When Pi is Not 3.14 | Infinite Series | PBS Digital Studios

[@Eggerhexe]

After some careful research, I came to the conclusion that birds don't use the Euclidean metric when they fly. They use their wings.

[@shailushiritchie974]

#dadjokes

[@agrajyadav2951]

I am offended 😡

[@Fire_Axus]

SGSE

[@YohanBarrosdeOliveira]

wise comment!

[@deleted-something]

True

[@stoat7]

The most mathematician thing i have ever heard: "The real world isn't exectly my thing" ^^

[@the1derpface]

Gnosticism in a nutshell

[@super-awesome-funplanet3704]

How is it?

[@gwebangetforever]

School Maths: Pi=3.14159. Street Maths: Phew=3.17157. (only for 1D)

[@SawtoothWaves]

Oh my god it's a circular square.

[@1ToTheInfinity]

The Brony Notion It is a circle but not in our euclidean world and also I didn't know you liked maths!

[@jackharmony3827]

The Brony Notion You can't square the circle? well take that, greeks!

[@gustavrsh]

You mean a Parker Square?

[@TheMasaoL]

All these squares make a circle

[@khalilrahme5227]

The Brony Notion you know you shouldn't be here when they are telling you a circle is a square.

[@0NlRAPTOR]

My favorite part of you video is when you said something like, "that is in the real world and as a mathematician. I dont really deal with the real world." made me smile. I love mathematicians. They do all the fiddlie bits so I can think about the cosmology.

[@Fire_Axus]

YFAI

[@Cinqmil]

Brilliant! Did she just squared the circle? :D

[@feynstein1004]

All we had to do was change the damn metric, CJ.

[@guyedwards22]

Having two insanely high quality mathematics channels (this and 3Blue1Brown) actively producing videos is a dream come true for a lot of us out there. Excellent material Kelsey!

[@yash1152]

wow, I am coming to this channel only now and that too, by someone's reference in comments. Never heard it before. While 3b1b i know from quite some time now

[@GuyWithAnAmazingHat]

This channel has over taken PBS Space Time for my favourite educational channel, and sorry Numberphile :p

[@adityakhanna113]

Wow! I was going to say pretty much the same thing! Probably because I love math more.

[@gauravmanwani9148]

GuyWithAnAmazingHat I'm sure about PBS Space Time! Sorry Matt! But not about Numberphile! Brady is still working!

[@manueldelrio7147]

well, this aint like the cake... you can enjoy all of them!

[@hellNo116]

dude go watch 3blue1brown. his linear algebra course is freaking amazing!!!!

[@gauravmanwani9148]

john smith I would thank you if i hadn't been! Grant really deserves a grand applause! Whether it be for 3B1B or Khan Academy! Though i am 15 and in class 10 now, i am looking to start linear algebra courses in March. I love his other videos though!😊

[@GellyGelbertson]

4:10 I think (1/2, -1/2) does not belong - it's of distance 1, not 3, from the origin in the taxicab metric.

[@kwinvdv]

She did say "One and a half comma negative one and a half". So I would think that the person responsible for the graphics messed up.

[@Keronin]

It was an error by the animator. When it appeared on screen, she said "1½, -1½"

[@CheCheDaWaff]

If you listen, she says (1 1/2, -1 1/2), which would be right, so it's definitely a typo.

[@pbsinfiniteseries]

Good catch. It should be (1.5, -1.5)

[@theironblitz]

Yeah, I caught this too. I think the simplest fix would be annotating 3 in place of the 1. Yay, improper fractions. =P

[@StephanePare]

I loved this, but when you said this L^P measurements are used in tons of different real world applications, I would've loved to hear one or two mentionned.

[@Fire_Axus]

YFWBI

[@StephanePare]

@@Fire_Axus I'm sorry, what does that mean?

[@Fire_Axus]

@@StephanePare YoFeWoBeIr

[@charlesx6569]

I'm not ashamed to admit, my motivation to watch these vids is half only half for the information provided; the other half is more related to checking out the lady presenting. I'll admit it, I've got a thing for smart chicks. I've subscribed just so I can satisfy both interests.

[@mikeyoung9810]

Love math, love spacetime and now infinite. And Kelsey Houston-Edwards. Amazing.

[@GregoryMcCarthy123]

So glad I found this channel. The woman explains things very clearly! Thanks!

[@bxdanny]

On the Commodore 64, the value of pi represented by the pi character was slightly LESS accurate than the one you could get if you used the expression 4*ATN(1). But it displayed looking as if it were more accurate: 3.14159265 instead of 3.14159266. That's because the algorithm that turned the internal binary floating point numbers into decimals for display wasn't perfect. So they chose the number that looked most accurate when converted (imperfectly) to decimal form, rather than the number that actually was most accurate, to be used for the pi character.

[@pforce9]

I remembered Pi as 3.14159286 The numbers you show are 3.14159265 instead of 3.14159266. I got mine from a table, probably my high school algebra book in the early 60's. You any idea why our numbers are different?

[@adamlovelace7572]

Squaring the Circle must be much easier in some of these metrics, lol.

[@chibi013]

I want to take this video and go back in time to high school and show it to my philosophy teacher who said "you can't imagine a square circle".

[@dannymunoz8027]

I just discovered this series. I'm in love with it! Please keep it up!

[@happmacdonald]

Favorite Metric? Hausdorff.

[@froop2393]

I am learning how to use tensorflow and numpy... and i'm struggeling about L2 and L3 distances. Now i understand what they are good for. Thank you very much!!!

[@ChilledfishStick]

People have already mentioned this, but at 3:59, there is a typo in the second point. Also, this was very fun and educational.

[@brianpso]

God I love thiss channel! I forgot I subbed, so seeing this video on my sub box was such an amazing surprise! Great stuff as always.

[@JakeFace0]

So what's the formula for the L^p circle constants? I didn't become a mathematician for decimal approximations!

[@satyu131089]

The perimeter in each metric is a certain integral if you work it out. And not all integrals are evaluable in terms of elementary functions and constants.

[@danielm.mclaury3202]

2 ∫ (1 + (x^p/(1 - x^p))^(p-1))^(1/p) dx, where the bounds are from x=0 to x=1.

[@JakeFace0]

Oh right. If it did have an evaluatable value, then pi would be algebraic which we know it isn't. satyu131089

[@danielm.mclaury3202]

I'm not sure what you mean by that. I don't think this function is elementary, but non-elementary functions can easily take on algebraic values at integers.

[@satyu131089]

+SafetySkull I don't think that's the way to think about it. The key idea is that not every function has an antiderivative in terms of elementary functions. So, even though an integral may converge to some number, that number need not be related to any of our beloved numbers in a simple way.

[@GilbertTang]

The more I see my children interested in watching folks like Kelsey, the more faith I have in the future of humanity.

[@orbital1337]

4:40 You're kind of cheating here: the length of a Euclidean straight line in some arbitrary metric is not necessarily the distance of its endpoints.

[@zairaner1489]

It is in these cases though

[@orbital1337]

Sure, but that's something that needs to be proven.

[@DrGerbils]

+Raphael Schmidpeter +orbital1337 Orbital is right, although I would have worded it differently. What she's measuring at 4:40 is a circular arc, the length of which is almost never the distance between its endpoints. An arc that is 1/4 of a circle in the L1 metric happens to coincide with one of the line segments through the endpoints of the arc. So she arrived at the correct answer purely by accident.

[@zairaner1489]

orbital1337 Ok now I have to admit I was absolutely stupid and didn't actually look to what you are referring (I assumed something else because I thought about that for a moment). I'm confused now though. How do you define the length of the line then when not through the distance of their endpoints? Through integration?

[@zairaner1489]

In that case forget I said anything. Yes, that is indeed far from obvious

[@alvarocafe]

just discovered this channel, this is genius!! bravo!!

[@lineikatabs]

I've learned so many cool things!!!

[@makermark2159]

Fascinating video. This brings to light the significance of the basic plains we work on in high school and college. Thank you.

[@Kram1032]

So what happens if you go for p-adic metrics? - For starters, what even is a circle in those?

[@tsss1179]

I was also thinking about this

[@freshrockpapa-e7799]

... Watch the video?

[@Djorgal]

A circle will look like a fog of points covering the entire plane. There'll be point from that circle as close as you want from any spot in the plane.

[@zairaner1489]

It gets a little bit more complicated because you don't even have a normal real plane in the p-adic metrics. When we just take a rational plane... I still don't believe you can draw the "circle".

[@amaysaxena2]

Eric Pive she didn't mention p-adics. she mentioned p-norms, completely different.

[@ViewtifulSam]

Loved this episode, I'm so glad this new channel is turning out to be this good!

[@JM-us3fr]

My favorite metrics are the p-adic because prime numbers are awesome

[@mokopa]

This beguiling Mathematical wonderment, of which I did not know, is as pleasing to ponder as it is to listen to deeply beautiful music.

[@Abdega]

"All these squares make a circle…"

[@jjmah7]

This is great! She's easy on the eyes and I actually MIGHT have learned something.

[@Swiftclaw123]

8:16 details not in description..

[@MusicKnowte]

The amount of times I have returned to this video to understand different metrics on R^n for my analysis and topology classes.....

[@daniellassander]

This is so fascinating, i wish i learned that mathematics was so much fun when i was in school!

[@mattcay]

I've found this channel recently and I really liked it. I also love how much you interact with others, answering questions in episodes, as well as responding to comments :)

[@touisbetterthanpi]

I was expecting this video to go in more of a geometry of the sphere or hyperbolic space. Like what is pi if you measure distance along the curved plane.

[@pierrecurie]

In that case, Pi is a function of how big your "circle" is. For spheres, it starts at the usual pi and decreases to 0. For hyperbolic space, it starts at the usual pi and increases without bound.

[@General12th]

We usually work out perimeter and area in terms of pi. But we can go the other way -- define pi from perimeter and area. In hyperbolic geometry, the area of shapes is proportional to (pi - stuff). So pi is different, but also kinda not.

[@LuisAldamiz]

Me too: that was exactly my hope when I clicked. Very disappointed.

[@LuisAldamiz]

So, Pierrecurie: when is it that Pi becomes exactly two and one? I'm guessing such kind of configuration is particularly interesting: a circle and its diameter or radius being exactly the same length... how's that even possible?

[@JohnnyAdroit]

@@LuisAldamiz Imagine the Earth is a perfect sphere. All points on the Equator are equally distant from the North Pole. If you measure distance along the Earth's surface, then the Equator has a diameter of 1/2 of the Earth's circumference, because the distance from the North Pole to the Equator is 1/4 of the Earth's circumference. The circumference of the Equator is the Earth's circumference, so pi here is 2. If you increase the distance so that the circle is a southern line of latitude, the circumference of the circle starts getting smaller, so pi gets smaller as well.

[@coffeeandproofs]

I remember studying normed linear spaces when I wrote a research paper on geodesics! For those interested, a linear space R is said to be normed if each element x in R is assigned a non-negative number ||x||, called the norm of x, such that (for a real number a): 1. ||x||=0 if and only if x=0 2. ||ax||=|a|•||x|| 3. ||x+y||

[@alexandrugheorghe5610]

Feels like you could collapse all of this in just a single function.

[@twilightknight123]

My favorite metric? The Schwarzschild metric. Gotta be honest, I'm a bit disappointed you didn't go into talking about the effect of curvature ( the value of pi on the surface of a sphere is less than on a sheet of paper, for example). But oh well.

[@BareClause]

That's not a metric in the analysis sense.

[@mattscatterty]

Ooo, my ears perk when I hear Schwarzschild! But I've only ever heard it in the context of the Schwarzschild Radius, which is the radius that any given amount of matter requires to be condensed into/beyond in order to form a black hole. What is the Schwarzschild Metric? I imagine a somewhat related concept to the Schwarzschild Radius, except one of our math rather than astrophysics.

[@twilightknight123]

AB CD Not exactly. It is true that the circumference of a circle (assuming for a fixed time) remains the same as the euclidean circle, however because it defines a curved space-time, the radius is not the same as you would assume under the typical notion of distance. In fact, it is slightly larger than you would expect. Take, for example, the distance between the ISS and the moon. You can view their orbits as circles of radii A and B respectively. However the distance between them is not B-A as you would expect, but is slightly larger. ***** How is it not a metric is the analysis sense? The entire point of the Schwarzschild metric is to define the proper distance between two points under the curvature of space-time in general relativity. That's the definition of a metric "in the analysis sense", is it not? Matthew Scatterty The Schwarzschild radius actually comes from the metric itself! The metric is what we use to define the curvature of space-time around a massive object (such as the earth, sun, a black hole, etc). The radius is found from this metric by assuming this is a point-like massive object with a particle infinitely far away. As this particle free-falls towards the massive object, it's speed increases (and a bunch of other things happen because of general relativity, but I want to keep this explanation brief). The Schwarzschild radius is the distance from the massive object that the particle never "technically" passes and where its speed reaches the speed of light. Now of course it DOES pass this radius, but the rest of the explanation is based on the point of observation and my comment is long enough as is. Hopefully this was a satisfactory explanation though.

[@mattscatterty]

Wow, that was SO much weirder and cooler than I thought it would be! That's interesting. It's given me a totally different way of looking at Schwarzschild Radii. It's like coming at it from the complete opposite direction.

[@coopergates9680]

In S^2 (2D spherical space), the diameter of a great circle measured from within the space is pi * radius of the sphere, and the circumference is 2 * pi * radius. Pi for that circle is only equal to 2. For smaller and smaller circles in spherical geometry, Pi tends back toward the Euclidean value.

[@Adam_42_01]

Absolutely fascinating video. This is one of my favorite channels, for sure.

[@rubikfan1]

that moment you realise this girl just told you a squer is a cirkel. and you agree...

[@schadenfreudebuddha]

i'm no rocket surgeon, but I didn't agree.

[@funkytom7709]

but the circle isn't a square ... because the shape of the square also change ! What she said is that shapes we use to see depend on how we imagine what a distance is, and you can change it !

[@catStone92]

well a square is a four sided polygon where all sides have the same length, and with the circle definition given in the video, when for L^1 and L^inf the figure is both a circle AND a square

Circle can look like almost anything if you pick the right way to measure distances ;-)

[@ffggddss]

+ Pedro G A 4-sided polygon with all sides equal (= an equilateral quadrilateral) is actually a rhombus. A rhombus with a right angle, has 4 right angles, and is a square. And yes, in the L¹ and L∞ metrics, the figure *is* both a circle and a square.

[@andrewharrison8436]

3:21 "Math isn't restricted to reality" - priceless

[@charliephilip6864]

If I would have watched this 2 days ago it would have been pi day 🤔

[@IshowFUNNYvids]

I love this channel! Its what I wanted out of Space Time: maths! Thank you so much!

[@timmytanga7458]

Love the haircut! And the math too I guess

[@TheLolle97]

awesome. I already learned how to calculate all these metrics in university a while ago, but you just showed me for the first time what they actually mean. Thank you.

[@maxrenaud7795]

I'm pretty sure this girl was my calculus TA freshman year of college. Kelsey I think? although tbh i dont really remember her name.

[@maxrenaud7795]

yup definitely her. Kelsey if you read this Hi lol. thanks for not letting Guckenheimer demolish me in calc... Cameron says hi too.

[@General12th]

hi cameron

[@p.as.in.pterodactyl1024]

She looks like a Kelsey for sure lol.

[@KarnKaul]

This channel never ceases to blow my mind. Just like Kelsey's organisation, delivery and panache make my knees shake (in a good way!).

[@TwaritWaikar]

Karn Kaul Umm btw, don't believe this please.

[@KarnKaul]

Why should you "believe" maths in the first place?

[@TwaritWaikar]

Karn Kaul I meant that do not believe them when they say that Pi changes its value in different metrics. They are using Pi a concept and not a constant irrational value(they said this on Quora). They could have just as easily defined another number that is the ratio of circumference over the diameter. They are not specifying this in the video and that is what is pissing off so many people.

[@TheGeneralThings]

What would the value of pi be in these contexts when "p" is a positive number less than 1? What about a negative number? Would pi start going back down to 3.14, back up to 4, beyond 4, or a different value altogether?

[@KHMakerD]

Cody Griffin I believe p is always assumed to be greater than zero.

[@RedTriangle53]

well, it would approach (1+1)^1/0, so it diverges and therefore cannot be set equal to any radius(at p=0 at least).

[@RedTriangle53]

and without taking the zeroth root it would be 2 = 1, heh

[@KazmirRunik]

Actually, Lp space is also defined for values below 1, but what she described is the context in which pi is between the stated values. The theorem she stated doesn't hold in lower Lp-spaces than 1. For example p=0.5 yields a pi value of pi + 4, while p=0.1 gives you s pi value of 33.866. You can also analytically extend it to some values below zero, giving you a pi of pi/2 for p=-1, while, for p=-2, pi completely falls apart to infinity. The gritty details are something she said she'd link in her description, so do stand by for when she does.

[@zairaner1489]

You don't get all the nice properties of a metric there, but it would still be interesting to see

[@bckzilla]

What an awesome channel to stumble upon. Thanks for that.

[@StuartDinh]

7:49 Looking for the Lp in which 'pi' equals p itself

[@drmedwuast]

Huy Dinh let me know when you find it

[@DrGerbils]

3.30524

[@112BALAGE112]

That is amazing. How did you get there though? Cold you please elaborate?

[@DrGerbils]

I cheated. I searched the web to see if anyone had computed pi for the various Lp metrics. Someone had. I found a forum in which someone cited C. L. Adler and James Tanton The College Mathematics Journal Vol. 31, No. 2 (Mar., 2000), pp. 102-106 for the equation pi = 2/p * (integral from 0 to 1 of ( [u^(1-p) + (1-u)^(1-p)]^(1/p) du) I guessed at a reasonable value for p, plugged the integral into the Wolfram Alpha integral calculator. I took the resulting value for pi and plugged it in as my new p. I iterated until the result converged to 3.30524. It's only an approximation, of course. The formula above computes pi based on the definition pi = C/D. If you define pi as the area of a unit circle, then pi = p at 3.66197. The area of the unit circle for the Lp metric is 4* the integral of ((1 - x^p)^(1/p))dx from 0 to 1.

[@joopie99aa]

Steve's value is close, but not correct to the suggested precision. It should be 3.30522, or using more decimals: 3.3052189294288623 This was found by numerically solving the relevant equation (and integral) in Mathematica. For the curious, the equation is: ∫0->1[1+((x^p)/(1-x^p))^(p-1)]^(1/p) dx = p

[@ponpetr]

Thank you. The series is awesome. Regarding the question about the drum sound difference and how physics manages to answer it: there are many different things about the drum, the way sound starts playing, where we listen to it, what is between the drum and the listener, what is around them, what is used to listen to the drum, and so on. Each of these is going to affect the answer, but with a different level of magnitude until you reach the point where you take into account more and more things, but the answer stays the same. Before doing any mathematical modeling you need to decide and describe how you are going to test your answer in real life with what set of parameters, where and when. After that, you start from creating the list of such things and estimate the level of magnitude for each. After you are certain about the top level things you can build the first model with very few, maybe even only one single most important and powerful thing from your list. You analyze the model, find conclusions from it and then go and build the next mathematical model, that is more precise. Say you continued for a while and on the model number n with a precision of n you see that your conclusions stay the same as in n-1 model. Overall, you try to find to what conclusion all your models converge with increased precision. Sometimes you will see that by going to n+1 or n+2 precision models your conclusions suddenly change, despite having the same conclusions at levels n-1 and n. Because of this, you need to see if there are any correlations between different things and how you model them at that levels of n-1, n, n+1, n+2 and so on in general. Once you are done with all of these you can imagine more complicated experiment settings and repeat the process until your experiment is very close to the real settings you are interested in. The fun part is that you can make predictions and compare them with the experiment on every step of your way. This is how a theory of certain phenomena is created in physics in a nutshell. It is a long, interesting and challenging process. Sometimes it is ignited by a single experiment that does not fit the common theory predictions, or just radically new and unrelated to the common knowledge (physical theories). Sorry, the comment is becoming a bit complicated, philosophical and long =) Let's return to the question about drums! First, we rephrase the question to a testable scenario. For example like this: you bought one pair of speakers and your friend bought another one. The only difference between them is in the shape of the membrane of the dynamic (that is the moving part of the speaker the makes the sound). Can you personally distinguish them by playing music from your phone via both sets of speakers or are there certain membrane shapes that you will not be able to distinguish (your friend can make any shape of the membranes of your speakers and his, but they have to be distinguishable with your eyes at the distance you are going to hear the sound from them in the room of your home, and, by the way, you can select any set of sounds you want to test the speakers)? You can make any experiments you want other than the real scenario above before you give the answer. Your answer is going to be compared with the experiment. What should you do to give the right answer? The strategy is different depending on what you already know about sound, speakers and your hearing/vision abilities. If you know, modeled and tested many many things on that topic, then you most likely will be able to build a mathematical (!) model with good enough precision and give the answer that is correct right away. But usually the question is going to be new to you, or somewhat new, otherwise it is no longer science (it is engineering instead). So most likely you will have to make lots of mathematical models, and run lots of experiments before you can give the right answer. You can begin with very simplified experimental settings that are easy to model and test (you test experimentally all the time in order to improve your theory). For example with such settings: Let's imagine that we are going to use drums with a surface area of 0.01m^2 and weight 10g. These drums are made of hard metal, for example, from steel, and suspended in the air with a very light and super strong thread. The sound is generated by hitting the drum in its center of the mass by a metal ball with weight 10g and speed of 1m/sec. You change different shapes of the drums including the shapes with equal eigenvalues and see if you can model the experiment and give the right answers about it (correct predictions). You continue developing your sound theory with making your experiments closer to what your main question is about until you are confident with your predictions. Finally, I think you will reach the conclusion that there are always many ways to built a set of sounds during the speakers' analysis that generates a noticeably different set of sounds you hear, even when you have the same eigenvalues of membranes. Please correct me if I am wrong. One of the ways is to play a tone that is corresponding to one of the eigenvalues (one of the main tones) and then you can move the tone very little around it back and forth with an ever increased speed of tone changes. When there is a difference between the tone played and the main tone, it causes lots of waves on other eigenvalues. But that process is not immediate because energy needs time to dissipate to other main tones. For each main tone, energy is distributed unevenly across the membrane. Moreover, different regions of the membrane can accept and transfer energy differently because of this. As a result, you will hear exactly the same eigenvalues and exactly the same sound for the soundwave that is a little bit off from the main tone, but when you do change the tone, the sound relaxes to the "stable state" using the path that is unique to the membrane shape, not only to its eigenvalues. The process of that relaxation is very quick and most likely you will not be able to hear the difference between membranes based on just these two different paths. But when you start changing the tone around the main tone and repeat it with an ever increased speed you will cause waves for certain eigenvalues to be highly enriched and at the same time highly depleted for some other eigenvalues due to correlations between energy transfer and the speed with which you change the tone at a few distinct moments along the way. You will hear these as a slight nuance difference between two speakers. This change will be something like "the first speaker sounds a bit metallic and cold at the beginning and very warm and woody in the end, while another speaker does not have any strange things, other then it feels a bit too loud in the middle of the experiment". These nuances will be stable for the same speaker over the same set of sounds and how they are played in time so you will be able to pick your speaker out of many. It is very interesting to hear your thoughts because these is not my field of expertise. The cool thing is to make an experiment to test all these =)

[@adityakhanna113]

0.999999..... th!

[@copperfield42]

I see what you do there XD

[@special-delivery]

Aditya Khanna This is probably the most nerdiest 1st comment xD

[@TS-kt3nf]

lol

[@adityakhanna113]

Yevhenii Diomidov I had a doubt too, but "th" sounded better

[@bored_person]

Ordinals only apply to natural numbers.

[@BenChieeal]

Love your videos. You remind me of the guy in Space Time. Great info, great presentation, and good looks to boot! :D

[@theshuman100]

so you're not going to tell me how to calculate the circumference of a euclidian circle without pi.

[@ReaperUnreal]

You know, I never really considered that the ratio could be different. This was fascinating.

[@thisaccountisdead9060]

If you take any random irrational number, for example 2.6435......., then subtract the whole digit to give 0.6435...., then invert it to give 1/0.6435..... = 1.5540, then again subtract the whole digit to get 0.5540, and then invert that to give 1/0.5540 = 1.8050...., and just keep repeating the process, are there any properties you find with certain numbers or is there always going to be a random pattern?

[@thisaccountisdead9060]

I suppose the average value of all the numbers you get could be interesting...?

[@justunderreality]

With rationals this will always become 0 since you can represent all rationals as a fraction (and then greatest common denominator keeps getting filtered out). With irrationals it's not that easy since they don't follow the same structure. Consider the golden ratio (call it "gr"): (sqrt(5) + 1) / 2. This number has the property such that 2> gr > 1 and gr -1 = 1/gr. In summary this has a repeating pattern. Using this as an example, I think it's safe to say it would always be random.

[@ptyamin6976]

huh, let's think about that

[@thisaccountisdead9060]

I have no idea - but I'll take a look, thanks. I have looked into the golden ratio and fibonnaci series quite a bit - the single digits of the fibonacci series repeat themselves every sixty terms (and also you get an intersting correlation if you reverse the series and put it side-by-side with the regular series, as well as the number 0 repeating every 15th term and the number 5 repeating every 5th term).

[@tadashimori]

It's not random. For sure there is some sort of pattern for some numbers because if your starting number is Sqrt(2), after your first iteration you'll have Sqrt(2)+1, and it'll get stuck on this number. It'll get in a cycle for sqrt(3) also. Probably you'll get a cycle for any square root due, you can prove it by rationalising the numbers, I'll try to find out if this is true for all numbers. For PI, it seems to be quite random, but with an average value of 7~12, I'm not sure of it yet.

[@brandonneth7707]

This channel is everything I want and more!

[@HemmligtNavn]

Welllllll, I need to disagree a little here. It is simply an issue with definitions. Pi is DEFINED to be the ratio of the circumference of a circle to its diameter USING THE EUCLIDIAN distance measure not just ANY measure. Saying that you can 'redefine' Pi by using a different measure isn't completely correct. What you CAN do is to re-interpret the equivalent of Pi on Euclidian space with a non-euclidian length measure. However, that is not Pi, it is something else......

[@zairaner1489]

I agree.

[@LuisAldamiz]

No ven using Euclides: using a piece of rope!

[@truebaran]

9:22 it is amazing that the mentioned paper by Milnor is actually ONE page long

[@lucidmoses]

My favorite metric is the liters. :p

[@rafacrazyboy]

It's great to discover this series while taking math analysis.

[@guillaumebourgault5532]

I have a question... Pi is irrational, i.e. it cannot be expressed by the ratio of two whole numbers. Yet, we always define pi as the ratio of two numbers. Somebody pointed to me that if the diameter is a whole number, it just means that the circumference is an irrational number. But it looks like a non-answer to me. How do you get to the irrational circumference number then?

[@jaimeiturriaga3111]

This was actually proven by Lambert back in the 18th century. math.stackexchange.com/questions/895611/lamberts-original-proof-that-pi-is-irrational The fact that either the diameter or the circumference are rational, but not at the same time, is a noticeable side effect, but is considerably more difficult to prove without having proved that pi is irrational.

[@guillaumebourgault5532]

FrostyDynamic Thanks! That was helpful!

[@BareClause]

It is defined as the ratio of two numbers, but those two numbers are not integers, Recall that a number is rational if it is the ration of two -integers-.

[@retepaskab]

irrational numbers aren't impossible numbers...

[@grandmaster-grouch]

Guillaume Bourgault i always considered pi as our understanding of infinity. So far its proven to be the only thing we know that has no pattern in number sequence. Pi still isnt fully understood and still in quantification. We are arguing as if there is an answer. Lol im glad weed is legal im cali now.

[@jacksonforner4868]

This video is so amazing. You are doing great things for the math community!!! Keep it up

[@-_Nuke_-]

Pi is 3,14 ... only because we have 10 fingers! If we had say 12, then Pi would have been 3,18480 ... instead! That's because we developed base 10, because it was very much easier for us to count up to 10 fingers. Maths today show, that other bases, like base 12, is actualy ALOT handier! Base 10, is one of the worst bases we - today - could use for our math problems! Just some food for thought! (ofcourse by changing bases, only the symbols change, not the actual value itself)

[@mikeo759]

I blame the French for base 10.

[@mikeo759]

Pi is 11.001001000011111101101010100010001000010110100011... in base 2

[@-_Nuke_-]

orochimarujes I believe yes

[@-_Nuke_-]

Gabriel cazorla persson true

[@-_Nuke_-]

I should add here, that by changing bases, only the symbols change, not the actual value itself - that remains the same of course...

I just discovered your channel recently and I have to say that you (and probably your team also) do an AMAZING job!!!! You explain things so clearly and the topics you choose are so fascinating. Thank you, keep going.

[@SebastianLopez-nh1rr]

what about curved space? you know, like Spacetime

[@deraj00]

I think you mean Non-Euclidean Geometry and trying to figure that out has become a bit of a head scratcher for me. It seems like the answer is yes but the more I look the more confused I become.

[@ModMINI]

It's sort of easy to visualize in 2 dimensions. Imagine a circle on a piece of graph paper. We know how far the points are in L2 or Euclidian geometry. All points on the circle are equal from its center. Now imagine that the space time (or piece of paper) is warped. If the paper is curved, the points around the circumference of the circle will be will be closer to each other as measured through the 3d space. (there is a shorter line between the points than as shown on the paper. Another common way to think about it is a triangle or whether parallel lines never cross or diverge. In flat space, the three angles of a triangle add up to 180 degrees. In negatively curved space, it is more than 180 degrees, and parallel lines diverge. In positively curved space, it is less than 180 degrees and parallel lines converge (like straight lines on a globe).

[@deraj00]

***** Hm, I knew about the bit with the triangles before but I kinda didn't even remotely consider the first part of that...

[@KevinSmithGeo]

The circumference of a circle in a curved geometry is not a linear function of the radius. The euclidean circle constant (whether you use pi or tau) is still involved though and is the limit of the ratio as you approach radius 0 (Curved geometries look flatter the smaller the section of them you look at)

[@coopergates9680]

Switched 'em. The sign of curvature is the sign of the change in the angle sum relative to 180 degrees. Negative curvature (e.g. hyperbolic space) causes triangles with internal angles that sum to < 180 degrees, and it can even tend to 0.

[@Saphirefenix]

OMG I love that this show exists.

[@The_Rising_Dragon]

I thought you were gonna tell about the Indiana Pi Bill. (that one time when Indiana tried to officially change the value of pi to 3.2).

[@pbsinfiniteseries]

We definitely thought about including that. It's hilarious! Maybe that guy in Indiana was just measuring using the L^3 metric. (Or whatever p makes pi in the L^p metric equal to 3.2 -- it's probably a little smaller than p=3.)

[@The_Rising_Dragon]

+PBS Infinite Series What I personally found weird about the bill is that they almost succeeded in passing it. If it had passed though, it would have been treated like an invention rather than a mathematical truth. Which while also being amusing, would have made it almost unusable to the general public and the world. Thus, it would have generally become obscured after a few years. And, I don't think that is what you do mathematics for.

[@fx4d]

The taxicab measurement of Euclidean space and how that affects distance has been nagging me for a while. So glad to finally get a well-explained answer (that it's using a different metric) to help clarify things. That 3 - 4 range thing is fascinating.

[@taylorakinser]

So does that mean it is possible for pi to equal infinity?

[@pbsinfiniteseries]

Not exactly. For most of the "reasonable" metrics, the value of pi is between 3 and 4. See: www.researchgate.net/publication/242075312_On_the_Perimeter_and_Area_of_the_Unit_Disc

[@AxiomTutor]

It's interesting to think about the discrete distance metric which says that two points are a distance 0 if they are the same point and a distance 1 otherwise. In that case what is a unit circle?

[@justunderreality]

If I understand right, a circle in your discrete metric is every point except the center. As I understand it, Pi is either non existent or trivially 1 = 1/1 in this metric.

[@AxiomTutor]

That's right, a circle is every point except the center. Then you have to define what the circumference of a set is, in this case trying to figure out just what the "boundary curve" is and how to define the length of a boundary curve given any distance metric. In this context it's ... hard.

[@iankrasnow5383]

Pi would approach infinity if you could conceive of a circumference that approached infinite length (probably a circle with a fractal boundary), and/or if the diameter that approached zero. I'll think closely about this, whether you can come up with a system to make a circle with either property.

[@RichardMeikle]

This was immensely satisfying. Subbed!!!

[@pogan1983]

My favourite metric is half a litre of beer times 4. ;p

[@MrDeltaWorld]

I miss this show.

[@MaxLohMusic]

A guy on Quora claims you deleted his comment which explains that pi doesn't actually change when space changes because pi is used for so many other constants in math which have nothing to do with how we measure a circle. My question to you is: In a universe with altered space and circle ratios, what happens to those other functions which use pi? Does pi change to the altered pi in all uses of pi, or does it remain 3.14... for all non-circle-related functions?

[@einherjar77777]

Love this series. Well made, well presented, interesting topics.

[@thomasjmcfarlane]

The video does a disservice by promulgating a misunderstanding about how π is defined. The value of π can not be changed by changing the way distance is measured. The value of π is a fixed constant, period. It is the ratio of the circumference to the diameter of a circle in the Euclidean plane with the standard Euclidean metric. In non-Euclidean space or with other metrics, the definition no longer applies and one is no longer talking about π. One is talking about something else. Calling it π only confuses people.

[@tekublack]

Thomas McFarlane thank you! This vid is complete crap

[@tekublack]

master mansson so what your saying is. When you say pi you use it correctly unless you use it wrong then you should open your mind

[@martinepstein9826]

There's a great blog post where the author argues that defining pi in terms of circles actually misses the point and you're better off defining pi in terms of the period of the exponential function. This is the reason pi is so pervasive in mathematics, not because of circle circumferences. This definition also prevents confusion about whether pi changes in different situations, as a benefit for the less formally trained. affinemess . quora . com/What-is-math-pi-math-and-while-were-at-it-whats-math-e-math

[@nate4511]

mind blown....I love this!!! PLEASE KEEP UP THE GREAT VIDEOS

[@Raiment57]

This is so annoying! Pi is a NUMBER, just like 2 for instance -- you can NOT give it a different value. One of the things that equals Pi is the ratio of a circle's circumference to it's diameter IF you're talking about a "normal" circle. If you change your metric you change the value of that ratio BUT you do not change the value of Pi -- the ratio is just some other number. Pi has a life of it's own away from anything to do with circles and if you try changing it's value you'll break a LOT of other maths.

[@Roshkin]

Well that's the thing. Let's look at the math it breaks and see if anything cool happens

[@LuisAldamiz]

Assuming you can claim a square is a circle and not burst in laughter, you can apparently do that. Reminds me of a (bad) joke on economists: three academics are lost in the desert and are very very hungry, all they have is a can of beans but the opening device is broken. The chemist ponder how spitting on it and letting it bake under the sun, they might open it by corrosion in a couple of months, the physicists ponders how applying small well calculated bites to specific points they could maybe open it in just a few days. Then the economist comes with his proposal: "let's us suppose we have a can opener". Mathematicians seem to be doing the same: they have totally lost contact with any sort of reality. Pi is the ratio between a string and the mark you can draw using it and a central peg (or a drawing compass or any other similar device). There's no other Pi, much less a square is a circle.

[@ScottGardne123456789]

love these videos, every video and every application I try to conceptualise infinity and I'm always left in awe

[@BrentLawson]

Interesting but still nonsense.

[@goodcontent4392]

Brent Lawson how is that nonsense ?

[@yamansanghavi]

This video blew my mind... Pi is minimum at our euclidean metric.That kinda seems that there has always been a specialty in our euclidean metric that pi is minimum and i didn't know it yet... Thanks a lot to Infinite series.

[@rushilu3315]

This is probably my favorite video so far! I love this channel! I have one issue though: That diamond isn't a diamond, it's a tilted square. While this may be a semantic, I feel that it is an important distinction, because a diamond shape implies a kite figure.

[@MarioFanGamer659]

Isn't "diamonds" a synonym for rhombuses (of which squares are)?

[@1D0N]

No idea what I just what, but her telling it makes it good.

[@AbuSayed-er9vs]

Speechless!!Mind blowing!!!!!!

[@8mrLuka8]

Loved it! Great video, great explanation!

[@meatrace]

This channel is hitting it out of the park, keep it up!

[@ossiebird0]

I did this at Christmas, (√-1) 2(√2^4) Ʃ (4/1) - (4/3) + (4/5) - (4/7) + (4/9) - (4/11)... I love your videos, now you've just made me hungry for more. :)

[@OdysseyWorks]

The Schwartzschild metric is my favorite :D

[@Immerglad]

This is really nice, never thoihgt of it! (subscribed)

[@killsobmeanie2300]

Hurray, more math and science ! Thx Kelsey! Thx pbs !

[@oli2.019]

Just found this channel. Awesome.

[@hydrogenkhan6423]

Great lecture. I have been working on compressive sensing for long times and it has norm concept. Like one, two and infinity norm. This lecture explains this concept very well. Thanks

[@civilsavant6072]

This stuff is so cool. I wish I could have gone to school and learned it all when I was little. Maths education is SOOOO important!

[@peopleoftheworld6]

I really like this channel, please keep up the videos!

[@gmozzi5827]

This is so amazing!

[@SolveEtCoagula93]

Wow! So inspiring and fascinating. Thank you!

[@adamcwatts]

Brilliant video! One of the best math videos I've seen

[@AllanKobelansky]

Very well done. I enjoyed this.

[@RandyFortier]

The taxi cab metric is also called the Manhattan distance, making your example very apropos.