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P8: Tangent Tango (Morikawa's Recently Solved Problem)
395 Plays4 years ago
Join Sofía and Gabriel as they talk about Morikawa's recently solved problem, first proposed in 1821 and not solved until last year!
Also, if you haven't yet, check out our sponsor The Great Courses at thegreatcoursesplus.com/breakingmath for a free month! Learn basically anything there.
The paper featured in this episode can be found at https://arxiv.org/abs/2008.00922
This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. For more information, visit CreativeCommons.org!
[Featuring: Sofía Baca, Gabriel Hesch]
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Transcript
Introduction & Problem Overview
00:00:06
Speaker
I'm Sophia. And I'm Gabriel. And welcome to Problem Episode 8, Tangent Tango, where we're going to be talking about Murray Kava's unsolved problem. This was solved during the COVID quarantine. It's fascinating. I love the fact that people attempted these unsolved problems and they made some progress.
00:00:24
Speaker
Oh yeah, there's been a lot of weird academic stuff going on during COVID and this is a pretty cool one.
Community & Support
00:00:31
Speaker
But before we start, we have a few plugs. So if you want a poster that has a lot of cool math diagrams on it that you could put on your wall, it's a 36 by 24 inch poster, we have one and we are selling it on facebook.com slash breakingmathpodcast for $19.65, which includes the shipping and handling, which is $4.50. You can get that at facebook.com slash breakingmathpodcast.
00:00:53
Speaker
And again, it's a flyover of tensor calculus, which is pretty darn cool. It's the mathematics used in Einstein's general theory of relativity. So it makes a great birthday gift. Oh, yeah. And then we have our Patreon, if you want to support us monthly. It's just $1 or more. You could have our episodes with no ads and outlines. If you want to have updates about the show, you can find us on Twitter at Breaking Math Pod or Facebook.com or Breaking Math Podcast.
00:01:22
Speaker
We also have our website breakingmathpodcast.app, which we're going to be updating pretty soon. And if you want to email us, just send us an email at breakingmathpodcast.gmail.com. Yeah, with any ideas, comments, or questions or corrections.
Sangaku: Historical Context & Problem Introduction
00:01:35
Speaker
So without further ado, this problem comes from a type of math known as Sengaku, which has its origins in Japan. Basically, Sengaku consists of geometric arrangements often involving geometric objects tangent to each other in artistic ways.
00:01:49
Speaker
And it was presented as gifts to Buddhist temples in Japan during the Edo period, which is basically between 1603 and 1870. Japanese math was distinct for quite a while. For example, Sekitagakatsu, who lived from 1642 to 1708, was a Japanese mathematician who invented something very much like calculus, researched Bernoulli numbers, and this is all separate from the West.
00:02:15
Speaker
One thing about Japanese math is you have the Sangaku which they would be put on like Gabriel said tablets and presented to Buddhist temples. And one example of this kind of problem is let's say you have two rolling pins that are resting on a table of different sizes.
00:02:30
Speaker
If you want to squeeze in another rolling pin between them in the kind of triangular space between the table and the rolling pins, the size of that circle is given by 1 over the square root of its radius is equal to the sum of the inverse square roots of the two bigger rolling pins' radiuses.
00:02:50
Speaker
That's the, um, the Sangaku, uh, for example.
Geometric Problem Description
00:02:53
Speaker
And today's problem is a specific Sangaku problem proposed by a man by the name of Jihei Morikawa in 1821. Sophia, would you like to explain this problem? All right, sure. So let's say we have a line and you have two circles of any size resting on that line. So the two circles are touching one another. And they don't have to be the same size circle, right? One can be larger than the other.
00:03:17
Speaker
Yeah, and the only stipulation is that the two circles are touching one another and they're touching the line. So right now we have a point where the two circles touch each other and two points where the circles touch the line. So one point for one circle and one point for the other circle. So it's just tangent to the line. Very good, and I understand in this problem we are interested in the little space between the circles and the line. How would you describe that shape?
00:03:43
Speaker
It's kind of like, I think you could call that like, I don't know, an inverse lunoid or something. Interesting. Yeah. That is what we're interested in this particular problem. Yeah. And the question is, what is the minimum size of the square that is tangent to both the line and the two circles? So I think it'd be good at this point to talk about what tangents are exactly. And it's pretty simple. It's like if two geometric shapes touch each other at exactly one point, they're tangent to each other at that point.
00:04:11
Speaker
Correct. And in this case, if you have a square that is flat on the line, it'll be tangent all along that particular side. So it's not always only one point. It's also just a given segment.
00:04:23
Speaker
Yeah, and this square, if you think about it, if the corners are touching the circles, for example, then it could be tangent at those two points and wherever it is on the bottom. It could also be tangent to the side, and it could be also tangent to both a corner and a side. A way to imagine this is if you have a coin rolling off of a table with a really hard edge, right before it starts rolling off the table, you could see that the corner of the table is touching the coin.
00:04:50
Speaker
And the coin is resting on the table. Yeah. And for the sake of this problem, this was mentioned earlier, but I think it's worth stating again in the audience's mind, the audience is allowed to resize the square of any size at all. And essentially the square can be rotated into any orientation. And that's actually part of this problem as well.
00:05:13
Speaker
So, and the question is, what is the minimum size of the square? This problem was proposed in 1821, and it wasn't solved until June of last year. It was solved by Jan E. Hawley and David Crumb.
Solution Approach & Methods
00:05:28
Speaker
That's right. And the paper can be found. We'll put the link to the paper in the show notes so you can go through it step by step if you'd like. And I believe it was solved with a computer. Is that right? Part of it was, yes. One of the steps involved finding Galois groups and roots and stuff. But a lot of it was done, it seems, by hand. And I think it's kind of interesting, too, because
00:05:52
Speaker
It relies on the algorithm used being reliable, and it's an algorithm called MAGMA, and I believe it has been shown to be reliable. Very cool.
00:06:02
Speaker
So first what we can do to solve this problem is we might be a little bit daunted because let's say the left circle has a radius of 10 and the right circle has a radius of 7. It seems like we have quite a few different configurations that the circles can be in. But we can actually, without laws of generality, we can set the smaller radius to 1 and the other one to a number greater than 1.
00:06:22
Speaker
So let's say you have a radius of like one half and three quarters, you can resize that to one and three halves. And without loss of generality, the circle, the larger circle is going to be on the right. And the reason we could do that is because if you just flip the problem around, it's the same problem. Good.
00:06:38
Speaker
Next, we prove that for every angle that we want to rotate our square at, there's only one position in size that has the square at that angle that is tangent to the two circles and the line. And the way that this was done in the paper is basically showing that if you wedge it in there, that you can't really rotate it in any way, basically showing that it's completely wedged in there.
00:07:06
Speaker
Again, what we're going to be doing here is giving a big overview of how this is done, just so you understand perhaps what a long-form proof is and the type of steps taken. And I think this is a remarkable proof because it takes so many steps. And it seems like it almost must take so many steps, especially when you have things like squares that are a little bit detailed compared to a circle, right? Like have the corners and all that.
00:07:32
Speaker
And so, yeah, so what this has the effect of doing is now the only parts of the problem that we need to consider are the angle that the square is at and the size of the circle on the right. And and that makes the problem already a little bit less daunting. Right.
00:07:50
Speaker
Yes, absolutely. In fact, the next step is simply enumerating the different configurations that exist. The paper uses the two points on the square touching the circle and how they are tangent to the circles that they each touch and where the tangents are. In fact, it gives a diagram of every possible orientation.
00:08:10
Speaker
And you can imagine the ways that they're tangent. There's only three ways that the paper I uses for the way that they're tangent. There are points that are tangent to only a side. So this would be if like the circle touches the middle of one of the segments between two corners, right? There's one where it's like to a side and a corner, which is like what we said earlier with the coin rolling off the table example. And then there's just the one where only the corner is touching the circle.
00:08:40
Speaker
That's a very, very subtle difference, but a relevant one nonetheless. Yeah, and if my intuition is right, these authors probably tried a lot of different ways to categorize these orientations before landing on this. This is the kind of thing that you do with a big math proof is you just kind of try different things.
00:09:00
Speaker
Those three categories aren't the only things that we have to consider. We also have to consider the height of the corners. So if they're the same height, that's a different category than if you have the right over left, even if the same types of corners are touching. So you can imagine that as being like the two corners touching the circles versus the square wedged at an angle
Configurations & Mathematical Proof
00:09:22
Speaker
where both the corners are touching the circles.
00:09:24
Speaker
Now I think there are three orientations that we have to consider here, as you said earlier, the orientation in which the two corners are at the same height, the orientation where the right corner is oriented above the left, and then the invert of that, where the left corner is above the right.
00:09:41
Speaker
And if you think about it, a square is symmetric. If you rotate a square 90 degrees, you get the same square, right? Correct. So we only have to consider 90 degrees or pi over two radians of rotation or tau over four. Tau has the utility of being one full rotation versus two pi.
00:09:59
Speaker
Yeah, you start with configuration one, two, and three is what they called it. And I suppose you could start with a few other ones, but this kind of makes it neat. And the configuration one, two, and three is where the square is sitting flat on the line below. But configuration one is where both the top corners are touching the circles. Configuration two is the same thing except for the
00:10:21
Speaker
The left side is touching the circle also on the edge and the corner, so it's a tangent right there at the corner. And then configuration 3 is where it just touched the left side in the upper right-hand corner. And as you rotate the square, you get all these different types of configurations. For example, if you rotate it a little bit from configuration 3 to configuration 4, you still get it touching the side, but now the left corner is below the right corner because it's rotated a little bit.
00:10:45
Speaker
All in all, there are a total of 19 configurations if we follow these rules. And in this paper, they are ordered by angle, and they start with configuration one, then they go to configuration two, and so on and so forth. Yeah, and what's interesting too is that it's kind of a graph, meaning like you go from configuration one to six, two also goes to six, but three goes to four and five before it goes to six.
00:11:09
Speaker
The paper has this on page four, and it's going to be in our Patreon outline as well. Yes, yeah. And actually, I really do enjoy the challenge of having to explain something where somebody has to envision in their mind what we're talking about. Actually, that's a good exercise. I mean, that could almost be breaking math's mission statement. Yes, exactly, exactly. Breaking math, making things hard for ourselves.
00:11:33
Speaker
So basically what you'd show is that there's only two valid configurations and this was done by eliminating configurations a few at a time using some clever geometric methods. All of them are pretty much stuff that could be solved by actually Euclidean math pretty easily. So honestly like the entire problem up to where we're going to be talking about the actual side length, a lot of it can be done by really ancient mathematics and that's pretty cool too. That is such a recent proof of that.
00:12:03
Speaker
Yeah, you know what's interesting is looking at this diagram. I actually thought of a way to solve this creating a program That's sort of like a complete circuit where the light would only be on when all angles are touching You know what I mean, and it just tried through combinatorics every possible arrangement of that square I don't know if that's possible or not, but I don't know just a thought you know
00:12:23
Speaker
Oh yeah, I mean, but then you run into the problem of how do you make the thing rigorous. I mean, I think that's, that'd be definitely good for kind of like an empirical look at it, which I'm sure is what they did at some point too, because I mean, when I've solved problems myself, a lot of times I just plug it into programs and see what everything is kind of like and then go from there. But I mean, there's so many ways to a problem and I've said this on the show before, but solving a math problem can be like trying to get to the
00:12:47
Speaker
store where you end up at your friend's house first, then you end up on Mars. Then you end up in a universe that goes backwards in time before you finally make it to the store. Yeah. Yeah, that's true. That's true. Just looking at these diagrams though, it looks like, like a rolling dice in an alley, you know, with all these. Oh yeah. With the square rotating like that. Yeah. Yeah. So I just thought, you know, would there be some way to, you know, model
Computational Tools & Validity Testing
00:13:12
Speaker
that anyways, not to take away from the paper.
00:13:14
Speaker
Oh yeah, no. And I mean like it could have even been that this was verified by a computer, which a lot of stuff is. But I mean it would honestly be interesting to ask them. So configuration 6 and configuration 19 are the only ones identified as valid. And configuration 19 is only valid when the circles are the same size.
00:13:33
Speaker
So configuration 6 is where the two upper corners of the square touch the circle, and the circle is angled towards the smaller of the two circles. So it's angled to the left. Configuration 19 is the same, but the circle is angled to the right, and it only occurs when the circles are the same size. And if you think about it, they're very, very similar, configuration 6 and configuration 19. And one more important thing is that the sides are not tangent to the circles in these configurations, only the corners. Okay.
00:14:02
Speaker
Now, the next step is to identify what size the minimum solution is. And this solution involves magma.
00:14:15
Speaker
Magma is a computational algebra system Which takes in like algebra and does fun stuff with it sure acronym is it an acronym? I believe let me see we should look that up seems an acronym Nothing, it's actually called magma. Yeah, okay, so apparently it's not an acronym But still that's an awesome name for a mathematical tool magma. Oh Yeah
00:14:40
Speaker
So the minimum side length of the inscribed square is the minimum value of a certain function that was identified. And the function is the square root of...
00:14:58
Speaker
So what does that mean basically? That is the configuration of the square. So we already know the configuration that the square is in and we know that it touches two sides. So this is the formula that describes a square that touches at two corners in the way that it does.
00:15:19
Speaker
And so finding the minimum side length, you have to find the minimum value of that function. And that was done using magma. And it turns out that it's a root of a god-awful polynomial. It's a 10th degree polynomial where you define seven other polynomials.
00:15:42
Speaker
and it's F1 through F7, where it's F4 times quantity F1 times F7 plus F2 times F6 minus 2 times F3 times F5 in quantity squared minus quantity F3 squared times F4 plus F5 squared minus F1 times F4 times F6 minus F2 times F7 in quantity squared.
00:16:07
Speaker
And to define F1 through F7, actually, you know what? You could just find it on the paper, which again is linked in the show notes. But it's a horrible function. So now the question is, is there a closed form radical solution for this minimum side length?
00:16:29
Speaker
Now they actually show the non-existence of a radical solution. And you might think of what is a radical solution? A radical solution is just writing something down in a way that uses square roots and cubed roots and 800th roots and 24.3rd roots, just things like that. And it uses rational numbers and roots.
00:16:52
Speaker
and division and multiplication, addition, all that. And they prove that it doesn't exist. And you might think, why doesn't it exist? Why can't you just take the root of a polynomial?
Conclusion & COVID Context
00:17:02
Speaker
Because you might remember from school the root of a quadratic can be done using the quadratic formulas.
00:17:08
Speaker
The cubic formula actually exists, too, for a cubic equation, and the quartic exists. Once you get to the quintic, there is no general solution, which is shown using Galois theory, which we should do one of these days. It connects group theory and field theory, where group theory is like, it shows symmetries, like, you know, if you rotate a triangle 120 degrees, you get the same triangle, and you can flip it and all that. And field theory, which is like,
00:17:36
Speaker
rational numbers, the field that we're concerned with here is the field of the radicals. And you could show that this polynomial does not reduce using that. But we're not gonna go into that quite because it would require an entire episode on its own. But I think it's just interesting to note that perhaps this is the reason why this wasn't found until recently is because a lot of ancient math relies on constructable numbers in a way that doesn't involve roots.
00:18:07
Speaker
Wow. What a way to spend a COVID quarantine. Yeah. I mean, yeah, totally. Like, uh, I think it's pretty, uh, pretty cool. Um, and again, uh, that was, uh, solved by Jan E. Holly and David Krum. And you can find their, um, paper on archive.org or ARXIV. And the paper's reference number is 2008.00922V1, uh, in, uh, 25th of July, 2020. And again, it'll be on the, um, show notes.
00:18:36
Speaker
All righty. That's been fun. Yeah. And, um, again, uh, you check out our sponsor, the great courses plus. Yeah. It's actually a free access for a month. If you guys go through our leak and actually that's really a great way to help this show out. Oh yeah. And it's pretty cool. Like I've been, I've been using it like, uh, it has been pretty great. Um, and you can do that at the great courses plus.com slash breaking math.
00:19:01
Speaker
Yes, that's a great way to help out this show. You got a whole month for free and there's so many topics on there, everything from cooking to yoga, to history, to astronomy. I mean, what, what a way to educate yourself and have some great ideas for conversations at your next cocktail party when COVID is over. Yeah. You could bypass the not being able to talk to people again because of being in our isolation. Yep. Till next time.