New Recipe for Pi - Numberphile
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- Опубліковано 14 жов 2024
- Interview with Sinha & Saha at • Pi-oneers (interview w... and the extra physics bit is at • New Pi Formula (the ex... --- More links & stuff in full description below ↓↓↓
This video features Tony Padilla: www.nottingham...
Arnab Priya Saha & Aninda Sinha are based at the Centre for High Energy Physics, Indian Institute of Science - chep.iisc.ac.in
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And on arxiv: arxiv.org/abs/...
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What flavour?
Lifelong Pi mathematician here. It does have to do with circles. Chudnovsky specifically has to do with circles, in the complex plane, using hyperbolic geometry, using 163i as its basis. Ramunujan's is the same, but for 1i. You can make single series reps of pi with ramanujan sato series with all the Heegner numbers 1,2,3,7,11,19,43,67,163. You can make infinite ramunjan sato series if you allow multiple sums. There is ALWAYS a circle, lol.
Bro half the "mathematicians" on numberphile are so confidentially incorrect
@@aniketdhumal2692They're not confident though? Does "I don't know" and "maybe" and "I think" sound like someone brimming with confidence or someone making an educated guess? FWIW, pi rarely has to do with circles. Even in complex analysis, where you can correctly claim pi being a factor of a residue is due to integrating over a circle, it's just not that helpful the deeper you go.
Frankly, since numberphile typically only talks about recreational mathematics and undergraduate mathematics, it's extremely rare for them to even have to chance to be wrong about something. It really says a lot about you if you can comment something like that in spite of this.
@@SilverLining1 you say this but there's obvious problems in many of these videos. Heck Mr. Parker is known to make mistakes. Kinda cute how you say this is easy maths and still have shitton of faults every other video lol
Does this mean the formulae converge to 1/pi when an infinite number of terms are taken?
I'll take your word for it.
7:32
- Tony: the 105th trillionth digit of PI is 6.
- Brady: good to know
I loved that moment.
I was 10% sure it would be. How nice that it has been confirmed..! 😇
With a certainty of 50%, the next one is between 0 and 4 included.
How do they know this chudnovski formula doesn't deviate at 89 trillion digits or something? Don't you need another algorithm of equal or superior accuracy to verify?
I think you only need around 62 digits of PI to calculate the circumference of the universe from its radius and only be off by a Planck length. More digits are just for bragging rights, and measuring computer speeds. Nothing wrong with that of course. 😅
Also, the universe is expanding, so we'll need another digit in about 86 bn years.
Not sure if my math is correct:
14 bn LY is about 10²⁶ meters.
10²⁶/10⁶² = 10^-36
Planck length = 1.6 * 10^-35
I never want this channel to end. Brady and I are roughly similarly aged, and if I'm refreshing my YT subs page at 80 and there aren't any new videos, I'm just gonna lay down for good.
Best math channel in my opinion.
I went to school with brother Arnab, was two batch junior.
He was already a local legend in that area when it comes to maths back in 2009
A legend, you mean :)
@@DadgeCity No, an urban legend.
He roams the underground pipe network at night like an alligator
@@Irondragon1945
And was always thirsty for novel math problems
What's a batch junior?
@@RonJohn63 Indian way of saying, Two grades behind 😂😂
The cool thing about the Indian series 1 - 1/3 + 1/5 ... is that it had an error term that vastly increased its usefulness. After ten terms, the sum is way off (3.04) but the error term ((n^2+1)/(4n^3+5n)) zooms it right to 3.14159270
Where does that error term come from?
@@AureliusRmathologer has an excellent video explaining this exact thing, do check it out!
@@AureliusR the Madhava_of_Sangamagrama wiki page speculates it was from working with continued fractions
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In earlier days of PC programming (80s 90s), the BASIC did not have pi included. To get it with the program prcision, we used the atan(1)*4
Which is rather ironic since AppleSoft BASIC has 1/2 PI and 2PI constants in ROM.
@@billferner6741
Same same with FORTRAN-77.
And where did it get atan(1) from?
Most likely a function call for some n iterations of the taylor series expansion of arctan(1)
@@MichaelPohoreski LOL @ you not knowing what "ironic" means
Every time I think of a question the camera man asks it, it's so helpful
Yes, or when I haven't thought of it, it's always a great question
Except for my question "where that graph peaks at lambda = 3 and a bit, is THAT pi ???"
I’m unreasonably happy that the length of this video is 14:28
Last time I was this early the Parker Square was just an erroneous attempt at a magic square
Lo
And this is just Parker's Pi
At least he never lied to us about things being equal to -1/12. I think.
@@The.171 Since I'm German, YT offered me to translate your comment. Apparently "Lo" in English is "It" in German xD
@@Nachiebree it kind of is
IIRC, if you had two circles, each the size of the universe, one based on pi and one based on an approximate value of pi, you only need about 60 digits of approximate-pi for the two circles to be exactly the same. Any theoretical difference would be smaller than the Planck length, the smallest possible distance in the fabric of space.
Planck Length is *believed* to be the smallest possible distance but modern Scientists have nothing even _close_ to measuring this (several order of magnitude off.)
When working with machinery, it's four places. That's because of material breakdown and at the microscopic level the line is still jagged, but not to a point where it structurally matters.
@@orlock20 You wouldn't happen to know what tolerances are mil-spec grade by chance? Not looking to reading AS9100 spec. :-)
@@MichaelPohoreski The highest accuracy mentioned for anything was .00001 and that was used as a joke.
@@MichaelPohoreski it is not smallest. it where grabiti and other forces, are equal or smth. it is never ending zoo of "particles", that do not really excist. and are figment of imagination.
I'm incredibly shocked that science (/maths) journalism would overhype and completely misrepresent a technical result. Well, ok, not that shocked. Not shocked at all really.
Don't be shocked. Here you are on the video.
Yeah that shouldn't come as a surprise whatsoever. I hate journalists so much it's unreal.
You should be hating the advertisers and executives who have turned journalism into the farce it is to get more outrage bait and clicks for revenue
From what I recall, it was only hyped in India. It is an interesting aside and the authors downplay its overall importance. It is novel, and a little interesting, but currently does not offer much, compared to Chudnovsky.
6:41 640320^{3k}. Shades of exp{π√163}~~640320^3+744.
Hey you're right! Lol!
(just kidding, no idea what you said)
@@donweatherwax9318 😂
interesting observation, when viewed in WolframAlpha with /input?i=exp%28%CF%80%E2%88%9A163%29-%28640320%5E3%2B744%29 it gives error of only around 10^-12.
Not a coincidence, the results are connected!
@@vaakdemandante8772 It's called the Ramanujan constant, it's actually a very famous result. It's connected to a lot of deep areas, but Numberphile have a vid introducing it (called "163 and Ramanujan Constant"). The Wikipedia page on "Heegner numbers" is also worth a look to go deeper and work out what to google if you're interested.
I imagine the optimal lambda values require pi in the first place by beeing a transcendental number themself, which would require knowing pi in the first place to calculate (or better: approximate) that optimal lambda...
Edit: Nevermind, they definitly *are* transcendental numbers, since they are simply rational multiples of pi.
Only for the truncated series. I don't think there's a particular reason that the rate of convergence as a function of lambda should have an extremum at values of lambda that are finite expressions involving pi. But, they might.
@@SilverLining1
Well, if you don't truncate the rate of convergence is pretty much irrelevant, because without truncation it will simply give you the exact value of pi regardles of lambda. How fast it converges does not matter when the process is infinite...
It would be cool if you found a formula that gets close to finding the optimal value of lambda given an approximation of Pi, so that you could just alternate between the two formulas to get closer and closer.
The Madhava series is a Taylor series for arctan(1). I wonder whether this new representation is also a Taylor series of some kind
It is, but Madhava wouldn't have known that
Add a factor x^n to the sum and you have one.
@@landsgevaer Bingo. You can turn any series into a function. Whether that function turns out to be useful for anything or related in a meaningful way to existing functions is a different matter all together...
@@trueriver1950 Madhava did know about the power series expansions of sin, cos, arctan. Madhava founded the Kerala school of mathematics where they discovered differentiation, integration and power series expansions.
@@trueriver1950 Powell's Pi Paradox
Brady was correct that there must be an irrational value of lambda which yields pi exactly with 0 approximation error, but we will never be able to find it
If anything, finding the perfect lambda would require its own method of approximation, and knowing how Pi is it would somehow be defined in relation to it. Finding Pi with Pi is self-defeating, and otherwise improving an approximation with a second harder approximation isn't much better
@dunda563 yes, I was thinking this is essentially circular reasoning
Yeah, doesn't this method really just provide a convenient container for the magic constants in the other methods? I'm sure you could improve those series further just by finding the right constants, which would be the same as searching for ideal values of lambda here?
Successive approximations as a solution are not necessarily useless. See: Kepler's equation
I have a formula for Pi that is a really close approximation after one term, but never improves after that.
If pie is the meal then the series is the recipe.
Lovely way of describing a math formula.
Tony's enthusiasm and his ability to communicate a complicated subject to duffers like me, make him a must watch.
i love your editing style and how your videos stayed consistent throughout the years. great work!
Brady, the gawking rabble demands an explanation of where Ramanujan's and Chudnovsky's series come from.
This!
Sure, the equation is fairly simple, precise, and accurate; but WHY does it work?
@@Talon19, and how did they arrive at it???
Not sure if this true but I have the legend that Ramanujan saw this formula conjured into his mind. To be honest somehow I find this much more believable than actually the guy trying to find a formula with a summation and factorials. But man I see your point, it is mind bending to think how these formulae came!
Ramanujan himself had said the formula came to him in his dreams
Shut it mortal, It was revealed to me by the universe.
I suppose, from a computer scientist's POV, that the real question about any of them, is not how many iterations it takes to get n digits of pi, but how "expensive" the mathematical operations are. E.g. factorials become more expensive than exponents. Thus, one needs to figure out how many additions, subtractions, multiplications, relatively expensive divisions and exponents are required for at least n digits.
11:04
I saw a different video where another commentor pointed out that lamda =1,2,3,... Has increasing accuracy for pi, but lamba=iinfinity converges super slow, so its nice to see the "best" values for lamda to converge quickly.
Something that wasn't addressed in this video is that (from their appearance at least), the first 4 terms of each series are way way different in how complex it'd be to compute them. I wonder which series fares better with e.g. 1 minute or 10 minutes etc. of computation time, since some of these contain huge factorials or exponential terms.
I wonder how Madhava found that the limit of 1-1/3+... is π/4. I mean, it is, but the series converges so slowly that he couldn't have got all that close to π/4 with as many terms as he'd be able to sum using 14th-century tech.
I believe, persistence
If you look further, he actually used correction terms that basically boost the accuracy of the result.
This can be seen in action with:
4(1/1-1/3+1/5-1/7+1/9-1/11+1/13) = 3.2837...
4(1/1-1/3+1/5-1/7+1/9-1/11+1/13-1/(13*2-2)) = 3.1408...
Madhava didn't manually add up numbers, he discovered power series of trigonometric functions. Plug x=1 into the arctan series atan(x)=x-x^3/3+x^5/5-x^7/7… and voilà.
3 Blue 1 Brown proved the Madhava series in a easy to understand way in 1 of his videos
We are actually talking about the approximation of decimal places of pi.but what is the exact decimal places of pi ? How is it even calculated? How we can prove that the approximations actually approximating the decimal expansion of pi
Paper change interludes always make me happy.
he eh ehe e eeeeehe
@5:19
My favourite Ramanujan approximation is π ≈ (355/113)(1 - 0.0003/3533), with relative error less than one-half part per quadrillion. I found this years ago (don't remember where), wrote it in my reference book, and have not found a source.
3:14 (Well, about there, but I fancied using that timestamp 😉): How is Ramanujan's formula useful in practice? That √2 at the beginning seems to make life difficult, as the square root needs a good approximation before you can even start.
Not being a mathematician, I never noticed that. But darn tootin', I think you're right. Using one irrational number to define another one? Is that even valid? Far from it for me to question Ramanujan, but you really got me thinking.
Well, it only worked for a certain number of digits anyway, it was only ever spitting out approximations. That sqrt is only an issue if its infinite right? Otherwise it just needs to be close enough to get those few hundred correct digits, which is more than enough for any use case for pi really.
the good old 355/113 compared to the often taught 22/7 has a deviation from pi of 2.66/10 000 000 compared to 1.26/1000
which translates to a watch running less than 8.5 secs wrong per year compared to more than 11 hours
This is great because chef John at food wishes posted a recipe for peach pie on the same day.
In this video Tony showed several methods for approximating the digit 4.
One bit I'm curious about, when he says they've set a record for number of digits of pi calculated using the Chudnovsky formula, how do they *know* for certain at what digit they've accurately calculated, if they know that it's an approximation? My assumption would be that in order to know for certain that a calculated digit is correct, you'd need to go some level further to confirm it? Is it basically that you would need a calculation using N terms, then calculate the N+1'th term and the position of the first nonzero term gives you the confirmation of the last known-accurate digit? If all it takes is a known calculation/approximation at N terms and adding the N+1 term, that makes me wonder at *actually* how much computation time it takes for a supercomputer to determine the next term, it must be way more than I'd initially expect.
There are other ways to get the approximation of pi. Search for Spigot algorithm
This is the first puzzle that struck me about this video. The second puzzle, even more puzzling, is why do they not even _address_ this puzzle? Provability is the obsession of Mathematics.
The third puzzle is the near-absence of puzzlement about it in the comment section.
They should be validated against each other… inaccurate use on earth tethered activity isn’t a problem, but celestial activities could be disastrous over great distances
At 11:42 Brady raised a good question that I dont think was answered. The fact that lamba goes from positive to negative indicates that there is a value for which the calculation provides a 0% deviation. Indeed, for a few values of lambda, the deviation is absolute 0. Or perhaps if we zoomed in more we would find some asymptopes that lamba tends to infinitely small?
indian Acharya Madhav was really shocking for me how he discovered and purposed the infinite series of all trigonometric functions at that time when calculus was not invented and so many undiscovered things ????
I can’t be the only one who’s wondering what happens if Lambda is equal to Pi.
He mentioned Lambda should be a convergent series. Pi is a divergent series.
@@i_rish_0 Numbers themselves are neither divergent nor convergent. He said that basically any number can be used for lambda, as long as the series converges. You can see from the graph that lambda = 0 does not yield a converging series - it shoots up to infinity.
isn't it on the graph, between 0 and 4?
What if you normalized the efficiency of the different approximations by looking at the number of basic operations (i.e. computation time) instead of the number of iterations?
i was thinking about that ...
I should have known better than to post before reading (the small number) of comments...
The most important thing obtained was the new form of geometric manipulation.
Yeah, that to me is the more interesting question, because I don't think anyone's seriously going to be doing these by hand (beyond perhaps a few iterations). So that suggests a representation can be considered 'better' either because it is more efficient in number of operations for a computer, or because it is more easily parsed/calculated by an unaided human.
The problem is that depends on your computer architecture, so it's not an abstract mathematical analysis. But considering that Chudvosky is a thousand billion billion billion times more accurate than Ramanudjan using the same number of terms (at n=4 anyway), you can afford spending a little bit more time per step.
"So the story goes back to another Indian…"
Me: "Yeah, Ramanujan!"
"around the late 14th century called Madhava."
Me: "Oh."
I'm paused at 6:36 -- while the Madhava series takes a lot more terms to get close, it is also significantly simpler than the Ramanujen and Chudnovsky series. Would be interesting to compare the actual computational power required to get to n-digits of accuracy.
One way to do this would be to time the program, and take the last value found before the given time.
Computer scientist here. The Madhava series takes way, *way* more terms and computation time for any non-trivial result. It converges extremely slowly compared to the Chudnovsky series.
In computer science terms, Chudnovsky converges with O(n(log n)^3) complexity. A million digits takes 216 million terms. Ten digits takes just ten terms.
Can't recall the exact complexity, but Madhava is far, far worse. Calculating just ten digits of pi requires billions of terms. And the difference grows faster than linear, so no amount of linear constant complexity would make up for the difference.
Someone can double check my math, as it's real early in the morning here, but having implemented both algorithms in the past, I'm confident the flavor is at least correct. I can tell you Madhava converges very, very slowly. It's possible to use correction terms which help things quite a bit (enough to feasibly get hundreds of digits) but it's still nowhere near Chudnovsky.
Hope this helps.
Madhava locks in the first decimal place at the 25th term.
It locks in the second at the 627th term
The third at 2,454
The fourth at 136,120
The fifth at 376,849
Regarding the complexity of the Ramanujen/Chudnovsky series, the terms are basically factorials and powers. This means you don't have to calculate each term from scratch. Rather, you can reuse results from the previous term, which will greatly reduce computing time.
It's incredible how can math notions help developing math.
I cant immagine working with plain text for every equation of complex operation to do.
I challenge anyone to find a series that approaches π more rapidly:
π + 0 + 0 + 0 + ...
I think you're only allowed to use known quantities... 🤔
Obviously it’s cheating to use pi to calculate pi. I’d go with tau/2 + 0 + 0 + 0…
@@unvergebeneidI've come up with a formula so neat that it coverges to trillion digits of Pi with its first term. And it's just an integer divided by 10^trillion, how cool is that?
As mr. Incredible rightly said, "PI IS PI!!!"
Unfortunately I think "approaches" pi disqualifies it since it never changes.
Interesting to see that we (still) use just infinite series to calculate digits for pi. Because I knew of the existence of two other modern ways:
Iterative algorithms. A computer program that takes initial values for some variables, and at each iteration changes the values for these variables to better approximate pi. In general each iteration doubles the number of accurate digits.
Spigot algorithms. They look like just infinite series. But their advantage is that one can compute any digit for pi without needing to compute any preceding digit for pi (in the right base what is not necessary base 10.)
Also, in the converging series I see in this video I see square roots, and they must be computed as well with high accuracy.
This was fun to watch! Thank you for the video. ❤
In Ramanujan's formula you need to know the expansion of the square root of two, which itself is an infinite series.
13:45 "pi-oneers". Haha I see what you did there
Wow! The numberphile logo looks more beautiful now!
I'll stick with 22/7.
Four ASCII characters - the shortest definition of pi you'll ever get!
22/7 is not actually pi
Me too
The natural set (2 direction) of x and y, R^(x,y)_0=10: 3/4*[4+[3^(1/3)/10*2-1/10]]=3.141337, 3-d space is rough.
(edit) You can subtract the exponential function of x and y, but you can't add it to make it truly two directional, so space 11 is a placeholder.
"They calculated Pi to 105 trillion decimal places" - I think they should stop at the Planck's lenth.
What exactly did you have in mind? Pi is a dimensionless number from pure math whose decimal representation has an infinite number of digits. Planck Length comes from physics and has the dimensions of length. Its numerical value depends on the units chosen. If you choose Planck Units, the value of Planck Length is exactly 1. But the value of Pi doesn't change-it remains 3.14159… regardless of whatever system of units you choose in physics.
knowing a lot of digits of Pi probably isn't going to help you solve real world problems, but you can use it, for example, to research properties of Pi itself
the graph shows you can choose lambda to make your approximation as good as you want, even better than chudnovsky, since the graph crosses 0! clearly it crosses 0 for values which are as hard to compute as pi is in the first place, but one can compute an approximation and run with it to get a very quickly convergent series
if these "optimal" lambda values are related to Pi, you can iterarively reuse your calculated Pi digits to produce more and more accurate lambdas
@@vsm1456 that's an intriguing idea
HOW THE F DID RAMU COME UP WITH THAT CRAZY EXPRESSION! ... NEW VIDEO NEEDED!!!!
The goddess told him
Smartest guy of the 20th century, killed by British food
@@ssl3546 was he killed by it or did he die to escape it?
Thanks a lot, there is no scratching sound from brown paper anymore.
7:28 I wanna know what k value they went up to in order to get 100 trillion digits
Honestly, the computation time with increasing k for this equation looks kind of nasty, so maybe not even that high.
Why? We have Stirling's approximation for factorials
Chudnovsky formula calculated an average 14.1816 decimal digits of pi per iteration
"There's a few things we're probably going to have to explain"
Understatement of the day
That “good to know” from Brady left me cracking 😂
In 1972, working on my Masters, I used a computer running on punch cards to calculate pi as far as it was able, so this video is very near to my heart. I'm amazed at the tiny laptop you are using - mind giving me the brand?
I don’t know about the computer’s brand but I am almost sure that the program is Maple in case you were wondering. Best.
@@SergioGomez-qe3kn Thanks, yes that's the program.
Love the title ❤
The difference between a string theorist and a string phenomenologist is that a string phenomenologist focuses on connecting string theory to observable phenomena. So their goal is to bridge the gap between abstract string theory and experimental or observational data in physics, particularly in areas like particle physics or cosmology. They explore how string theory could potentially predict measurable outcomes or observable effects and look for ways to test these predictions with experiments or observations. This involves working with specific models or scenarios in string theory that could align with the real-world physics we observe.
So in essence while both study string theory, a string theorist is more focused on its theoretical and mathematical foundations, while a string phenomenologist is interested in applying the theory to real-world phenomena and finding ways to test it experimentally.
IISC representing india ❤❤
Discussions on math pages suggest that when you use more terms you get better result with larger values of lambda.
What perplexes me is how these series have such arbitrary-looking integers in them. How the heck do you find these integers?
I like how they've tip-toed around the fact that Ramanujan's and Chudnovsky's series have terms which grow insanely fast, which would easily cause overflow if one tries to compute them naively past first, say, 50 terms
Missed an opportunity to bring this video out on July 22nd.
I did calculate Pi once by using triangles and sum them up. It was something like the sum of(r*dx)/2 where x is the circle divided up to infinity. It was about 50 years ago when I studied to become an electronic engineer and had some mathematics. I have long forgotten it though. I am sure some mathematician can write it up correctly immediately. The result should be close to correct.
Do all of these example all converge exactly to pi at the limit? Or are some just good approxiomationsas?
Yes, they all converge exactly to pi at the limit.
These can be mathematically proven just like how we proved pi is an irrational number in the first place.
Exact value of pi is not needed (and impossible to get in the first place) in those proofs.
Pi-ist logic:
>define the circle and all circle-related formulae using the radius
>define circle constant using the radius*2
Interesting that the series with this lambda is somehow derived from string theory. Why didn't you talk about how they discovered this series? Just writing down different series that converge faster or slower without any explanation is honestly pointless.
0:18 Tony Padilla is my favorite string phenomenologist
1:41 - Madhava of Sangamagrama, eh? Did he know that other vowels were available?
my indian last and middle name are pretty lengthy and the only vowels are As which constitute every other letter
@@theacorn7240 Is there a reason, seriously, for this? Google isn't helping me here.
It's a transliteration thing. All indian names with an 'uh', 'aa', 'ae' and many such sounds gets simplified simply to just a in English because Hindi has around 14 vowels @@leefisher6366
@@leefisher6366our names stem from Sanskrit.
@@leefisher6366 So basically you're looking for the "reason" why names in a 7000 years old Eastern civilization don't follow your Anglo standards?
I discovered a formula for pi/4 that is an alternating infinite series of powers of pi with rational coefficients where the powers of pai vary from 2 to infinity. The implication of this in that if we multiply each side by 4/pi, then any rational number can be represented by an alternating infinite series of powers of pi with rational coefficients when the powers of pi vary from 1 to infinity.
7:37 that's the most mathematical conversation i've ever seem
Finding the formula for pi was just a side quest for them
New Flavour!!!
What about the Newton method. He used the integral of the binomial expansion for exponent 1/2 back in 1666. And, of course, he invented integration.
So the ‘state of the art’ series has a lot of intermediate calculations, compared with then Madhava series. I presume the state of the art is still the best, since they use it for computation, but it would be interesting to compare convergence based on computation time, rather than by number of terms.
Yum!
I get so excited to test out this summation after hearing about it a few days back, but then I get to time 1:24 , pause, and I'm like... well what's lamda? Who's solving for second variables in a summation? That's not fair. _Now I'll have to watch the rest of the video._ 🤪
So how do we know the exact value of pi to compare these recipes to?
I've been wondering the same thing!
Man, the dirt on the monitor triggers me hard. Apart from that, a great video
Did you know that the volume of a pizza radius "z", thickness "a" equals pi.z.z.a
This is amazing info 🍕
The amount of computing time the calculation takes is much more interesting than the number of terms. You can make a slow series converge faster by combining terms.
"...two Indian string theorists..." \*proceeds to pronounce Sinha as if it was Brazilian Portuguese\*
I think it'd be interesting to compare the various approximations in a computational sense. Like ripple-carry vs look-ahead, or certain matrix formulations which at first glance change an Order of an algorithm - but then you find you're doing more per step, or need more memory. I bet there would probably be some "constant" or even "constants" where you can relate some measure of computational complexity per step, accuracy/deviation, and "family", where a "family" of solutions are more-or-less the same, more effectively separating the concept of effectiveness.
Edit: I'm curious in the String Theorists derivation. First thought is 'wait Pi can be termed as the length of a "line" intersecting all the points at the same distance in a curved space' or a different geometric interpretation. There might be a similarity to how we have no equation for the perimeter of an ellipse. Just take it to extremes: an ellipse has 2-3 values: separation of centers and r, points and r, etc , but can we do MORE? Can we *require* two independent values e.g. they're not simplifiable, but both can take any value and produce pi?
Why thumbnail is Indian pi
Because the guys who discovered the new series are Indian
Because the people who discovered the new formula were from India
In fact 3 of the 4 formula we discuss, including the new one, were discovered by Indian mathematicians/scientists.
An interview with the latest ones can be found at: ua-cam.com/video/2lvTjEZ-bbw/v-deo.html
Oh thanks
Glad to see Tony is still around!
Wouldn’t there be an infinite amount of ways to represent pi? Math is just counting an infinite amount of zeros.
Of course there are. The one presented here also is itself an infinite number of ways to represent pi. But not all of those infinite representations are useful.
When you wrote the first series on paper, where you stopped could have been using odd numbers or primes for the denominators.
From a 10,000ft perspective, It looks like the chudnovsky series is embedding static information about pi into itself to improve its accuracy. My guess is that we could increase the size of the static values within a new series to speed up the calculation.
11:03 This graph implies that there are exactly four different lambda without any approximation error, which means that your could calculate pi exactly in just four terms. The only problem is oft course to get those lambdas, as they are most likely also irrational.
Damn.
Love how Sinha & Saha are humbly saying : hey guys we never said this is a revolution
I love that Tony's ∑s are way sloppier than mine ever were.
/a vindication of sorts
//ok not really but i feel a bit better about everything
So the ramanujan sum of all integer to infinity (-1/12) could be represented by -pi sum of k=0 to infinty ...... (Chudnovsky formula)
A fairer comparison would be to count not "number of terms", but "number of flops".
You could trivially get a "faster" series by bundling, say, 10 terms of the current best series into one massively complex term of your new series, but that isn't really an improvement computationally.
Oh, that's going to be fun to play with!
*edit:* 17 hours later-- Finally had a chance to poke around, this is neat! I caught you say that lambda has to converge, so in Python speak, I've been trying silly things like lambda = 3 - random()*2**-n and it works since that does technically converge to 3. I'll be trying for other functions to see if I can get something useful.
I have always been amazed by the concept of irrational numbers. How is it that a number that never ends is able to define a finite thing? The area of a cicle with a radius of 1 is pi, the ratio of width to length of A4 paper is 1:square root of 2. Is this just an artifact of our base 10 system and is there a different base number that would make these numbers end?
For most practical purposes the approximation of the inverse of pi that is easiest to deal with is 113/355. It is elegant, and easy to remember. And ordinary folk aren't looking for the kind of accuracy that these complicated formulae are capable of giving after a half dozen or a dozen iterations. And it is accurate to within 8.5 millionths of a percent. And I cannot imagine taking the value generated by their formulae for the inverse of pi and calculating pi from it. For my approximation you invert the basic formula. End of story.
It's insane how they even came up with these series!
I feel like the Kolmgorov complexity of some of those series expressions might be greater than the complexity of just writing out the digits directly.
"Does it have anything to do with circles" - Bloody good question.
Comparing number of terms is not fair: one should compare the number of operations. The series that require fewer terms have much more complicated expressions per term which contain many more operations.
This strategy is bomb!!! I won 3/4 just testing it out!!! Thank you for sharing!!
It's definitely an INTERESTING way of determining pi - a formula with a free parameter? Ha! Cool.