
1:29 “Everyones favourite number [the circle constant divided by two]” 😡 not mine
this makes so much more sense than anything school ever tried to do with maths.
amazing work!
You can get the circumference of the observable universe to a planck length with only 64 digits if I recall
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?
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.
This is fantastic, thank you!
I realized the same but going L/R down the Stern-Brocot tree. This is prettier.
Please make more!
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.
Oh, it seems obvious now. The number of discs covering a given point is exactly it's denominator. Lovely.
beautiful and thought provoking, thank you so much!
11:15 wouldn't there be 4 R's so negative numbers would start with L and numbers less than one a single R like in the stern brocot tree.
Please find that "other time" to tell the story of hyperbolic geodesics and Farey sequences. Thank you!
An eyes openning video.
I love your room
"Why does it have this amazing 292 early on?" (13:38) Seems to me it must be "Because it can't be otherwise". No matter what crazy universe one finds themselves in, mathematical truths always apply.
I feel the urge to correct you here, but that can be a slippery slope so please keep in mind that I'm trying to expose my view on the subject instead of imposing it.
Questions starting with "why" can be interpreted in various ways. You seem to have understood it like when people say "Why does the golden ratio have so many properties", implying "it must be magical ! / it must have been chosen by a god !". In that case "why" implies that a choice was made, with an intention. The usual math answer is "it's just a number", meaning that it doesn't need to be chosen to have interesting properties, they emerge from math definitions, and if we really wanted then any number (maybe not all transcendentals ?) can have interesting properties if we dig deep enough. I believe that was your intention when you said "it can't be otherwise" however that sounded dogmatic, and seems to shut down any attempt to study the number further.
Instead, I think "why" here means "can I find a neat explanation for this property ?". It's not about finding a spiritual significance to the number, but rather looking for a reasoning that makes us go "oh yeah it makes sense". Maybe even "oh so this can be applied to something else...".
And then about crazy universes, I don't know how you can claim anything about them since I presume you can't visit them either. I see what you mean about math being self-contained and independent of the world but that's not a truth, that's an opinion that can't be tested. Many people think instead that math is closely related to the world by the fact that it emerges from our brains and their sensory experience of the world. We could claim that until we got an idea of what's a circle, pi just didn't exist. Again that's just an opinion, a way to see things. In that context, mathematical truths don't always apply, because until their subject is defined, they are nonsensical.
Or maybe your comment was an "anthropic" joke, because a "crazy universe one find themselves in" implies that "one" is a human, so the universe can only be ours until proven otherwise.
@@ghislainbugnicourt3709 I don't care if you find me dogmatic. I try to speak as unambiguous as I can and let people respond how they like.
The "Why" was not mine, but the video author, and I simply added my answer.
As for math in other universes, all mathematical truths must hold because mathematical truths transcend physics. I don't even know what it would mean for any universe to have different mathematical truths. It's not that I know so much about math or physics. It's because the question is a category error. But try to describe such a universe if you can.
Not me googling "fairy subdivision of the real line" thinking it was another piece of colourful maths terminology (like the Friendly Giant!)
So it seems we will need a method like this origin perspective when we get faster than light travel, to avoid stars. It would be interesting to apply this to a 3D star map to find the origin of the universe , if there is one.