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): >>>>>
As a mathematician, I am impressed and inspired by the beautiful and understandable visualization of some pretty deep mathematical concepts. Great work.
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!
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!!!)
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.
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 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!
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.
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.
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
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!
🎯 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
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 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.
The point at 12:00 is fascinating - does this mean there are certain rational numbers that are, inherently, good landmarks for reals? What the irrational we are talking about is didn’t seem to matter to the number of lines hitting it
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.
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!
If you just have L/R options, would there be a significant difference between using fairy* subdivisions instead of binary ones? Also, the fractions you put on the grid don't quite make sense to me. (1,1) is further from the origin than (0,1) and (1,0), so it seems like it should represent 1/2 instead, since it looks smaller, with (1,0) representing 1, (1,2) and (2/1) representing 1/3 and 2/3, (1,3) and (3/1) representing 1/4 and 3/4, and so on. *[sic is what you get for being called Farey and not spelling it out :P]
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.
13:15 This is beginning to remind me of Mandelbrot set zoom visualisations in the way they generally seem to sort of swing back and forward between quite chaotic areas and areas that could be perhaps described as 'metastable', seeming like a repetitive pattern has been reached only to then fall back to a chaotic one. This similarity feels tantalisingly meaningful, although I couldn't tell you what I think it means. I wonder if the change in size of the L and R numbers (using that as a possible representation of the level of chaos at a given resolution) as we home in would follow a discernible pattern?
This is probably more tangential to what you are saying, but there are remarkable patterns to continued fractions "generically". Khinchin's constant and the Guass-Kuzmin distribution gives hints to that. They basically say almost all numbers (except for measure zero set of numbers) have a lot of control on the behavior of the continued fractions. For example, roughly 40% or L's and R's exponent will be 1 for most numbers (it is unknown if pi is one of those numbers) even though apriori there is no real reason to think that--you can just write down any list of numbers to get a continued fraction but there is a "preference" for smaller numbers. This preference for smaller numbers appearing in the continued fraction expansion is also implicit in the existence of Khinchin's constant--which is the limit of the geometric mean of the first n numbers in the continued fraction, as n goes to infinity, of almost all numbers (that is right, almost all numbers have the same exact geometric mean!).
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'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!
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.
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?
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.
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.
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.
"It's a lot simpler to do addition on decimals" - can we do addition on continued fractions directly? I.e. find an algorithm that takes one potentially infinite sequence of Lefts and Rights and a second potentially infinite sequence of Lefts and Rights and, without detour through a decimal (or your favourite base) expansion or for rationals through regular fraction arithmetic (which will then have to be reduced first) yields the sequence of Lefts and Rights which correspond to the sum, product, or fraction?
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.
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.
Am I right in saying that there isn't a pattern to the continued fraction basically by definition of an irrational? In other words, if we knew a pattern, we would have a closed form solution for instantly calculating pi to an arbitrary degree of accuracy, which is the same as saying it's rational (contradiction)? Now then I guess what remains is proving that pi is irrational.
🎯 Key Takeaways for quick navigation: 00:01 📊 The real line contains both rational and irrational numbers, with intervals around rationals containing irrationals, and vice versa. 00:29 🔄 Decimal expansions are just one way to organize real numbers; there's a geometric approach based on continued fractions. 01:41 🥧 Decimal expansions represent real numbers as sequences of rational numbers getting closer to the target number. 03:38 🧮 The quality of a rational approximation to a real number is based on the cost (denominator size) and benefit (approximation quality). 05:13 🌟 The Recluse Theorem visually shows how rationals and irrationals are organized in the real line, with irrationals covered by infinitely many disks. 08:38 🔀 The Fairy Subdivision concept breaks the real line into intervals using mediant rationals, providing a different way to organize it. 11:09 🚗 Pi's continued fraction expansion is a unique way to address it, with driving directions consisting of Ls and Rs indicating left or right at each step. 13:39 🤔 The continued fraction expansion of Pi has some large coefficients early on, like 292, making it an exceptional approximation. 14:21 🌌 Hyperbolic geometry plays a role in understanding continued fractions, but it's a complex topic beyond the scope of this discussion.
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.
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.
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
>
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
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): >>>>>
As a mathematician, I am impressed and inspired by the beautiful and understandable visualization of some pretty deep mathematical concepts. Great work.
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!
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!!!)
So thrilled you won!!! I've watched most of your videos and I really appreciated your way of explaining! Good job!
Congratulations on being one of the winners! This was such an interesting video!
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.
whatttttt this is the most exciting math video that ive seen!!!
Great visuals! Really enjoyed this explanation.
i watched your video a few months ago and ive been thinking about it constantly, its changed the way i view number! super thanks!
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 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!
A real pleasure. Thank you!
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.
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.
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
Really fun video and great music choice! Thank you!
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!
i am SO happy ur video won!!!!! this was so so sooooo good
🎯 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
Absolutely wonderful video! This is really cool and I think definitely deserves strong recognition in the #SoME3 comp. Thank you for making this!
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.
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
What a great and well-explained video!
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.
Thank you very much for the visualizations!
I've noticed that my favorite visualisations also inspire a metaphysical terror in me. These were very good on that account!
The point at 12:00 is fascinating - does this mean there are certain rational numbers that are, inherently, good landmarks for reals? What the irrational we are talking about is didn’t seem to matter to the number of lines hitting it
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.
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!
Thanks so much for this fascinating conference ! I loved already continuous fractions but you gave me further reasons to keep on...
If you just have L/R options, would there be a significant difference between using fairy* subdivisions instead of binary ones?
Also, the fractions you put on the grid don't quite make sense to me. (1,1) is further from the origin than (0,1) and (1,0), so it seems like it should represent 1/2 instead, since it looks smaller, with (1,0) representing 1, (1,2) and (2/1) representing 1/3 and 2/3, (1,3) and (3/1) representing 1/4 and 3/4, and so on.
*[sic is what you get for being called Farey and not spelling it out :P]
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.
Great video. Very briefly I thought this was going to veer into p-adic numbers.
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.
A new classic here! I've had this video in my Downloads for some time.
Absolutely awesome. Definitely my favorite some3 video so far.
13:15 This is beginning to remind me of Mandelbrot set zoom visualisations in the way they generally seem to sort of swing back and forward between quite chaotic areas and areas that could be perhaps described as 'metastable', seeming like a repetitive pattern has been reached only to then fall back to a chaotic one.
This similarity feels tantalisingly meaningful, although I couldn't tell you what I think it means. I wonder if the change in size of the L and R numbers (using that as a possible representation of the level of chaos at a given resolution) as we home in would follow a discernible pattern?
This is probably more tangential to what you are saying, but there are remarkable patterns to continued fractions "generically". Khinchin's constant and the Guass-Kuzmin distribution gives hints to that. They basically say almost all numbers (except for measure zero set of numbers) have a lot of control on the behavior of the continued fractions. For example, roughly 40% or L's and R's exponent will be 1 for most numbers (it is unknown if pi is one of those numbers) even though apriori there is no real reason to think that--you can just write down any list of numbers to get a continued fraction but there is a "preference" for smaller numbers. This preference for smaller numbers appearing in the continued fraction expansion is also implicit in the existence of Khinchin's constant--which is the limit of the geometric mean of the first n numbers in the continued fraction, as n goes to infinity, of almost all numbers (that is right, almost all numbers have the same exact geometric mean!).
Holy smoke didn't know there is such deep connection between the reals, projective geometry and complex plane.
Amazing content. What a great motivation for continued fractions. This has to be my favourite #SoME3 entry.
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 didn't expect to watch this whole video but I did. Congrats kAN!
I think this is a WILDLY helpful video. Awesome job.😎
Pls keep going on the hyperbolic geometry suff. I addicted to your video now!
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!
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!
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.
Didn't think I'd see one of my old professors on TH-cam. Nice video.
Oh man, the cliffhanger! Can't wait for that video :)
This is awesome!
The Sylvester-Galai Theorem in Euclidean Geometry describes the existence of Irrational numbers! Whoa!
Please find that "other time" to tell the story of hyperbolic geodesics and Farey sequences. Thank you!
Wow. That was amazing. Thank you for sharing
This has made me very happy. Fabulous
I never thought of continued fractions as binaries.
Great work. I world like to watch more on continous fractions.
Thank you so very much for giving the Reals some Voice.
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.
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?
definitely me favorite video of #SoME3 so far
1:29 “Everyones favourite number [the circle constant divided by two]” 😡 not mine
Neat! Continued fractions remain mysterious to me, but this is a great geometric connection.
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.
Wow great video. I like the rethinking of the most fundamental concepts like that and the visualizations
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.
What a lovely and interesting video, thanks!
I've heard about all these things separately, but never together, and with these visuals! Great work :D
I love youm om! :)
Well presented. I've come across these representations but never truly understood them until now 🙏
This was brilliant! Really enjoyed it.
Outstanding! Subscribed!
Wow awesome video! Wld love a series on hyperbolic geometry and continued fractions
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.
Incredible video. Thank you!
this makes so much more sense than anything school ever tried to do with maths.
"It's a lot simpler to do addition on decimals" - can we do addition on continued fractions directly? I.e. find an algorithm that takes one potentially infinite sequence of Lefts and Rights and a second potentially infinite sequence of Lefts and Rights and, without detour through a decimal (or your favourite base) expansion or for rationals through regular fraction arithmetic (which will then have to be reduced first) yields the sequence of Lefts and Rights which correspond to the sum, product, or fraction?
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.
Really nice presentation ✌🏼🐻❄️
Wow! great explanation and visualization - thanks!
11:04 ~= the video "Infinite fractions and the most irrational number" at 7 : 41 mark on Mathologer channel :)
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.
Am I right in saying that there isn't a pattern to the continued fraction basically by definition of an irrational? In other words, if we knew a pattern, we would have a closed form solution for instantly calculating pi to an arbitrary degree of accuracy, which is the same as saying it's rational (contradiction)? Now then I guess what remains is proving that pi is irrational.
Please make more!
You ma'am, are a genius!
🎯 Key Takeaways for quick navigation:
00:01 📊 The real line contains both rational and irrational numbers, with intervals around rationals containing irrationals, and vice versa.
00:29 🔄 Decimal expansions are just one way to organize real numbers; there's a geometric approach based on continued fractions.
01:41 🥧 Decimal expansions represent real numbers as sequences of rational numbers getting closer to the target number.
03:38 🧮 The quality of a rational approximation to a real number is based on the cost (denominator size) and benefit (approximation quality).
05:13 🌟 The Recluse Theorem visually shows how rationals and irrationals are organized in the real line, with irrationals covered by infinitely many disks.
08:38 🔀 The Fairy Subdivision concept breaks the real line into intervals using mediant rationals, providing a different way to organize it.
11:09 🚗 Pi's continued fraction expansion is a unique way to address it, with driving directions consisting of Ls and Rs indicating left or right at each step.
13:39 🤔 The continued fraction expansion of Pi has some large coefficients early on, like 292, making it an exceptional approximation.
14:21 🌌 Hyperbolic geometry plays a role in understanding continued fractions, but it's a complex topic beyond the scope of this discussion.
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.
amazing work!
Thank you for this video. I had never seen the motivation for the mediant spelled out clearly like you did using the 2D plane.
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.
amazing! i want to see the follow up!
I think that may have helped me over the conceptual hurdle for "why do I need projective geometry?" TYVM
@9:11: "...let's just keep doing this forever."
Me: I accept your proposal.