Transcript
Speaker: If you are there, and I am here, we can measure the distance between us. And if we are standing in a room, we can calculate the area of where we're standing, and if we want, the volume of the room itself. These are all examples of measures, which essentially tell us how much stuff we have, or how big something is. So what is a measure?
Speaker: How are distance, area, and volume related? And how big is the Sierpinski triangle? All this and more on this episode of Breaking Math. Episode 57? Is that how much?
Speaker: I'm Sofia, and you're listening to Breaking Math. With me, we have on a Meryl Flaherty. Meryl, thank you for being on the show. Thank you. And you work at the University of New Mexico, correct? That's right. I am a PhD student and a teaching assistant.
Speaker: and today we're going to be talking about measure theory but first we're going to do a couple of plugs first of all if you want a poster which is an introduction to tensors you can go to facebook .com slash breaking math podcast the poster is there for fifteen dollars and fifteen cents which is uh e to the e dollars plus four dollars and fifty cents shipping and handling coming to a total of nineteen dollars and sixty five cents
Speaker: It also has a little introduction to Einstein's General Theory of Relativity, or at least I use tensors with it. If you want to support us monthly, you can do so on Patreon. For $1 or more, we have a tier where we deliver outlines, and if you donate $22 .46 or more,
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Speaker: All right, so we're talking about measure theory, right? So what is measure theory? So pretty much measure theory describes the size of things. So like if you have some set or some shape or whatever, how big is it and in what dimension?
Speaker: All right. So if we're talking about length, obviously, right, you calculate the distance, uh, using, you, you measure along the length, correct? That's right. And there's one dimension and right. So length is just one dimension. So it's what your measure is in that one dimension. Yeah. So, uh, the measure of a lava, like a path into dimensions would be zero, right?
Speaker: Right, so the circumference of a circle would be a one -dimensional measure because a circle itself is what we call a one -dimensional manifold.
Speaker: Yeah, definitely. And if we took the circle instead, though, as the points within a 2D space, we would have to measure be measuring the area right if we'd use the 2D measure, right? Right. And so in that case, it would be different. Yeah. So it's like, say we take the circumference of a circle. So that's a one dimensional measure. But let's say that we take the area within that circle. Now, that's a two dimensional measure.
Speaker: Yeah. And the three -dimensional measure is like, like the volume that a sphere occupies. A four -dimensional measure might be like, I don't know, like some kind of hyper volume. It's pretty intuitive. I mean, and also one thing that we noticed with measures is that if I have a sphere, right? And I fill it up with a bunch of cubes, all those cubes will add together to equal the sphere, right? Right. Provided that you keep making the cubes smaller and you keep giving us more and more cubes.
Speaker: Yeah, so that's what we mean by the measure of something. It is how much stuff exists within something. And there's measures that exist for things that aren't just strictly Euclidean sets. There's like measures for fractal dimensions as well. And we're going to talk about those as well.
Speaker: All right, so length, area, volume, those are all part of something called Lebesgue measure, right? Yeah. And that's kind of a generalization of measure or more specifically a generalization of integrals.
Speaker: Yeah, because we're going to talk about later how we can have a fractional dimensions, like the dimension of a Sierpinski triangle, which is that triangle where you keep cutting out the middle triangle over and over again, kind of like a fractal version of the Zelda icon. The drive force. Yeah, but a whole number of dimensions in Euclidean space have a measure that can be characterized all by Lebesgue measure.
Speaker: So first of all, we're going to talk about Lebesgue measure in one dimension. So first of all, we're going to talk about what it means to have a subset of the real numbers. So a subset of the real numbers might be 0 to 3, right? Yeah. But I don't know. Do you want to give a more colorful example of a set within the real numbers?
Speaker: So there's all sorts of ways you could go with this. You could do something like negative infinity to zero non inclusive. And that's all of the negative reals. Or you could do something like negative three to negative two union with let's say like zero to one. And so those are two just disjoint sets that are joined together.
Speaker: Yeah, and you can even have like infinitely detailed sets, correct? Like if we took the length from zero to one and took out the middle third over and over again, I forget the Hilbert thing. The Cantor set? Yeah, not Hilbert side. But yeah, so all of these are subsets of the real line, right?
Speaker: Ultimately, they are. And to measure them, what we have to do is we have to define the Lebesgue measure. So the Lebesgue measure is defined, we're gonna just do the mathematical definition real quick, is defined as the infimum, which is the minimum basically, like the minimum of all possible things within a certain definition. And that definition that we're using here is the sum of lengths of some set of intervals that cover what we're talking about.
Speaker: And covering just means that these lengths all together, like all the union of all of them. Well, let's talk about the union real quick. It's going to be important. So what is the union between the closed interval from 0 to 1 and the closed interval from 1 to 2?
Speaker: So in this case, union means something different than what workers do to collectively bargain. In this case, it means that we're joining two sets together, basically, is that it's like you have a set. What were the numbers you said?
Speaker: 0 to 1 and 1 to 2. Okay, so let's say that you have one interval from 0 to 1, and you have another interval from 1 to 2. And is this inclusive or non -inclusive? Inclusive. Okay, so let's say it is inclusive. So 0 and 1 are members of that 0 to 1 interval, and 1 and 2 are members of this 1 to 2 interval. And so we would write that with square brackets for those intervals.
Speaker: Then so the union of these then would be it should just be the closed or square bracket interval from zero to two. Yeah, because the interval from zero to one contains every number greater than zero less than one and zero and one in the union of the interval from zero to one in the interval from three to four is just it would just be all the numbers between zero and one and three and four, right?
Speaker: Yeah, so so those numbers between one and three, then those are not part of the set. Yeah, so if we have the union between zero and one, and three and four, then all those numbers between one and three are not part of the set.
Speaker: Yeah, so like two and a half or something like that would not be part of the set, but like one and a half would be because one half is greater than zero and less than one. Exactly. And we can't really simplify that any further than that. We just have to say the union between the interval from zero to one and from three to four.
Speaker: To get maybe a clearer picture of this, we could use a physical example. So suppose there's some road that's infinitely long, but a lot of it has weeds, right? So we're going to say that this road is the set of all real numbers. And let's say we go like, I don't know, like a foot down the road, we're at the number 1, 2 feet is 2, and so forth, right?
Speaker: Right. And we're going to say that if there's a weed which could be infinitely tiny next to some section of road, then that section of road is within some set or the number it represents and so forth. Is that clear? Yeah. So what I'm getting is that this is a setup for sort of like the Diraclay function.
Speaker: I mean, it could be for the Dirichlet function if we had like these infinitely tiny weeds. The Dirichlet function is one where there's irrational numbers and zero where there's rational numbers. And you can do the integral of it. But I'm talking about more like just a set up like so if we're talking about the set between one and between negative one and three to the right of to the left of the point that was a zero one foot down that road there covered with weeds all the way to three feet on the in the other direction, right?
Speaker: Okay, so weeds at this point are being used as an analogy for intervals. Yeah, patches of weeds, etc. Exactly. Where an interval is from zero to whatever and so forth. So let's say there's two people with giant backpacks full of pesticides that start at the zero point and walk down the road in opposite directions.
Speaker: So they could spray haphazardly, right? Like they could spray back and forth. So we're going to say that one spray of this pesticide is enough to kill a weed. So the optimal spray of the pesticide will not spray the same weed twice, right? That sounds right. And we don't want to spray patches of road that don't have weeds on them because we'll still be wasting a pesticide.
Speaker: So let's say that we have some spray pattern that uses the minimal amount of pesticide, right? That is kind of like the Lebegue measure of the weeds. Or you can even talk about it as the time that the sprayer is on. Because remember, they're rocking down the road at a constant rate. So if the sprayer is on for like three seconds and they've walked for however distance, it'll be one -to -one correspondence, right?
Speaker: Yeah. And just so I'm clear on this, what we're looking at is we want the least amount of pesticide possible that will kill all the weeds. Yeah, exactly. Yeah. Minimum amount of pesticide or minimum time that the pesticide thing was on for. Okay. That makes a lot more sense, actually.
Speaker: Yeah, so you can look at the spray of the pesticide as the sets that are part of the covering set, right? Because when the pesticide is turned on, if you take that time and subtract different ones turned off and you add all those up, it'll be the total time that it's been on for. And also, if you take the pesticide, the amount that remains, if you take that from the amount that you started with and you add all those up, you get the amount that you've used in total, right?
Speaker: Right. So that's the analogy that you can maybe use to understand this a little better. And so let's say we have, um, so that's one dimension, right? Um, assuming that the road itself is one dimensional. Sure. Yeah. We're mathematicians. Roads are one dimensionals and cows are spherical in a frictionless, perfect vacuum. It sucks too, because I tried to go to the store the other day and I couldn't walk there. Just kept slipping. I wasn't even tumbling because my muscles couldn't, uh,
Speaker: Yeah, none of this works. But anyway, yes, it's math world. So let's say two dimensions, right? You can visualize this on the covering set of like, let's say a circle as a bunch of squares that cover it. So like, let's say we are in Microsoft Paint.
Speaker: or some kind of drawing program like that, and you keep drawing squares. If you draw blue squares over a picture of something red, like a skull or a circle, and if you cover it so there's no red remaining, then that could be considered a covering in two dimensions. The infimum of those is the Lebesgue measure. And in three dimensions, it would obviously be cubes, right?
Speaker: Right. And so what I'm getting at here is that. So let's assume that all of these cubes are equally sized and we're cube squares whatever are equally sized and we're using them to cover whatever shape whatever set we're looking at. Then we get a clearer idea of the size of this thing if we keep making the cubes or squares or whatever smaller and we keep counting how many of those there are.
Speaker: Yeah, because we can start off with some big cubes, right, that like block off a lot of the area, but then just kind of make them smaller and smaller and smaller, and fill up more and more of the area, yeah. So maybe the big squares fill up 95 % of it, the smaller ones fill up 99%, and so on, all the way up to 100, theoretically. As long as we're in an integer number of dimensions, right, we're pretty good so far. But let's say we have a fractal.
Speaker: So do you want to talk about what it means to have a non -integer number of dimensions? So before I do that, I do think I need to explain something about dimensions. So a good way to describe dimensions is that, so let's say that you have just a square and you multiply all of its side lengths by some constant, let's say r.
Speaker: then its area as you scale something by R is going to scale by R squared. Or with the cube, you scale the side length and the volume will then scale by R cubed. And that works with any shape, right? If I have two dragons that are copies of one another, dragon statues, and the length between one dragon's eyes and its tail is twice what another is, it'll have eight times the volume, right?
Speaker: Right, because eight is two cubed. So the thing about that, though, is let's say that we have a fractal shape and it can be anything, a Sierpinski triangle, the Cantor set. And the Sierpinski triangle just again is you take a solid triangle, you take out the middle third from that triangle, the middle like a triangle facing downward. So you have three upwards facing triangles.
Speaker: From each of those you do the same thing to what you did to the first triangle. So now you have nine little triangles. You do it again so you have 27 and so forth. Yeah. And the cantor set is where you keep removing the middle third of a line between zero and one. And then you remove the middle thirds of those remaining intervals. And you keep doing that on and on until you have a set that is infinitely many points but zero length.
Speaker: Yeah. So, I mean, it is kind of the one -dimensional version of the Sierpinski triangle now that I think about it. Something like that. So here's the thing though, is that going back to the Sierpinski triangle. So what happens as you keep cutting out more triangles from this is that you get something where, and you can use the series for the area or for the perimeter to calculate this, that it would have zero area yet infinite perimeter.
Speaker: Yeah. And the reason why is, um, I mean, we could do it, uh, super quick. I mean, you take out one third of something over and over again, it's going to go to zero, right? Cause there's no lower parts. There's nothing that it converges to, right? Right. It just goes to zero. Yeah. I mean, I guess it's a quarter of the volley area, but yeah, you take out a quarter every time and it keeps going to zero. Um, because it's three quarters to the infinity of the power, which is, I mean, just go on a calculator, keep multiplying it by three quarters. Um, and it'll go as low as you want.
Speaker: right. So what we have is a sequence of we have some real number to the power of n. And so what we get is that this number is less than one. So the more we raise n, we see that it eventually goes to zero.
Speaker: but the perimeter is the opposite way because one of the small triangles has half of the perimeter of the big triangle because it's a triangle half the size so each side is half as big so basically it's half of this the total length but there's three of them so we have to multiply by
Speaker: three halves each time, adding 50 % over and over again. And doing the same thing, raising that power until we go to infinity, we see that that perimeter itself also goes to infinity. So yeah, continue with what you were saying about the difficulty of measuring this.
Speaker: Right. So the way that mathematicians have gone about this is that so we can't define the size of this in length, and we can't define the size of this in area. And clearly, it doesn't have volume. So we decide instead, what if we try to find something in between this where we can measure its volume or content or measure, I suppose is the correct word for it here.
Speaker: So before we go on to that, I should give the formula or a formula for finding the dimension of certain things. So if I, if I multiply, so the seriously triangle is three copies of itself. Um, if you, uh, if you double it, right? Yeah. But a square doubling it is four copies, right? Sounds right so far. And the cube doubling it is eight copies.
Speaker: Yeah, so powers of two. Yeah, and that's a dimension three and dimension and dimension two for those two. And it turns out if you take the log of eight divided by the log of two, it's three because log base two of H is three. Another identity will do that some other time. And log four divided by log two is two.
Speaker: So if we, so it turns out that we could do log 3 because we tripled the stability triangle, right? Divided by log 2 because it's double the size. Then arrive at the dimension, you go to log 3 divided by log 2, which is 1 .58496, etc. But Meryl, tell us more about it.
Speaker: Okay, so Minkowski -Booligan dimension, also known as box counting dimension, is a specific way of estimating a fractal or fractional dimension, which go hand in hand in this case. The way that it goes is that you count a number of boxes that cover a shape that you're looking at. So let's say we're talking about the coastline of Indonesia.
Speaker: So let's say that we have boxes of some certain length and we'll call it what I think what people usually use is Epsilon and some very small number Delta for those as far as boxes go daily.
Speaker: Huh. Well, it doesn't matter. We're just assigning letters to things. So what happens is let's say that you have a bunch of boxes of length Epsilon and you can cover the coastline of Indonesia with these. So there's a certain number of boxes of length Epsilon that you need to cover that. And so Epsilon it's a, it's a number that gets smaller, right? It's like a limit, right?
Speaker: Yeah. So let's, let's say we start off with Epsilon being one foot. Uh, what would that look like if we were to physically do this to Indonesia? So I might not start at one foot, but let's say like, let's go for something really big. Yeah. Let's go for, let's go crazy. Let's go a hundred miles. So it's like, you're just going to get these big chunks of coastline and it's not really going to give you an accurate description of the length of coastline.
Speaker: And it's a minimal covering, right? Because we could theoretically put boxes all over the coastline, like hundreds of thousands of these big coastline boxes. But we're talking about the minimum number that fit within. Yeah. So just don't let your boxes intersect. Just try to get the smallest number of boxes possible that are 100 mile side length to cover this coast.
Speaker: Now, what we want to do is we want to start making them smaller and we'll get a more accurate representation, right? Oh, yeah, absolutely. I mean, I suppose what you're getting at is like if we made them like a mile big, then we'd start getting like coves and individual islands. Right. And so as it as we keep making these boxes smaller, we keep getting more and more details. And what we see is individual coves and coastal and grains of sand, right?
Speaker: Right. So this progression of boxes as it gets smaller, right? Each covering. So we have a covering for a hundred mile big boxes, a covering of one mile big boxes. And as we get to micron and nanometer boxes, these are all to calculate a dimension, right? A number instead of a collection of things, right? We just need one simple number from it.
Speaker: Okay, so a really rough description of how we get dimension out of this is how many more boxes. So what is like, how does the number of boxes scale with the refinement of the size of the boxes? So as the boxes get smaller, how many more boxes do we need to cover this coastline?
Speaker: Yeah, so the Minkowski -Boulogon dimension of the coastline of England is approximated to about 1 .25, for example, which means it's a little bit more one -dimensional than the Sierbinski triangle. Right, so it's a little too complex to just use length to calculate.
Speaker: Yeah, which is fascinating. And Madeleine Broad decided to publish a paper called What is the coastline of Great Britain? How long is the coastline of Great Britain? And everybody was like, what the heck are you talking about? Like, just look it up in an atlas. And then didn't he publish another one? He's like, no, really, how much is that?
Speaker: And I guess to go back to the analogy, it's like, sure, let's say that you take a yardstick and you just keep measuring and adding up till you've gone all around Great Britain. Then it's like, you'll get some value. But let's say that you just get a footlong ruler now and you don't know. Let's get stupid with it. Let's just make it an inch long ruler and someone.
Speaker: And you'll notice that this length that you get will keep increasing and increasing the smaller of this ruler you get. It'll eventually, let's say that, you know, someone out there for some incomprehensible reason had a millimeter long ruler who would be, you would get a value that is way greater than you did with your foot long ruler. That in turn would be greater than the meter long ruler.
Speaker: Yeah. It would just keep, it would eventually go to infinity because you have something that is self similar and infinitely complex and the length would go to infinity. Yeah. Or I mean, very, very high. There would be a finite number of particles and wouldn't, if we measured their electromagnetic boundaries, there must be some smooth function. Okay. We'll use applied mathematics terms. We'll just say it gets, gets arbitrarily large.
Speaker: So one way to measure the amount of stuff that's within something that works for fractals is through the Hausdorff measure. So let's say we have a covering, right, of this thing that we're taking the Hausdorff measure of, and we're adding up. So let's just imagine that these are balls in whatever dimension that we have, right? So if we're doing the Sierpinski triangle, the balls would be embedded within two dimensions, right? Yeah, even though that they would be log two of three dimensional balls, but
Speaker: Yeah, that's a little counterintuitive. Yeah, and that's the that is the power you and raise it to. So basically, like, we're imagining that these are like little like cubes or little balls or little whatever. It doesn't really matter. So that they cover this set that we're measuring.
Speaker: As the balls get smaller and smaller, all we have to do is take their diameter and raise it to the power of s, where the diameter gets smaller than some number delta, and we'll get what's called the Hausdorff measure. So it's the infimum of the sum of the diameters of the individual sets of some set of sets that covers a set that we're measuring, and each of these is raised to the power of some number s. But if s gets too big, then the Hausdorff dimension is zero, right?
Speaker: Right, and I think I have an easier way to visualize this. Please. Yeah. So let's say that we have a square, then we know that it has an area, that it has a dimension of two, right? But let's say that we try to take a volume of this square, then, well, we haven't extruded it out into three dimensions, so it's just going to be zero. Yeah.
Speaker: And it doesn't matter whatever the dimension is, if it's larger than two, it's that measure is going to be zero. And same thing for like if you have a cube, then if you try to take some sort of four dimensional measure, or it doesn't matter, let's say we have a 3 .3 dimensional measure, it's still going to be zero because it's greater than three.
Speaker: Yeah, and in the same way, a line that is one -dimensional, that is infinitely thin, will have no measure in two or three dimensions. Yeah, so it would have no area, no volume, no four -dimensional measure. But yeah, so this is the Hausdorff measure. So let's take a simple fractal, right? The Sierpinski triangle. Meryl, you're an educated grad student. You should know the Hausdorff measure of the Sierpinski triangle.
Speaker: If I'm not mistaken, it's log 2 of 3. Well, that's the Hausdorff dimension. But the Hausdorff measure, this is weird about the supremacy triangle, because the Hausdorff dimension is known, right? And the Hausdorff dimension comes from the Hausdorff measure. But we only know that through properties of the Hausdorff dimension. Basically, the Hausdorff measure is a surprisingly current problem. It's something that has a bound between 0 .77 and 0 .819161232881177.
Speaker: Or there's a paper, which is in a paper by Peitemor, but there's another paper by Heiyu Wang and Hingua Wang, which gives another estimate that's a little bit closer. But the point is, is it's an unsolved problem. And it turns out that the reason why it's still unsolved, so a level one Sierbinski triangle is just a triangle, right?
Speaker: Um, I think that would be the zero iterations actually. Yeah, that's right. Yeah. Zero iteration. Yeah. That's just a triangle, right? Yeah. The first iteration is you kind of triangle out of the middle. So, you know, you get your typical legend of Zelda try force.
Speaker: Yeah, and then you cut it again. You get a triforce made of triforces. Cut it at level three is a triforce made out of triforces made out of triforces and so forth. So it turns out that counting the number of triforces within a specific disk or like our radius is
Speaker: not a solved problem. And it's kind of easy to know that it's not solved because, I mean, just think about the regular grid, right? Like, if I get an infinite graph paper, the number of points within a circle that I draw is not going to be something that we know currently.
Speaker: So this is known as a Gauss circle problem, which basically states that the amount of points, it's a question that is still unsolved as to how quickly the amount of points within a circle of some radius are gross. And the points within a Sierpinski triangle are distributed a lot more complicatedly than that. And so the fact that it's not known is pretty expected. A level 30 Sierpinski triangle has 159 trillion
Speaker: 310 billion, 733 million, 867 thousand, and 10 triangles that need to be considered while doing a simulation. And I know that when Pei Ted was doing their simulation, it took like four days at this resolution, and it triples each time, which shows the difficulty in finding these numbers, and also the fact that lower bounds and upper bounds currently have to be estimated by slightly different means.
Speaker: The paper by Heiyu Wang and Hinghua Wang, we're trying to get more information on that. Currently it's only in Chinese, but I'm trying to get some help with that and hopefully we'll have an update with you on that. I just found it kind of interesting that it's so hard to measure this, you know? Yeah, but it also makes sense because it is a very complicated sequence that we're looking at.
Speaker: Yeah, just the regularity makes it seem like it should be much easier to figure out stuff about it. But yeah, I guess it is counting those things. It's just one of those weird like sometimes you turn a corner in mathematics and you just hit a wall, you know. Yeah. And that's the thing is like infinitely complex things like this can just come out of the simplest of rules. And making a Sierpinski triangle is about the simplest rule you could come up with. You're just cutting triangles out of triangles.
Speaker: Yeah, and it doesn't help that we've just started in the last 150 years to study this. Like, mathematics takes quite a long time. Yeah, for us mathematicians, 150 years is like 5 seconds. Yeah, but we chip away continuously at this gigantic sculpture.
Speaker: The amount of stuff within something is the task that gets harder the more complex of shapes we look at. But this just means the measure theory is a fertile ground for further development. I'm Sophia and this has been Breaking Math. With us we head on today Meryl Flaherty who is a grad student at the University of New Mexico and teaches there.
Speaker: Any impressions about the show today? Any further thoughts? There's just a lot on this subject, and I could talk to you about it for days, especially because fractals in particular and fractal geometry are a special personal favorite of mine, and I could just gush about it if I wanted to. All right. Was that a little lame? It was a little lame. I don't care. We're doing math, of course, for lame.
