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

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

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

Oh my god it's a circular square.

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.

Brilliant! Did she just squared the circle? :D

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.

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

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.

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!

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.

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.

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

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

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

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 :)

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".

Favorite Metric? Hausdorff.

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

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

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.

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

I've learned so many cool things!!!

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

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!!!

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.

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

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||

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.

"All these squares make a circle…"

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

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

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

8:16 details not in description..

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

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

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

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.

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.

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

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

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

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.

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.

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

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

0.999999..... th!

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

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

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

Love the haircut! And the math too I guess

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

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

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?

My favorite metric is the liters. :p

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

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

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?

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?

This channel is everything I want and more!

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

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......

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)

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

This was immensely satisfying. Subbed!!!

what about curved space? you know, like Spacetime

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).

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

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

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

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

The Schwartzschild metric is my favorite :D

OMG I love that this show exists.

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?

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.

Loved it! Great video, great explanation!

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.

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.

Interesting but still nonsense.

Wow! So inspiring and fascinating. Thank you!

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

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. :)

This is so amazing!

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 =)

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

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

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

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.

Yes, some colleges and universities do, in fact, teach classes on "Taxi Cab Geometry".

Very well done. I enjoyed this.

Speechless!!Mind blowing!!!!!!

Just found this channel. Awesome.

WOOOW! This is just awesome!!

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

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.

Regarding "hearing the shape of a drum" in higher dimensions... It much simpler to show that there are multiple topologies of a vibrating, two-dimensional membrane embedded in a higher dimensional space that have identical power in identical modes. However, eigenvalues and eigenvectors are properties of the membrane, and not something we could hear or record as sound. The waves on the membrane have to propagate to the recording device through the n-dimensional space. In three dimensions, you can set a simple relationship between the drumhead and the microphone: for example, the microphone can be set on a line normal to the membrane that passes through the midpoint of the membrane. With one microphone, however, there is an obvious ambiguity- if we have a non-circularly symmetric membrane, the recorded sound would be invariant to rotations of the membrane about the line between the midpoint and the microphone. To break that degeneracy, we could add an additional microphone. Just like your binaural hearing gives you the ability to discern the direction sounds come from, a second microphone would help lock down that degree of freedom. So you are starting to “see” the membrane with spatial information and frequency information. Adding two more microphones perpendicular to the first two would help resolve 2-D shape information from the 2-D membrane . In higher dimensions, sound would propagate differently because it would spread out into more dimensions. If that can be ignored, then you should still be able to use 2 sets of stereo microphones to reconstruct the 2-d membrane because it is the same 3-D problem just embedded in a higher dimensional space. The issue really comes when the 2-d membrane is allowed to bend into all of the other dimensions. Now the line extending normal from the membrane doesn’t have to point into the axis we previously placed the microphones that helped provide spatial information. Assuming there is a location that stereo microphones can be put in the added dimensions so that they would ‘see’ the sound waves propagating from the membrane in that dimension, then you should be able to recover spatial information in that dimension. So I think a hand waving answer is that you could reconstruct a 2-d vibrating membrane in N dimensional space provided you have at least N stereo microphones distributed into each dimension… with the big caveat that there are probably pathological topologies that would require many more microphone positions than dimensions.

I am reminded of story that is attributed to Abraham Lincoln. He supposedly asked someone the question "If you count a tail as a leg, how many legs does a horse have?" When he got the response of "5", he said that calling a tail a leg doesn't make it one. And calling a diamond a circle doesn't make it one.

That is an excellent presentation. I was expecting you to go to the sphere metric, but this was better to show the nature of the abstraction and the general result. A very good work.

I love this new channel :) Keep up the good work!

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

This blew my minddd haha love it 🙌🏻