When Pi is Not 3.14 | Infinite Series | PBS Digital Studios
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You’ve always been told that pi is 3.14. This is true, but this number is based on how we measure distance. Find out what happens to pi when we change the way we measure distance.
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Previous Episode - Can We Hear Shapes?
• Can We Hear Shapes? | ...
Mathematician Kelsey Houston-Edwards breaks down what happens to pi, and circles as well, when we change the metric we’re using.
Sources:
[Pi] is the Minimum Value of Pi by Charles Adler and James Tanton
Link Unavailable
Thanks Kelly Delp!
Planar Isospectral Domains
math.dartmouth.edu/~doyle/doc...
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Made by Kornhaber Brown (www.kornhaberbrown.com)
Comments answered by Kelsey:
Chris Morong
• Can We Hear Shapes? | ...
Calvin Jones
• Can We Hear Shapes? | ...
Taylor Kinser
• Can We Hear Shapes? | ...
Spencer Key
• Can We Hear Shapes? | ...
Thiemo Krebsbach
• Can We Hear Shapes? | ...
After some careful research, I came to the conclusion that birds don't use the Euclidean metric when they fly.
They use their wings.
#dadjokes
I am offended 😡
The most mathematician thing i have ever heard: "The real world isn't exectly my thing" ^^
Gnosticism in a nutshell
How is it?
School Maths: Pi=3.14159. Street Maths: Phew=3.17157. (only for 1D)
Oh my god it's a circular square.
The Brony Notion so that's how you square the circle
The Brony Notion It is a circle but not in our euclidean world and also I didn't know you liked maths!
The Brony Notion You can't square the circle?
well take that, greeks!
You mean a Parker Square?
All these squares make a circle
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.
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.
Brilliant! Did she just squared the circle? :D
All we had to do was change the damn metric, CJ.
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.
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.
It was an error by the animator.
When it appeared on screen, she said "1½, -1½"
If you listen, she says (1 1/2, -1 1/2), which would be right, so it's definitely a typo.
Good catch. It should be (1.5, -1.5)
Yeah, I caught this too. I think the simplest fix would be annotating 3 in place of the 1. Yay, improper fractions. =P
This channel has over taken PBS Space Time for my favourite educational channel, and sorry Numberphile :p
Wow! I was going to say pretty much the same thing!
Probably because I love math more.
GuyWithAnAmazingHat I'm sure about PBS Space Time! Sorry Matt!
But not about Numberphile! Brady is still working!
well, this aint like the cake... you can enjoy all of them!
dude go watch 3blue1brown. his linear algebra course is freaking amazing!!!!
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!😊
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!
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
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 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?
So what happens if you go for p-adic metrics? - For starters, what even is a circle in those?
I was also thinking about this
... Watch the video?
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.
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".
Eric Pive she didn't mention p-adics. she mentioned p-norms, completely different.
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".
Squaring the Circle must be much easier in some of these metrics, lol.
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.
It is in these cases though
Sure, but that's something that needs to be proven.
+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.
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?
In that case forget I said anything. Yes, that is indeed far from obvious
Favorite Metric? Hausdorff.
Love math, love spacetime and now infinite. And Kelsey Houston-Edwards. Amazing.
People have already mentioned this, but at 3:59, there is a typo in the second point.
Also, this was very fun and educational.
So what's the formula for the L^p circle constants? I didn't become a mathematician for decimal approximations!
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.
2 ∫ (1 + (x^p/(1 - x^p))^(p-1))^(1/p) dx, where the bounds are from x=0 to x=1.
Oh right. If it did have an evaluatable value, then pi would be algebraic which we know it isn't. satyu131089
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.
+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.
3:21 "Math isn't restricted to reality" - priceless
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||
Feels like you could collapse all of this in just a single function.
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.
"All these squares make a circle…"
My favorite metrics are the p-adic because prime numbers are awesome
The more I see my children interested in watching folks like Kelsey, the more faith I have in the future of humanity.
that moment you realise this girl just told you a squer is a cirkel. and you agree...
i'm no rocket surgeon, but I didn't agree.
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 !
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 ;-)
+ 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.
8:16 details not in description..
The amount of times I have returned to this video to understand different metrics on R^n for my analysis and topology classes.....
So glad I found this channel. The woman explains things very clearly! Thanks!
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.
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.
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.
Me too: that was exactly my hope when I clicked. Very disappointed.
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?
@@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.
If I would have watched this 2 days ago it would have been pi day 🤔
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 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. :)
I've learned so many cool things!!!
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.
That's not a metric in the analysis sense.
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.
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.
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.
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.
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.
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.
yup definitely her. Kelsey if you read this Hi lol. thanks for not letting Guckenheimer demolish me in calc... Cameron says hi too.
hi cameron
She looks like a Kelsey for sure lol.
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?
Cody Griffin I believe p is always assumed to be greater than zero.
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).
and without taking the zeroth root it would be 2 = 1, heh
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.
You don't get all the nice properties of a metric there, but it would still be interesting to see
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!!!
This beguiling Mathematical wonderment, of which I did not know, is as pleasing to ponder as it is to listen to deeply beautiful music.
so you're not going to tell me how to calculate the circumference of a euclidian circle without pi.
7:49 Looking for the Lp in which 'pi' equals p itself
Huy Dinh let me know when you find it
3.30524
That is amazing. How did you get there though? Cold you please elaborate?
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.
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
This channel never ceases to blow my mind. Just like Kelsey's organisation, delivery and panache make my knees shake (in a good way!).
Karn Kaul Umm btw, don't believe this please.
Why should you "believe" maths in the first place?
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.
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.
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?
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.
FrostyDynamic Thanks! That was helpful!
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-.
irrational numbers aren't impossible numbers...
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.
This is so fascinating, i wish i learned that mathematics was so much fun when i was in school!
"Pi is the minimum value of pi." That's not a sentence I ever thought I'd hear.
I like the Karlsruhe metric because it messes with your brain a bit while still being relatively easy to visualize. But I guess a circle is the same as a disc under this metric, so defining pi makes little sense.
0.999999..... th!
I see what you do there XD
Aditya Khanna This is probably the most nerdiest 1st comment xD
lol
Yevhenii Diomidov I had a doubt too, but "th" sounded better
Ordinals only apply to natural numbers.
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?
I suppose the average value of all the numbers you get could be interesting...?
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.
huh, let's think about that
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).
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.
I just discovered this series. I'm in love with it! Please keep it up!
I must admit I cracked up when she said that the minimum value of Pi was (drum roll) Pi.
Love the haircut! And the math too I guess
what about curved space? you know, like Spacetime
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.
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).
*****
Hm, I knew about the bit with the triangles before but I kinda didn't even remotely consider the first part of that...
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)
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.
When a square is oriented to have one of it's corner at the top, it transform instantly into a diamond. But if this is one side at the top, it stays a square. Fascinating.
Worth mentioning is that axes matter in tax cab geometry. Point A can be 3 from point B, which is 3 from point B and they meet at a 90-degree angle, yet this does not imply that we know the distance to point C. It COULD follow the formula of dAB + dBC, but it could also take on other values, ranging from 3 to 6.
My favorite metric is the liters. :p
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......
I agree.
No ven using Euclides: using a piece of rope!
just discovered this channel, this is genius!! bravo!!
Imagining p=0 always makes my brain hurt. Oh no, it's happening again. >_
Burak Bağdatlı it just doesn't exist
Isn't possible it will be singularity
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).
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.)
+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.
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)
I blame the French for base 10.
Pi is 11.001001000011111101101010100010001000010110100011... in base 2
orochimarujes I believe yes
Gabriel cazorla persson true
I should add here, that by changing bases, only the symbols change, not the actual value itself - that remains the same of course...
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.
9:22 it is amazing that the mentioned paper by Milnor is actually ONE page long
So does that mean it is possible for pi to equal infinity?
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
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?
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.
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.
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.
My favourite metric is half a litre of beer times 4. ;p
Loved this episode, I'm so glad this new channel is turning out to be this good!
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
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.
Thomas McFarlane thank you!
This vid is complete crap
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
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
All this arcane math talk, and NOT ONE SINGLE PIE!!! Dammit, I'm hungry now!
[01:59] Actually, mathematicians call it, the "Manhattan Norm..." whereas any "taxicab norm" would allow for the fractal nature of finding the fastest path through traffic and undependably-depending on entropic tips, and taps, and the unreliability of detours.... Moreover, taxicabs exist in the 'Frisco Norm' too accounting for hills adding distance.
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.
Well that's the thing. Let's look at the math it breaks and see if anything cool happens
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.
Interesting but still nonsense.
Brent Lawson how is that nonsense ?
This is great! She's easy on the eyes and I actually MIGHT have learned something.
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 =)
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.
When is Pi not 3.14? Always. Pi is ALWAYS not 3.14. Pi is pi, by definition. That said, I could listen to Kelsey all day long.
I love this channel! Its what I wanted out of Space Time: maths! Thank you so much!
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.
If I would have seen this in high school, this would have probably prompted me to annoy my math teacher with more questions ;)
love these videos, every video and every application I try to conceptualise infinity and I'm always left in awe
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.
You know, I never really considered that the ratio could be different. This was fascinating.
OMG I love that this show exists.
The taxi cab metric is also called the Manhattan distance, making your example very apropos.
What an awesome channel to stumble upon. Thanks for that.
Dear Mathematician Kelsey Houston-Edwards: on the map of Manhattan, you seem to have placed the National Museum of Mathematics at the approximate location of the Museum at the Fashion Institute of Technology! They're both great edutainment attractions, and they both generally have the same "Taxicab Metric." Plus, to add to the NYC theme, I feel that your Noo Yawk accent has really come through in this video!
This channel is everything I want and more!
The ratio of her cuteness to the inverse of smartness is very high.
That comment is way more ambiguous than it appears at first glance.
Third channel where I hit the like button first, then watch the video. (The other two being PBS Space Time and Physics Videos by Eugene Khutoryansky)
Great. Now all these students will be arguing with their math teachers when they get a question wrong. "I was using the taxi cab metric, so I did it right."
Fascinating video. This brings to light the significance of the basic plains we work on in high school and college. Thank you.
I miss this show.
The Schwartzschild metric is my favorite :D
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.
Isn't "diamonds" a synonym for rhombuses (of which squares are)?
When Pi is not 3.14
*laughs in Engineer*
draw a circle on a chalk board with chalk, now determine (measure) exactly how wide the circle is. To do this you need to know how thick your actual chalk line is, so you can find where to start and finis your measurements (on the outside of the line? Inside of line?). How exact (small), can you go for your start and finish mark? Infinity small, so Pi doesn’t really exist, but it’s close enough for most calculations.
In Germany, we often call the 1-norm "Manhatten norm". Never heard anyone call it Taxi Cab norm before. ^^
Absolutely fascinating video. This is one of my favorite channels, for sure.