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Love Triangle and Other Maths (with Matt Parker)
3.5k Plays4 months ago
In this episode, Gabriel and Autumn interview mathematician, comedian, and author Matt Parker about his latest book, "Love Triangle." They discuss the unique page numbering system in the book, which is based on the sine function, and how it adds an extra layer of discovery for readers. They also explore the use of triangles and quads in 3D modeling, the concept of Perlin noise, and the perception of randomness. The conversation touches on the intersection of mathematics and creativity, as well as the practical applications of mathematical concepts in various fields. The conversation explores various topics related to mathematics, including the analysis of the Mona Lisa, the use of math in playing pool, the discovery of new shapes, and the application of math in various fields. The speakers discuss the motivation behind exploring these topics and the interplay between math and art. They also provide advice for science and math content creators on YouTube.
Keywords: mathematics, book, Love Triangle, page numbering, sine function, triangles, quads, 3D modeling, Perlin noise, randomness, creativity, practical applications, mathematics, Mona Lisa, parallax, pool, shapes, Fourier analysis, YouTube, physics, AI, machine learning
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Transcript
Introduction to Breaking Math Podcast and Guest
00:00:00
Speaker
Welcome to Breaking Math, a podcast where we explore the fascinating world of mathematics and how it intersects with our everyday lives. I'm Gabrielle Hesh, and joining me as always is my co-host, Autumn Thaniff. Hi, everyone. Today we have a special guest with us, Matt Parker, a mathematician, comedian, and author. You might know him from his popular YouTube channel, Stand Up Maths, or his bestselling books, things that make and do in the fourth dimension and humble pie.
00:00:27
Speaker
not a year to talk about his latest book love triangle how trigonometry shapes the world where he dives into the quirky and unexpected world of geometry relationships and the never-ending quest for perfection in shapes it's a fascinating read that combines humor math and storytelling in a way only matt can We'll be chatting with Matt ah about the inspiration behind Love Triangle, some of the mind bending concepts he explores, and of course, how math can make you fall in love with triangles. So grab your compass and protractor because you're in for a fun and enlightening episode of the Breaking Math Podcast.
Innovative Book Design with Matt Parker
00:01:04
Speaker
like
00:01:12
Speaker
I have one big question. Where did you get your page numbers from? yeah Yes, so for people who are not familiar with the book, the page numbers start on page 0.0000000 and not the first time I've started a book on page zero. My very first book, i I've gradually done more and more ridiculous page numbers and the very first book I had to negotiate to start on page zero instead of page one, but they obliged.
00:01:38
Speaker
ah The page numbers then very gradually go up because they got like six decimal places of precision. They very carefully go up to one and they come all the way down to negative one and then they go back up to zero, pretty much zero by the end of the book. It's because each page you're looking at is the sine function of the page number.
00:01:56
Speaker
as a number of degrees. I did the calculation and I was going through each one and I went, excellent this is off by a smidgen. Great. Yeah, yeah, yeah. Well, I had to stop at some point the number of decimal places. So how how many pages in did you go before you started going, okay, we'll start with these numbers. Oh, I looked at it immediately for the index and I went, okay.
00:02:19
Speaker
What mathic thing? I went sine, cosine, tangent, and I was like, all right, first I went for degrees, then I went radians, and I went, hmm, what is he doing? that you I like to. So, ah like not to not to derail your question immediately, but a little people ask, because I'm famous for doing things online, um'm doing YouTube videos and the like, and they'll be like, well, why do you write a book?
00:02:46
Speaker
And one of the things I like about the book is it's a physical object and it's got page numbers and it's got an index and it's got all this fun book stuff. And so I always like to have to to mess with those given I have the opportunity. And so I want the experience of using the page numbers.
00:03:01
Speaker
to give the reader ah an additional insight emotionally into something in the book. And so in this case, I was like, oh, this is great because you got that discovery of what are these numbers mean, and then you're doing your own little working it out, and then you're like radians or degrees, and then they're non-unique because, you know, it's a sine function, so it goes through the same value more than once. As I really liked,
00:03:28
Speaker
the idea that, as people are reading the book, they're getting all the same upsides and downsides to using a signed function, but in the page numbers. And um ah Riverhead, the publishers of the book, very were very, very supportive of my ridiculous ideas. And I wasn't allowed to make the book exactly 360 pages long, I did ask, but I was allowed to use the signed function of the page numbers.
00:03:51
Speaker
my hey I absolutely loved it. I calculated what it was. thirty It went from 34 to 36. And I think that was the page that threw me off. But I did it the first time and I went, something's awry.
00:04:06
Speaker
I'm going to have to make, because also in there I've got some references about rounding. yeah When you did log tables back in the day, the the the pros and cons of how they're rounded and ways that they were used to to spot people breaking copyright, like copying someone else's log tables. And so I tried to put some of that into the page numbers as well. What I haven't done, I will admit, is gone through and verified that the exact page numbers I sent in are the ones that they put in the final book.
00:04:35
Speaker
And I considered writing some terrible Python code to take the PDF layout file they send me, extract the numbers and check, but I haven't done that. So if anyone finds a mistake in the page numbers, let me know and I will, I'll pass that up the chain of command.
Humor and Easter Eggs in Math Books
00:04:50
Speaker
Okay.
00:04:50
Speaker
but that Before we begin this interview, I told you, Autumn, that I would only interrupt you if I had a really good reason to. And when you guys talk about Easter eggs, such as these ridiculous page numbers where they're they are trig functions, there there's there's one that that my former co-host, the late, great Sophia Baca has wanted to do. Sophia has loved making fun of page numbers where there's a page inserted that says, this page intentionally left blank. What kid does not pick up a book and and pick that up and say, why?
00:05:18
Speaker
Why? It's so good. you're you're You're wasting it. So then, Sophia got into this idea of self-reference and went crazy with it. Sophia would continue to put Twitter or X tweets or posts and Facebook posts where they'd put in parentheses, this post intentionally left blank. And then we did something ridiculous that is not good for our branding. We have an episode of the Breaking Math podcast that you can find to this day called This Episode Intentionally Left Blank. Do you want to know what's in the episode?
00:05:47
Speaker
that Is it like five minutes of silence? No, it's 60 minutes of silence. That's right. I'm sorry. Sorry static. It's static. It's just static. Yeah, but it actually knows playing. Yes. but It's been left blank. That's so good. Yeah. yeah And like, you but like what what is this? And then they created this episode. Yes. We, we, we, and I'm sorry, our listeners, this was just that may have bugged them, but you got to understand, you know, it's, it works.
00:06:13
Speaker
We're making fun of the whole idea. So I love the idea of communication about mathematics where you've got Easter eggs embedded into your own communication. Do you know of any other fancy ways of embedding mathematical ideas as Easter eggs into things?
Intersection of Math, Art, and Fashion
00:06:28
Speaker
Oh, that's great. I mean, yeah, I mean, do you have a favorite autumn? I don't want to come because I have all my favorites are ones I've done and I need to think one that someone else has done.
00:06:38
Speaker
Oh God. No, no. If you want to tell us your own, go ahead. I do a lot of mathematics integrated art or art integrated mathematics. So I would work up very complex origami problems with my students before, yeah and I would break them down into very simplistic ways of folding or talking about applications for concepts. So think about a map. Yep.
00:07:07
Speaker
And this will even go back into paralleling into some of the stuff that you're doing with, uh, in the book. You have your map and you can just fold it into one sheet and you take it apart and have a big piece of paper. How do you do that mathematically? So those are like my little Easter eggs when I'm teaching a lot of complex art concepts outside. There's a name for that folding pattern. I think it begins with M. I can't, don't have the top of my head. That's it. I, um, I was hosting a math fashion show.
00:07:37
Speaker
a couple weeks ago. And so a friend of mine, alien McDonald was like, what are you wearing? What are you going to wear when you're hosting a math fashion show? I was like, I don't know, like my normal outfit. And she's like, no no, no, no, no. So a friend of hers named when when designed me a shirt based on that folding pattern and Fibonacci numbers. So the shirt folds out and then you put it on and it ends the Fibonacci know in mountain valley pattern that in the shirt. That is brilliant. Yes. It was incredible. But also someone else at a conference recently gave me mathematical remittance, which I wasn't intending to show, but they're on there. Like I just was dumping stuff out of my case because I'm on the road at the moment.
00:08:20
Speaker
Look at these, they're like procedurally generated mittens. So for people who aren't watching the video, they've got like lines and zigging backwards and forwards in a seemingly random way and they've been generated algorithmically. And I guarantee, um this was a teacher named Emily, that probably the seed of this is like a phrase in binary or something like that. I'm i'm almost certain there's like a um ah hidden Easter egg and in the starting of this, but I haven't, I haven't decoded it. Okay. Math fashion show. That sounds incredible. Where, where, where did the math fashion show happen? That's a good, yeah, you're right. It didn't just happen, you know, on its own. There's a thing called the bridges math art conference, which is a conference for people who are either artistic mathematicians or mathematical artists. And it's kind of the, here's what Autumn was saying. It's like the the the union.
00:09:16
Speaker
The intersection, no, the union of people of math and art and people who talks about recreational math or interesting things I've come up with in terms of how it can be visualized and artists talk about math that they've had to use in their artworks and I have a huge amount of fun going to the bridges math art.
00:09:34
Speaker
conference. Depending on they they're ah how old they are, we'll have different levels of online ah presence. But I don't know if the conference itself has that much online. but If you search for if you should search for bridges, aren't math, you'll come across it. But I don't think they're putting much out there. They've been going for a while. I want to say early 2000s.
00:10:13
Speaker
I have a conference over here. Awesome. Okay. Yeah. I'm going to look that up after this episode because that sounds incredible.
3D Modeling and Geometry in VFX
00:10:18
Speaker
I want to blow up our social media with that. I just do personally, but I'd have to talk to myself. Oh, it's great. I'm a huge fan of their work and some of the attendees just do incredible stuff.
00:10:29
Speaker
OK, fabulous, fabulous. Now I know, Autumn, that I had interrupted a few times there. Did you have more you wanted to ask before moving forward? Go go right ahead, Gabe. OK, awesome, awesome. OK, so i ah in recent years, I worked in computer modeling. And I learned so much that I didn't know before. And I love that, because when you think you know everything you know like about binary and you know compression and all that, and then you get into a new field like 3D modeling, you learn so much more.
00:10:56
Speaker
ah Specifically, I, how do I say this? I was building 3D models for a certain client that needed vehicles and buildings ah done in anything like Maya or Blender. And then you texture them with anything, you know, like substance painter, any of that stuff. And then just like learning how 3D rendering works and learning how 3D modeling is done with triangles and with ah quads. um I knew this intuitively, but I realized as I'm reading it,
00:11:24
Speaker
I then learn it and consider it in in a different part of my brain. There's like the practical side of my brain and there's the reflective part. And I'm like, oh, you know, you you word it in a way that I knew intuitively. Anyways, I was wondering if for our audience, if you could tell us a little bit about, I want to say it was chapters four and five. I'm hoping to ask you if if you can basically summarize the way you did, how triangles are used in 3D modeling, ah as well as quads and what's yeah in that chapter.
00:11:53
Speaker
I'll give it a... and as a, you know, i'm I'm a lay nerd. And so for me, I get to dive into these areas for the first time. And so researching the book, I just kind of made a note, oh, I know triangles are used a lot, like triangle meshes in video games and VFX and the like. Because if you need to define a surface mathematically, so you can compute it in a video game, it's a bunch of points and you link them up into triangles. I'm like, piece of cake. That that that chapter will write itself.
00:12:21
Speaker
I'm speaking to a friend of mine eugenie von tonsoman who does a lot of VFX for film. and She's like, no, we we don't use triangles. We use quads. They're all quadrilaterals. They've got four corners. and four and i what Why would you use quads when you've got perfectly good triangles that are way easier to calculate right here?
00:12:42
Speaker
and She explained partly it's historical and partly it's still modern practicalities where triangles are just kind of hard to line up and when you're making a model of something in 3D that's got like edges and like features you want the edges of the shapes in your mesh to line up with the edges of the the the shape that you're modeling. And triangles just don't stack neatly, whereas you can have a nice row of quads perfectly neatly all the way around the very edge of something. And in the book, I use the example of a UFO. You can line all the quads up around the edge of the UFO, get a nice crisp edge.
00:13:19
Speaker
Um, but then once I, kind and also historically, it's been easier to generate surfaces using quads because early on when you'd never as much computing power, you can get a quad covered surface by multiplying two curves together and you get like their combined result as, as a surface. And that's always in quads. So I was like, oh no, my, she's like, Jane's just sunk my chapter. But then she's like, oh, but I guess.
00:13:48
Speaker
We don't require them to be planar, so they don't have to be flat. And triangles, and obviously I'm a big fan, you get flatness for free. Any three points, there's a nice flat triangle. Whereas a quadrilateral, you could have two opposite corners really high, and the one's very low, which means you're not going to have a flat quadrilateral. And and Eugenie points out that they don't care if they're flat or not. so actually Every quad in a quad mesh is actually two triangles back to back. So my chapter was saved. They are triangle meshes, but they're triangle meshes such that all the triangles are in pairs that form quads that then form up um the larger mesh. And so that that was my my journey of discovery as a tourist in the world of of mathematical 3D modeling.
00:14:35
Speaker
Nice, nice. And actually, what I read about, it was a problem that we exactly had. We were trying to model um a certain vehicle that has, and again, I'm being vague on purpose because I don't know about my and NDA and what I'm allowed to disclose here. We had a certain vehicle that had round parts and and and and sharp edges.
00:14:51
Speaker
And when you talk about a planar thing, think about, you know, a computer is trying to model something. The computer will simulate light and reflectivity. And when you talk about a planar surface, you've got, of course, as you said, four points that define a quad or a square. ah It knows exactly what to do about how to render the light when you've got a flat surface. But you take one of those points and you move it up and down so it's no longer in the same flat sphere.
00:15:18
Speaker
It has no idea what to do and what it and ends up doing is some weird approximation based on its parameters where like if you tilt the model suddenly it'll start flickering in that one spot or or the light will change also. It has no idea. So that's why when you're modeling something with a computer you you've got to be very very specific and and think about you know ah all all all of your vertices, all of your points, they've got to be in a flat surface here. But again, as you said earlier, sometimes you don't need ah them to be plainer. Sometimes you do. For my purposes, I absolutely did. And one other fascinating thing, I've used this in job interviews. They say, you know teach us something in 30 minutes. And and you know just off the top of your head, I always talk about computer um modeling. Have you heard about how they simulate ah smooth surfaces when you've got polygon, um jagged edges, um how they simulate a smooth curve? No. Is that like a ah bezel curve or something like that?
00:16:13
Speaker
Uh, I've used so that I don't want to say no, because I've used that term before. But like, if you have, let's just say that you have, ah you know, ah a low polygon count of something where you've got like a hexagon that is that's supposed to approximate something that should be a cylinder. um What you do is you you do something, it's a very simple operation, I could just say you're right click and you say shade smooth. But what it does is when the it computes,
00:16:38
Speaker
The computer knows how the light is supposed to bounce off every edge. What it does is it takes an average of the directions, and it has a simulation where where it where it smooths it artificially. where yeah It's still built with only polygons, but it just takes the directions, and it and it has an an algorithm where it where where it smooths it you know um per reflection point. So it it it it it fudges it, sort of. it like It's like blurring your eyes, sort of, if that makes sense. And it's- as you say you can just right click and pick thing yeah and a lot of students when they're studying math at school are like I don't need to know the math the computer does it for you but if you don't understand the math when you got your your vehicle model and as it tilts a bits flickering if you don't understand oh that's because there's some rounding in the approximation
00:17:28
Speaker
to to to simulate it as if it was flat. If you don't understand some of the math under the hood, it limits your ability to use the software. And I think that's such an important point. That it's not just a case of the computer does it for you. yes And you're just free to be artistic. It's still beneficial to understand the math.
00:17:47
Speaker
Yeah, it was also fun. and all This is a really, really quick one. There's something that that's called um either a normal map or or a bump map or or a height map where yeah if you imagine, you know in a 3D world where you make everything out of polygons, imagine if you've got like a a cannon blast in a wall, it would cause so many polygons to map that. It's ridiculous. They they they have a way of approximating that as well, where you make one really high polygon count thing and you can do it with a tool called ZBrush where where where every single surface is modeled using polygons and the amount of memory is ridiculous. However, you can take the information of ah the light bouncing off all those millions and millions of surfaces, you can bake it into something called a normal map.
00:18:38
Speaker
where just the information of the light bouncing off of it and not the information of how all those surfaces interact in 3D, but just how the light bounces alone.
Simulating Nature with Noise in Modeling
00:18:50
Speaker
You can take that and then map it onto a flat surface or even a surface with just like a couple of polygons.
00:18:57
Speaker
And the light behaves as it did with the high polygon surface, even though the actual behavior of the model ah is very, very low poly. that That's how they they do normal maps and bump maps, and you approximate something so so that it looks gorgeous, even though it's very low poly. Does that make sense? That's great. Yeah, you kind of you do the big calculation once.
00:19:17
Speaker
to get the answer for how the light's behaving, and then you can just remember that answer, yeah and not have to redo the calculation, and then apply it to all your low-poly versions. That's really nice. Yeah, yeah, yeah. I don't know. I personally enjoy learning about about how how approximations and shortcuts are done for the purpose of 3D modeling when you need something to look pretty. and Again, this is just for looking pretty, not for behaving accurately.
00:19:40
Speaker
that too entirely different. So physics and engineering modeling is different than visual or pretty modeling. And then the in this chapter, the other thing that I learned is the use of randomness in modeling and in computer generated thing. and And I learned about Perlin noise and about how you know you can map things to grids. I was hoping that you could summarize a little bit of that for our listeners. I think they'd enjoy it very much.
00:20:06
Speaker
Yeah, Perlin noise is the the the random noise. And by noise, we don't mean sound in this case. It's not like a podcast left intentionally blank. It's like visual. It's like the static for people from the past. If you remember a TV, it's like the ah visual noise. Someone named Perlin was working on the original Tron film and they got upset at how terrible the noise was.
00:20:31
Speaker
And because if you're rendering things in a computer, it can look very computery. And for a long time, VFX looked very VFX because it's a bit too clean, a bit too perfect. Whereas in reality, everything's got a patina, it's got some damage, it's got texture.
00:20:46
Speaker
And not only that, but if you're, so that's one thing, you want to apply the random noise to your surface to make it look real. But also if you're, like you're saying, you don't want to model a million polygons at once to generate a whole forest. Like you're like, I can't render every, I can't plan every leaf out. You can use the random noise to randomly decide when you're growing ah leaves algorithmically. Like you can just, you know, procedurally generate a forest. And pearllin wanted to come up with a nice mathematical way to do that. What they did ended up winning an Academy Award just for the noise, which I think is incredible. And um there's two versions. There's Perlin noise and there's Simplex noise. Both of them start with a grid. You assign random numbers to all the corners, all the crossing points in the grid. And then as you move a point around through the grid, you've got a way to calculate a new value for every in-between point.
00:21:42
Speaker
based on the distance to all the nearby corners and you use the distance to them in the value of the corner, you smoosh it all together in a big old function to give you an output. What's very nice about it is it's quick to calculate and as you move around in the grid,
00:21:58
Speaker
The influence from all the different corners kind of ebbs and flows like as you get closer to a corner, its impact increases. And then as you go away from it, it decreases very smoothly. And so you get this nice smooth noise. If you did random numbers, it's very jarring and it's not very natural.
00:22:15
Speaker
Smooth noise is very natural. And the difference going from Perlin noise to simplex noise is the opposite of what we've just been discussing. Perlin noise was based on a square grid. Simplex noise is based on a triangle grid. And the reason that's helpful is you need noise in different numbers of dimensions.
00:22:36
Speaker
And that's partly because you might be modeling something in more dimensions. And it's partly because as soon as you start moving something, you and you want the noise to change with it or to control the movement, you've now got time. So you might want four-dimensional noise or even five-dimensional noise. So you've got multiple types of noise and paths you can move through. it And it's absolutely fascinating. And in higher dimensions,
00:23:01
Speaker
triangles that become tetrahedron that become generalized tetrahedron have far fewer corners and are quicker to calculate than cubes because the square becomes a cube becomes a 40 tesseract a hypercube but and each time you go up a dimension With a cube, you double your number of corners, which stacks up very quickly. It's exponential. Whereas with triangles, you go up a dimension, you get one more corner every time. It's nice and neat and linear. And so because we suddenly needed higher dimensional random noise for video games and VFX and all these things, we switched to triangles because they behave better in higher dimensions. I love that a knowledge of higher dimensional shapes and meshes and lattices
00:23:46
Speaker
informs practical visual effects, well not practical effects, but practical application of digital effects. um ah I find that so fascinating that we need to understand higher dimensional objects to make movies.
00:23:58
Speaker
Yeah, no, no, that's awesome. Or even movies, but also, generally speaking, just storage. Like, if you need to store enough, you know, if you need to compress data and store it, because, you know, storage fills up fast, just knowing that, that you know, as you said earlier, you're a four-dimensional, I'm sorry, you take a triangle and you make it in 3D, it is the tetrahedron.
00:24:19
Speaker
That part is just astonishing to me. how How much less data, you know, a four-dimensional and five-dimensional triangle based shape takes than than a cube or even something else. And then how you use that to store information. That's huge. That's that's huge. and then This chapter got me thinking about even like like forensics. Now, I realize this is a little bit of peripheral ah from from just talking about um triangles. But but the the approximations of noise, I was astonished at how you know a grid is not noise. A grid is very ordered and repetitive, but you can still play with it and and and assign it different things where you have an approximation of noise. It made me think, okay, So if you wanted to like really use some forensics and figure out if something is like truly random as in nature, or if it's random, it made me think what would be the difference between a very convincingly random or a very convincing, uh, purlin and noise that's not truly random and an actual random thing. I don't know. I'm curious though. Now, you know, that's interesting. I imagine there would be a way of doing it there. Um, I've met forensic accountants before and part of their job is.
00:25:27
Speaker
teasing apart accounts to see if they have the right amount of randomness that you would get from organic expenses, costs, etc. Versus cooked books like artificial accounts, yeah where someone's making up random costs. That randomness is very different to natural randomness. And I imagine if there was someone doing like a video analysis of like, is this very convincing AI or is this a real video potentially? And this is obviously us just speculating.
00:25:57
Speaker
yeah potentially what you could do is try and analyze the noise in the video on the surfaces. Was this like mathematically generated nice neat simplex noise or is this like actual natural noise? And I haven't come across anyone doing that, but I i would say there's a ah decent chance that's possible and will be used before. I got a mind a mind blower here for you. This is something that is pretty well known at this point.
True Randomness vs. Perceived Randomness
00:26:24
Speaker
But what what really from a from a psychological perspective, the perception of randomness is also very different than true randomness. I'm sure you know about the original iPod and the complaints about the iPod shuffle. Do you know that story?
00:26:38
Speaker
Yeah, the original one was properly random, and people were complaining. They were hearing the same song again too soon. Yeah, yeah, because it's properly random. It's not the iPod. I mean, shuffle would be different, I guess. It wasn't the iPod random.
00:26:53
Speaker
so so So, what's funny is less random actually has the appearance of more random and there's there's you know reasons for that and it gets into the psychology of expectation and things like that. So, yeah, I don't know. All all of this just fascinates me. um And then the last thing I'll say, and I thank you for the for the time, Autumn, after this. I'm going to toss it to you ah for for a while. um Speaking of using forensics to to to determine frauds. from the you know real thing. My wife showed me a a show on Netflix. It was either a year ago or two years ago, I forget. It was about the fraud of the in in in Mormon history in the United States. um that There was a book, um oh gosh, it was involving a a salamander, a white salamander.
00:27:38
Speaker
And it turns out that there was a ah trader of Mormon artifacts this last century who was um extremely smart, complete fraud, but but his IQ was off the charts. And he would create these fake books, but then he would artificially age them using a ah like like a fish tank and electrolysis and humidity, where he would give it the appearance And you know for all intents and purposes, actual people who know how to age things are like, oh yeah, the exposure to the air, this only you know this kind of um entropy only happens after it's aged. It took new science, brand new science, brand new understanding of aging to prove that he was a fraud. Did you ever watch that show? No, but that's that's phenomenal. i guess the the the
00:28:26
Speaker
financial motivations for a good fraud, yeah ah pushing forward the barriers that, you know, new science to do good fraud needs new science to unpack it. But then I did, um, it's not quite that it wasn't a deliberate fraud. yeah I came across a painting, a copy of the Mona Lisa, um, when I was writing
Historical Forensics in Art and Trigonometry
00:28:44
Speaker
my book.
00:28:44
Speaker
And for a long time, everyone knew it was like it wass just a copy. It's just a copy of the Mona Lisa. Obviously, it's not worth as much by a long shot because it's just someone copied it. And then when they were cleaning it, they realized, hang on a second, and it's not exactly the same.
00:28:58
Speaker
Everything was subtly different, and they realized by using the mathematics of parallax, where you can analyze the viewpoint and how things in the foreground, as you move your viewpoint around, they change and shift different to things in the background. They were able to analyze the parallax in this copy of the Mona Lisa and realize it was painted by someone else.
00:29:19
Speaker
at the same time as Da Vinci to the side and a couple, I think a meter or two closer, they could pinpoint the exact location in the room. This other painter must have been looking at the same model painting at the same time. I think it's incredible by using math and, you know, geometry. We could determine this isn't a copy. It was one of Da Vinci's students in the same room painting concurrently with them, which is just phenomenal.
00:29:45
Speaker
Wow, and the motivation for that too. like you can like With the dude with with the the Mormon book, I imagine so many folks would just give up and be like, yep, it's real. The scientist guy said so. Yep, it's real. But then somebody else had to try themselves to say, is it even possible to to fra to make this up? And they I think they analyzed the the the statistical distribution of of the entropy. And they saw that when you do something quickly, the entropy happens in a different pattern than when you do something slowly over many years. But that wasn't known. He had to literally try to assume it could be done, try to do it, then under a microscope, and like, that motivation, like, he was driven, you know what I mean? Like, who would just give up and be like, oh, it's real, you know, like, I don't know, man. so And that's great because, you know, everyone else, if you you you make the fraud or you own the fraud or you want to sell the fraud to buy it, everyone's motivated to just come on, let's not.
00:30:43
Speaker
It's real, it's real. Who's asking more questions? The scientist said it's real. But to have that curiosity that, no, how could it be done and can I work it out myself and can I detect it? That's, yeah. I find that fascinating just thinking about We have the calculation, sometimes we put it out and we think that there's all these additional variables. You know, you have to think of things in like a closed ah or a black box, right? um One example that I know that you were talking about a little more in the book was about playing pool.
00:31:19
Speaker
Yes, right. So I know that you've gone in depth about that. But when you're playing pool and looking at um triangles and the trigonometry behind that versus the application in real life.
00:31:38
Speaker
How does that exactly work now? i I've seen the behind the scenes on a couple of your videos with that and was fascinated by it because I love playing pool myself.
00:31:51
Speaker
Ah, excellent. um Because I'm a late pool player. I love playing it. And that I always find it very pleasing that it it seems to be largely angles. Angles, trajectories, geometry. I'm like, I love it. I can do this. And so we decided to put that to the test. So I joined forces with another mathematical friend of mine, guy called Grant Sanderson, who does the 3Blue1Brown YouTube channel. if people Are people familiar with that?
00:32:13
Speaker
um Grant's great, also adequate at pool, like me. We made a good team. And then we we challenged a pool YouTuber, someone called Rollie Williams, who also does a fantastic climate change channel. Just, it was worth mentioning. I'm climate town, which is amazing. And a professional pool player, Jennifer Beretta.
00:32:31
Speaker
We played them. they they We didn't play a game. They set up one of their challenges because when they're training to be better at pool, they do these little practice shots over and over and over again, little training exercises. And they set up one for us where the goal is you've got to kind of bounce the cue ball off two of the cushions or the rails and then you've got to hit another ball on the far side of the table.
00:32:52
Speaker
And so Grant and I got out our measuring tape. We measured the distance from the cue ball to all the rails, all the cushions, and then we worked out exactly where it's got to end up. And then we put our cues down and we went and drew some diagrams and we approximated it as if it was like a particle of light bouncing off a mirror, like a perfect angle in equals angle out.
00:33:14
Speaker
situation, which, spoiler, is not. um And then we worked out exactly what we had to do. And I think the fantastic point you're making, Autumn, is this generalizes where it's all well and good to simplify a system to a point where it's easy to calculate, which is what we had done. But then you got to remember, hang on, I've probably thrown out a lot of nuance, which might be fine, or I might have thrown out something important.
00:33:42
Speaker
And any mathematical model, you've got to pay very close attention to what you're assuming, what you're ignoring, all those sorts of things. And we had ignored a couple of things. The friction on the table, for a start, but that's kind of fine. ah The friction of the cushions, now that's a problem. Because as a ball hits a cushion, and it gains spin. And we'd ignored the spin.
00:34:04
Speaker
And so we had a point particle bouncing perfectly as an actual ball with momentum and rotation and it hits the thing. It starts to spin and once the ball spinning. So the first collision we were pretty good because it came in with no spin bounced off the cushion. Great. But now it's gained spin. The next cushion it hits, that spin changes the angle it comes off at.
00:34:25
Speaker
and And so it opens up the angle. So because it's it's spinning kind of into the cushion, it makes it come off on a more shallow angle. And so our shot was way off because we didn't factor in the spin. And what was very interesting was talking to um Jennifer Barrett, the pro player, about what professionals do when they're playing pool.
00:34:43
Speaker
She was going, well, they'll often look at it ah a distant point in the room that they aim for. and They were using the same techniques we were using. They we like they were imagining the bounces off the cushions as if is Instead of light reflecting, and there's a whole mirror table that it's going into, and a mirror reflection, and another mirror reflection. And they're using those same mathematical techniques, but in a but without realizing that's what they're doing, which is great. like I don't know if someone came up with them originally mathematically or just they've evolved.
00:35:14
Speaker
People playing pool, but they're very math-based techniques under the hood, but they've got a layer on top of that where they understand the physics of how the table behaves and how even the humidity in the room changes the friction on the felt. It's phenomenal. And ah Jennifer showed us some some calibration
New Discoveries and Practical Applications in Math
00:35:35
Speaker
shots she does. This isn't in the book or or the video or anything else, by the way. This is behind the scenes. She's got a bunch of calibration shots.
00:35:41
Speaker
So when she's got to play in a new pool hall or a new competition, they get a bit of time on the table first, and she would do these set shots and look at how the ball responds, and she can then use that to calibrate exactly how much friction and spin and everything there is on this specific table and and cushions. and And then she can then play the game using her normal techniques, but she knows how to compensate for this specific physical reality of the table, which I found absolutely fascinating.
00:36:10
Speaker
i I actually find that to be fascinating as well and just seeing how we take all of these ah problems within physics and then we translate that from the math and you know, just seeing how that correlates to what you have in the book and how the book covers some of the interesting mathematical curiosities, right? So we have everything from the discovery of new shapes and polygons. And I'm a bit curious on what you found to be one of the most intriguing mathematical problems that you explored when going through writing the book.
00:36:56
Speaker
Yeah, that's a good question. And I would say I got distracted looking at where, how shapes are discovered, but it's like a passion of mine is like a lot of people kind of think that math was done by the Greeks and then we haven't really touched it. We just learned it now and like math in the past. And I really want to emphasize that it's an ongoing subject. We're still discovering new shapes to this day and we're still learning new math and coming up with new techniques.
00:37:23
Speaker
And I accidentally came across, I forget why I was researching like something thing about shapes with like eight corners or something and I came across a paper written in 1962 by someone called Donald Grace who was trying to find the biggest shape and by the biggest shape I mean if you had a shape with let's say eight vertices so you got eight corners and you can define it within a sphere just because you can imagine scaling it to be as big as you want but we want to say what's the most efficient shape if you can arrange eight corners on a sphere to form a polyhedron inside so what's the biggest eight corner shape inba basically
00:38:02
Speaker
And it's not a cube. Cube has eight corners and you think, well, that's pretty good. And actually a cube is the way that you can distribute the corners so that they're the greatest distance from each other. Like a cube is the most even distribution of the corners. But that doesn't work. That doesn't give you the biggest volume. And what Donald Grace did was fire up a computer and use some techniques at the time that were completely novel, but now they're very standard computer search techniques.
00:38:31
Speaker
to try and get a computer to search for the arrangement of eight corners such that you'd get the greatest volume possible. And they discovered a new shape. They discovered a shape that's got a bigger volume than any other shape with eight corners. And it was later proven that that that's the optimal. And there were two things I liked about that. And one is that some of these problems are still unsolved. There's always new shapes we don't know about. And even that that question of arranging the corners so that they're the greatest distance from each other. We don't have a way to solve that in general. Like, you couldn't give a mathematician, you're like, I want, I don't know, 50, 53 dots on the sphere as spaced as possible. They're like, I don't know, good luck. I can probably get a computer to try it. um But that's an unsolved problem. We can do it.
00:39:16
Speaker
if they match a platonic solid, like a cube or an octahedron or something. But not in general. And despite all the funding from Big Golf Ball, we've not found a way to arrange dots on a sphere, which I find phenomenal. um But also what I found incredible is I think this shape that Donald Grace found in 1962 using a computer, using the Burroughs 220 for retro computer fans,
00:39:41
Speaker
Is the first shape ever discovered by a computer and now modern like practical applied math industrial math. It's like using a computer is the same as breathing like everything's done on a computer you computer searches alike the bread and butter for industrial math research.
00:40:00
Speaker
Whereas Donald was just curious about if a ah certain shape existed. And as far as I'm aware, and I'd be very keen to hear if there's any prior examples, this was the first time someone ever then programmed a computer to search and find a solution to produce a shape, which I think was such a seminal moment moment in shape hunting in the human millennia long mission to find and document new shapes.
00:40:27
Speaker
Absolutely. Especially when you're looking at the advances of technology and how from then to now, if you're looking at the mapping of that and just seeing how that evolution goes for computer graphics and programming. And normally we take that triangulation into everything that we're doing from movement, right? So.
00:40:54
Speaker
When you're trying to program something just as natural as hair all the way to ah simulations of soft tissue robotic surgery with finite element analysis, you have to consider all of that programming and simulation bottling every single time.
00:41:19
Speaker
Yeah. And there's, it it's like we were saying before, there's a lot of interplay between that and art. And as you were mentioning, obviously you mentioned the finite um element analysis that's, you know, in engineering, that's just such an incredibly valuable tool, but it's not dissimilar to the meshes and things that are used in VFX. And so something we're using for visual rendering, it's not wildly different to practical.
00:41:43
Speaker
So when you're doing a lot of that for the VFX all the way to ah the finite element analysis, that's actually the same simulation software behind the scenes a lot of the time. And a lot of the people who do those graphics are the same people who are doing the video game graphics. Yeah. And and a lot of people are like.
00:42:08
Speaker
A lot of math people are like, well, what's the point of that? Why are you doing that? And it's very similar to art in that regard. It's because it's beautiful. It's wonderful. You want to do it. And actually at the Math Art Conference I was at recently,
00:42:20
Speaker
It's a group called Nervous System. They do some phenomenal generative um jigsaw puzzles and other phenomenal stuff. and they And this is going to be my terrible paraphrasing of what they do. ah ah Check out their work if you're at all curious about this. But they were using ways to ah procedurally generate. So like you you give the computer, like, generate something that kind of matches these requirements, and the computer would do it. And they're using something called Voronoi diagrams which is where you start with random points and then each point gets the region around it which is all the points closer to it than any other point and you get these very organic looking structures and they were doing that to like generate like a skirt like the mesh for a skirt by generating that using very organic looking results from very computer algorithms they were then involved in
00:43:11
Speaker
using very similar techniques that they developed for things like making skirts and fashion and and sculpture. They were then employed to help model human organs by using the same techniques to like, how would you build a lung from the ground up? And it's very it's a very similar kind of fractal organic pattern. And so the same techniques they developed for for artistic reasons, would they were then applying very practical ways to um generate and make artificial ah simulations and models of um human organs. And so I think it's, again, it's very hard to draw a line to say this math is useful, this math is pointless and just used for art or for curiosity's sake, because it'll it all bleeds from one to the other.
00:43:59
Speaker
I just thought us of the when you were talking about ver Vern, I'm sorry, say I can't say it right. I think it's Voronoi, but my pronunciation is famously flexible. I have seen that term on my Blender software. It's one of the procedural things you can drop in. And while you were talking about that, briefly in my head, this isn't probably not even accurate, but I thought that's seat that sounds like the simplex algorithm you described earlier for noise, but kind of in reverse, sort of, not not really. But then I thought, when you take a simplex noise, you start off with with the grid.
00:44:29
Speaker
how How doable, I wonder, would it be to do that process in reverse during forensics? That is, take a data set that is you think ah appears random and could you do like the simplex algorithm in reverse and just see if you could wind up at a grid and use that method to compare um simplex noise to organic natural noise or noise that are like a noisy surface that you would see in nature, then that brought up the further question of whether there really truly is actual randomness or whether everything is dependent upon some other ordered, I don't know, ordered source. Long set of questions there. so I'm going to park that second question for a second, but I think ah the first, they yeah, absolutely. I mean, and then again, this is me speculating off the top of my head. If you took a random, like a surface with like random noise on it.
00:45:20
Speaker
you could like pick all the maximum, all the local maximums for whatever the the noise, like the darkness of the pixels or something. And a Voronoi, there's probably better techniques, but it's not a bad way to imagine how you could reverse that. So a Voronoi would be to take all the maximum points, and then you draw a region around each one.
00:45:40
Speaker
which is all the all the pixels closest to that maximum compared to any other maximum and you might start to recreate an original mesh or grid that was used to generate the noise of course with the simplex noise or parallel noise They're scaling because of the values. And but that's also what nervous system were doing. They would then apply. So instead of having a real um homogenous, I really like the same in every direction situation. They could scale it in different directions or add different densities or distort it to get.
00:46:15
Speaker
um not just a pure random Voronoi looking series of cells mesh, but one that's got certain properties in different directions or bunches up in in different regions. So I imagine there might be similar techniques to reverse that. But you're right, I think, you know, if you look at random noise, potentially doing something like that, much if it reveals underlying structure, that might indicate it was artificially generated. But then again, I don't know enough about true naturally generated noise as opposed to just pure randomness, what that would display. It'd be like the book again. You'd have to analyze both cases, spot if there's any intrinsic differences, and then reverse engineer the math as to why that might be happening. yeah So, out of all of the topics within the book, ah what did you find to be the most surprising thing
00:47:11
Speaker
Oh my goodness, the most surprising thing across the board. Yes. That's interesting. what's the try I mean, the the the meshes the meshes not being triangle surprised me. That was startling.
00:47:24
Speaker
um Or something that you significantly left out and wanted to talk about. Oh my God. Well, that's, that's a very long list. I mean, the most surprising thing was that when I came up with the idea for the book, I actually pitched this book idea, a book about trigonometry about 10 years ago, when I was a very new author. It was like but my first ever book proposal.
00:47:49
Speaker
And I was like, oh, I think i think you know there's not any kind of real popular math, like like real not like a textbook, like a real general book for anyone about trigonometry. And I put together a proposal and my publishers, so I published with Penguin Random House, they were like, no we're not convinced. And then 10 years later, I'm like, you know what, I still think there's there's a book about trigonometry out there.
00:48:11
Speaker
I didn't know if the gap in the market was like an actual gap that needed filling or just it was a terrible idea and that's why no one had done it. But I was like, I'm going to do it. And I was surprised that I had.
00:48:24
Speaker
I was spoilt for choice. I was kind of wondering, because I had like a rough idea what the book was going to look like. I mean, still you start writing it and you're like, oh, I haven't got enough chapters or what am I going to put in there? I thought this chapter would be easy and I can't find enough examples. I had an embarrassment of riches. I was like, oh my goodness, there were so many amazing stories about geometry and trigonometry and things being used out there. The the thing that kind of got cut was when I realized I had so much other geometry and trigonometry stuff to talk about. I had a whole section at the end about Fourier analysis, which is decomposing signals and and and data into individual sine waves. And because I was talking about sine functions and the page numbers, or ah the page numbers are tackling a sine wave, which I find very pleasing. It's like the world's worst flick book.
00:49:12
Speaker
um that I want to talk more about now what we can do now we understand sine waves and that was the bit that then got a bit compressed partly because it was the bit that was least Perfect for writing, putting in a book because some things lend themselves to being bookified and some don't. And talking about sound and music and sound, what waves and images and stuff that was less likely. And so the one, so I was surprised that there was just so many things I could put in the book and the one bit I'm kind of sad didn't make it into the book.
00:49:46
Speaker
is I was writing a whole thing and looking into using waves and Fourier analysis to compress images. So like JPEG compression is, and I thought it was so amazing that an image is compressed using what we consider sound waves, but they're waves, they're just waves. and They don't care about sound.
00:50:08
Speaker
And so um that, I think, I've written a whole thing about it, but it was one thing too many for the book. It got cut like in the early drafts. I was very sad about that. um ah But probably the most surprising thing to ka turn change text slightly is that ocean waves are not sine waves. And so I've got the list of things that are waves. It's like sound, that's a wave, like electron.
00:50:33
Speaker
like quantum physics, that's, they're all sine waves and like light sine wave, everything's a sine wave. And then suddenly ocean waves are like, nope, not not sine waves. They are a different type of wave. And I was like, well, that's good. And I mentioned this in the book. I'm like, I'm glad there's at least one wave that's not a sine wave. It's fundamentally a different wave.
00:50:54
Speaker
Definitely.
Engaging Storytelling in Math Education
00:50:56
Speaker
The next book is going to be all about waves, right? that should be It really should be. here As an electrical engineer, I completely promote that. And also along the lines of what you're talking about with compression and with FOIA analysis, I want to go ahead and plug another creator. I don't know if you're aware of um art of the problem. He focuses on um um ah mathematics in electrical engineering and computer engineering with a lot of machine learning. One of his older videos is in collaboration with IEEE, and he he talked about a type of compression where the decoder is wrapped up in the the code packet along with the message itself, and it just like unfolds and decodes itself. It's amazing. i mean highly recommend the channel. yeah
00:51:37
Speaker
Oh, you you've seen it before. Yes. Yes. Oh, fabulous. He's a great guy. He's a great guy. Yeah, he was our first interview for this season. Actually, we talked about the mathematics behind chat GPT for as well as the measure.
00:51:51
Speaker
um There's a book on a brief history of intelligence. Great chat. Great chat. So yeah. um And until Mark comes out here and kicks me out, I'm just going to keep asking questions. Do you have any recommendations for science and math content creators ah and teachers who want to go on YouTube and make it? Oh, with the to to create or to absorb? To be successful. Like if you're a creator, like ah yeah what's the magic? That's a tough one. Yeah. the the I mean, I love.
00:52:17
Speaker
I absolutely love YouTube. I love creating videos. But I came out of, I was a high school math teacher for several years. And so, but also at university, I was into making short films. And I'm old enough. I remember when YouTube started and I was like, oh my goodness, this is perfect. Like the downside to making video before that was there was just no way to share it. I mean, there was rudimentary stuff online. I was like, this is great. And I can combine my passions of making videos with talking about math.
00:52:48
Speaker
The, just practically, the problem with YouTube is at the beginning it you you It's a lost leader, like you got to do it as a hobby to start with because I started doing YouTube in earnest and it took four years before it was paying for itself in terms of my time and effort being put into it. And I'm very fortunate, I had the privilege to spend four years doing that because my wife is a salaried physics professor. So I can i can do that, right? and And not a lot of people can just, you know, down tools doing other work and do something.
00:53:22
Speaker
so I would say my one bit of advice is you've got to enjoy doing it and it's sad to say it's got to start as a hobby. because it's not going to pay its way for a long time. And um that's unfortunate. Also, at the beginning, you're you're a one human band because you've got a light and edit and sound and everything else. um And so my vi because I'm in love editing, but I'm not good at sound, but sound is key. Like visuals, you can get away with terrible visuals on YouTube as long as people can hear you clearly. So I had to lean on some friends. Yeah, autumn is exactly what this is like. People will tolerate awful visuals that keep watching.
00:53:59
Speaker
But if the sound is hard to follow, they're gone. So I got advice from some friends and I just found it kind of standard plug and play. I do the same thing every time and it's good enough technique. But then I think philosophically in general, the reason why we do it is I feel like there's so many wonderful things in math. These wonderful moments when you discover something or you learn something new. And just the beauty of the patterns and the logic behind mathematics is incredible. But it takes a lot of background learning to reach some of those points. And not everyone has the time, energy, inclination ah to do to do all that work to achieve that at the end. So I always think of being a math communicator or or making content online about mathematics.
00:54:45
Speaker
is providing a shortcut. So I'm like, what is it about this bit of math that that inspires me or I find so pleasing? And what's the shortest path from where most people are to to to a rough approximation of that sensation? And so that's my goal, as always. How can I give people an insight into all the things that I love about math without them having to spend years getting a degree or studying it recreationally or whatever the case may be. And so that that's that's always my kind of um guiding principle. And in general, as a litmus test...
00:55:20
Speaker
I am my own audience. No, not exactly, but to a sufficient approximation.
Creating and Simplifying Creative Math Content
00:55:25
Speaker
And so I always think, what would I like to know? What did I enjoy the most about investigating this topic? And then I make the video I would have wanted to have watched before I started on researching it myself. And so that's, and it's not guaranteed to work. I just got very fortunate. It's a survivor bias. You know, there's a thousand of me who started out making YouTube videos.
00:55:46
Speaker
And I just got lucky that the type of stuff I enjoy making and and consuming myself, other people do as well. So I also take all my advice with a grain of survival by assault.
00:55:59
Speaker
I have ah i have one one one last thing I was hoping I could share. Autumn, I hope it's cool with you. I wanted i wanted to share our our idea of um how we're gonna make it big on on TikTok with TikTok shirts or whatever. I've wanted to do this animation where I'm sure that we all know the cultural impact of things like Dragon Ball Z, ah that are just these ridiculous cartoons or any other like like like fighting cartoon like or or game like Street Fighter II. What I've wanted to do is have a series of problems, math problems that look very, very difficult at first glance.
00:56:28
Speaker
but are actually almost trivially easy as long as you know what to do. And I wanted to have something show up like, ah you know, in Pokemon how it's like a wild whatever has appeared. Or in Dragon Ball Z, some threatening monster appears that looks all scary. And then, you know, the hero knows what to do. Like, oh, if all you do, you know, first of all is, you know, um multiply it by E or, or you know, like, ah you know, but multiply by this coefficient or do this one one one little thing. Or L'Hopital's rule, for example, it suddenly becomes much like less threatening. I want to have some kind of a, you know what I mean? like It's like Street Fighter where, you know, a problem appears, but if you know the special move, it's like up, up, down, down, left, right, L'Hopital's rule. Yeah. And yeah yeah, good. Love it. I'm on board.
00:57:14
Speaker
He throws that fireball and he says like, how you can, or whatever. yeah But like, uh, instead, low beat towels, rule, low beat towels, rule. factorize Yeah. Yeah. Just do something. And that comes from my own teaching experience. I was a big storyteller and I would have an assignment for my kids. I'll say, write me the hardest math problem that you can think of. And if they're like in fourth grade, they'll be like, what's a million plus a thousand plus, you know what I mean? And like, think it's some impossible thing. And then you learn that's just a trivial, repeated,
00:57:43
Speaker
thing, and then you learn a new concept. we we We did that, and we got as far as exponents. What is the scariest exponent that you can write? And then we would, you know, we'd come up with one, and then we'd show them the magic trick to to dis solve it. I just want to do that with animation, and I want to see somebody go, some student go super cyan, you know what I mean? and and then And then solve it using this trivial thing, and like, oh, it's that easy. So when I have all the time to animate, I'll have time for that. I just don't have time for it. So I know that feeling.
00:58:12
Speaker
Yeah, yeah. So I think that's it. But um it has been an absolute pleasure and i the book was very, very enjoyable. And I'd like to give you the last words, Autumn, since I took more than my share.
00:58:25
Speaker
Oh, I'd just like to actually thank you for coming on the show today. Oh, my pleasure. I have one big ending question, small but big. Okay. um What is the number one thing that you want people to know about the book, about mathematics, and just take away from it? Yeah.
00:58:46
Speaker
No, I think that small thing was my original motivation to do a book about trigonometry, which is I think it's shame that a lot of people think math is just what they were forced to do at school. And the reason I picked trigonometry and things like Pythagoras is I think that's the most advanced math that most people were forced to learn at some point. And so it's become this kind of mascot for math is hard and pointless and awful. And isn't it great? We can avoid it now. We're not at school. I think that's a real shame. So my one tiny big thing from the book is to reevaluate for people to reevaluate that what they thought math
00:59:24
Speaker
And specifically geometry and trigonometry was from school because it's not, I mean, being a teenager is the worst time to be forced to learn anything. I just want people to give it it give give math a second chance. It's not what you remember from school. Wonderful. Thank you so much.