Transcript
Speaker: In the study of mathematics, there are many abstractions that we deal with. For example, we deal with the notion of a real number with infinitesimal granularity and infinite range, even though we have no evidence for this existing in nature. Besides the generally noted, demi -rules, smaller things keep getting discovered and larger things keep getting discovered.
Speaker: In a similar fashion, we define things like circles, squares, lines, planes, and so on. Many of the concepts that we just mentioned have to do with geometry, and perhaps it is because our brains develop to deal with geometric information, or perhaps it is because geometry is the language of nature, but there's no doubt denying that geometry is one of the original forms of mathematics. So what defines geometry? Can we make progress indefinitely with it?
Speaker: And where is the line between geometry and analysis? All of this and more on this episode of Breaking Math. Episode 23, don't touch my circles. I'm Jonathan. And I'm Gabriel.
Speaker: And of course you could contact us at breakingmathpodcast .gmail .com. We do love to hear from you. Yes, and we do respond. We have a website at breakingmathpodcast .com and we do have a Patreon. Patreon has gone back to their old model and you can find us at patreon .com slash breakingmathpodcast. Yes, we're very happy about that. And yeah, we've got quite a few offerings up on the Patreon. So you said you said you should check it out.
Speaker: Yeah, definitely. And today with us, we have on... Adam. Adam! Now, Adam is actually my brother, and he is here as our guest today on the podcast. Adam, do you like math? Yeah, math is great. Awesome, awesome. Right on. History major, but... History major at Annapolis, Naval Academy class of 2008. 2008, that's right. Awesome.
Speaker: And as a history major, does, like, I'm actually curious about this, does geometry, when does mathematics show up in history? I know it's mostly wars and things, but does mathematics show up?
Speaker: Well, I don't know from the, I don't know, I had a hard time at college. I would say for me, I think what's most interesting is maybe just looking at data for like war and populations. And I think history is an art as much as it's a science. And so statistics is probably to me, what's most interesting in the study of history for trying to use history in a way to forecast the future. You can't break the future, but looking at historical trends and stuff like that.
Speaker: That's interesting to me because in my own study of mathematics, it seems like mathematics is so important for many of the events that happened. Like engineering is always dependent on mathematics. Of course. Oh, yeah. Oh, for sure. I didn't know in what context you're asking the question. I would agree with all that. And so, yeah, without further ado, I'm going to pose this question first to our guests. Why should somebody study geometry? I know you're not a geometer, but...
Speaker: I think maybe to me geometry is the most readily visible aspect of finding beauty in math. If you're out in nature and you look at like a conch shell and the mathematics of like the development of the shell or the pattern in like a sunflower when it grows or looking up at the stars. We're very visual creatures and before we understand the rigor of math the young child can appreciate the beauty of geometry in nature.
Speaker: Now, one thing you brought up just then is visual creatures. What's interesting about geometry is that, from what I gather, blind mathematicians tend to be geometers rather than analysts. Well, that's actually fascinating. I had no idea. Yeah, because apparently all the hardware still there in the brain is just not connected to eyeballs. Wow. That's fascinating. OK. But then here's my question. Why would they tend to be geometers as opposed to analysts?
Speaker: Well, one explanation that I read, there's two explanations. The one that analysis tends to be, I mean, when you write down like, you know, for like the quadratic formula or something like that, that's a simple version of analysis. And even then you have too many things. If I told you right now without you knowing it,
Speaker: Plus or minus square root of four a C minus B square or I got there backwards whatever the fact that I got it backwards it's it's big formulas take a lot of like memory and that's the thing that you can't refresh as easily with them without vision and
Speaker: That's one explanation. The other explanation that I heard was that geometry was that when the whole sense is being heightened because of the lack of another sense and that extending to the inside of the mind. Interesting. Wow. Wow. Okay. That's still quite fascinating. That's totally fascinating.
Speaker: Yeah, and why do you think that we should study geometry? OK, this is a great question. And actually, this is a question I really look forward to bringing up, especially for our listeners right now. Why have a whole podcast about geometry? One of the things that is an obsession of mine is education and the education system. Right now, in the American education system, I don't know about the world over, but generally in America, all high schoolers study geometry. And one might ask oneself, why?
Speaker: As an electrical engineer, I will tell you, I definitely used algebra. I definitely used algebra 2, I definitely used calculus, and certainly trigonometry. Geometry is just kind of out there. You've got these proofs and you've got shapes and discussions about angles, but aside from trigonometry,
Speaker: I would say I could have gotten my master's degree without knowing any geometry. So it raises a question. Is it still a necessary skill? Is it something that we ought to do?
Speaker: I think one thing that you brought up is that you don't tend to use a lot of geometry in the way that you think about electrical engineering. But I remember a lot in your control theory class, every single class that you've had that has matrices involved in it is at his heart a geometric.
Speaker: Real quick, let's just talk about matrices. So of course, matrices are most often seen, I would say, in any control theory class or any control systems class in engineering, also linear algebra. But again, it's not often talked about the relationship between linear algebra
Speaker: and geometry is not often talked about. I think you hit the nail in the head and I'm glad you did. By the way, I'm intentionally playing devil's advocate. I want the listener to really dig deep and think about a defense of geometry. So I'm glad you brought this up.
Speaker: Yeah, too often when we're doing a linear algebra, we look at it as an analytic system versus a geometric system. And I think sometimes that could be a, like for some people, that's not a big problem. Some people are just naturally better at analysis. I know one of my strengths is analysis. But geometry is an invaluable tool. It uses the hardware that's already there. And like any kind of, I mean, obviously Riemannian geometries, geometries, geometries in the name.
Speaker: but linear algebra is necessary for understanding of Riemannian geometry. And you can't have something necessary for something without being related to it. And it's one of those things that's kind of like going to the gym. If you go to the gym for two weeks, you're not going to see a single result after the fact. But if you go to the gym for two years, you're going to see massive results. And so when it comes to just developing yourself as a critical thinker, I think maybe something like the proofs that you learn are more useful than we initially think, even though they can be boring and frustrating.
Speaker: I think they also, by teaching someone sort of different forms of reasoning, that's more valuable than we might think, and applicable to things like perhaps speech and debate or just being a persuasive individual.
Speaker: You know one other thing I wanted to mention real quick was just when we were talking about okay So the title of this episode is called what is it? It says don't touch my circles Yeah, a quote from the famous mathematician Archimedes one of my all -time favorite and by the way, I'm always mispronounced his name Archimedes Archimedes I I've said it wrong cuz I were Archie old Archie. I mean if you're going technical is Archimedes
Speaker: Okay. I'm not going to go technical on that. I'm not very good at that. So obviously he was very interested in circles. And the story with that title is that there was a, I believe, according to legend, I don't know how true this is, but apparently he was so absorbed in drawing circles in sand that he didn't even notice when a Roman soldier had approached him. And the Roman soldier, I guess, had probably maybe stepped on some of his notes and Archimedes said, don't touch my circles. At which point the Roman soldier responded by killing him.
Speaker: Very sad. Yes, yes, very, very sad. It was a huge embarrassment for Rome because that soldier was not aware of how awesome and how renowned Archimedes was. I mean, Archimedes was actually on the list of people to spare. On this podcast, we're going to talk a little bit about early geometry and early geometers. We're also going to talk about the development of axioms.
Speaker: We're also going to talk about blossoming geometry or current developments in the field, as well as the future of geometry. There's actually a lot going on in the field of geometry that we didn't even know until we did the research. And in fact, I think we could do an entire podcast only about the future of geometry and where the field is currently going. So without further ado, the rest of the podcast.
Speaker: Like many forms of mathematics, early geometry was inextricably tied to mysticism and taxes. Geometry was used since early antiquity to measure fields and collect the appropriate amount of taxes on them. But, being a form of mathematics, it had an allure that drew people of a certain persuasion to it, as it does to this day. One of these early groups of people were known as the Pythagoreans.
Speaker: I think it's fascinating. This is one example of something that was deeply, deeply religious with a lot of, I don't want to say peculiar. What's the word I'm looking for here? No, you're peculiar is right. Okay. They had a lot of peculiar beliefs at the same time. They were so dedicated to this pure form of analysis that is
Speaker: mathematics. I find it fascinating in history when you've got a society that on one hand is extremely empirical and on the other hand is a complete cult.
Speaker: Yeah, and we actually have a list of rules right here that were followed by the Pythagoreans. And these are their religious rules, not their mathematical rules. Real quick, did we talk about this in our first episode? We talked a little bit about it on our first episode. We talked about the Pythagoreans and what they did, but we didn't go into much detail about their cult -like measure. These guys were interesting. It's almost like I'd love to see like an HBO special on the Pythagoreans.
Speaker: And the rules are very interesting. I can't really find a common theme amongst them. So rule number one, and these are from Bertrand Russell's history of Western civilization. So no eating beans. That's rule number one. No picking up things that fall. So I would not be able to have my phone right now.
Speaker: Touching white roosters. No breaking bread. Interesting also, no eating from a whole loaf of bread. I don't know how that works. If you can't break it or eat from a whole loaf. Over a crossbar, this seems to mean no stepping over the lines and that'd be almost impossible today with sidewalks. No stirring fires with irons. No making garlands.
Speaker: No sitting on a court measure, no eating hearts, walking on highways, sharing a roof with swallows, leaving an impression of a pot on the ashes of a fire, looking in the mirror beside a light, or not making your bed. Oh my, no, not making your bed. Yeah, that's like one of the 15 rules. You know what this reminds me of almost?
Speaker: So these were not the only group that had the sets of rules. I almost think perhaps like Leviticus, you know how there's a whole lot of rules in Leviticus or perhaps, gosh, how many rules? I think even in China, you know, like there's a lot of like even superstition with them, even from Confucius.
Speaker: Yeah, but the rules in Judaism seem to make, like, at least sense for, like, hygienic purposes, if not for any other reason, but I just can't find a common theme amongst these rules. Yeah, yeah, they're strange. They're no -eating beans, so somebody had a bad experience with beans and they... It's because they caught flatulence and that was considered evil.
Speaker: flatulence was evil. Okay. Okay. And yeah, I'm, I'm, I'm still puzzled by it. So it would be, I don't know. It'd be a hard, it would be hard being a path. Do you know anything about what would happen if someone was found breaking any of these rules?
Speaker: I know that if you broke one of their big rules, you could get executed. That's actually one of the reasons why Pythagoreans were no more after a while because they started taking over secular communities and forcing them to convert to Pythagoreanism.
Speaker: Okay. Wow. So execution and then, and then probably just other, other conversion and all that. Oh wow. Wow. Goodness. And it's funny cause on the other hand, they still had a lot of cool, uh, they preserved a lot of knowledge as well. You know, they've even got a theorem named after them or multiple.
Speaker: Yeah, amongst their primary contributions were the Pythagorean theorem, or at least the first nice proof of one. Most proofs had to do with showing examples of the Pythagorean theorem. And Adam, have you ever done a class where you had to prove the Pythagorean theorem? I can't recall, possibly, but nothing comes to mind. Yeah, what's cool about it is essentially,
Speaker: One of the ways to prove it, and we'll just do the setup on this podcast, and you could do the math in algebra at home. You take your right triangle, and you take four copies of it, and you arrange them so that they make a big square and a little square. So the outside of the square is the longest side of the triangle, and the second or third longest is the inside of the square.
Speaker: And basically, you could do this with construction paper and prove the Pythagorean theorem. And that's essentially the proof that they used. Yeah, you know, I think that that's like one of the first aha awesome little light bulb moments that I had. And it was in middle school at the time. I know, gosh, I taught elementary school for a little while, and there was some talk of the Pythagorean theorem in fifth grade.
Speaker: But the first time I saw that relationship between squares, you know, a squared plus b squared equals c squared, where a squared and b squared are the smaller sides, or rather a and b are the smaller sides of the triangle. And then the fact that together they equal c squared, that's pretty darn cool, actually. That's almost enchanting, actually. Oh, yeah, completely. They also discovered and tried to hide the existence of irrational numbers.
Speaker: Yeah, this is another topic that we talked briefly about on that first episode. This is fascinating So it was in fact a forbidden knowledge. Yeah, and I don't have the name of the person who disseminated the information in front of me, but he was executed
Speaker: The way I read this, and again, I'm not a historian, unlike Adam, I'll defer to you, but the way I read this is that it seems like when you've got a cult, they figure out something that works, and it's magnificent and awesome, and then anytime something comes along that breaks the rules or challenges the paradigm, then that has to be banished.
Speaker: Even to this day, a variance of the Pythagorean system are used by modern numerologists, which seems like a contradiction in terms of modern numerologists. Yeah, and again, let's clarify. So I'm not as familiar with even the term numerology.
Speaker: Numerology is like taking like, okay, your name is game. So G is the 11th letter, A is the first letter, B is the second letter, and E is the fifth letter. One plus two plus five plus 11 is 18, one plus eight is nine, and nine means something specific for your life. It's basically numeromancy.
Speaker: OK, well, are we going to have some numerologist emailing us who are angry? Bring it. OK, you know what? I like that. I like that aggressive stance. We're not standing up for fake news here in fakery. And they also discovered music theory. They thought that whole ratios were the essence of the universe because whole ratios of numbers are the most pleasing sounding to the ear, at least the Western ear. Wow, that was the Pythagorean's? Yeah. Yeah, wow, OK.
Speaker: It's fascinating. So they were so obsessed with these things that, I mean, you know, they discovered a lot with this obsession. Yeah. And just to rattle off the last two discoveries that I have written down here, they discovered that the evening star and the morning star were the same object, meaning Venus. And when you use the term muscle tone,
Speaker: literally refers to musical tone because it was part of their extended analogy. You know what I wonder now from all of this, I've been thinking a lot about machine learning and neural networks. I'm going to bring this, I'm going to relate this to the Pythagorean's obsession with whole numbers. It's almost like they were obsessed with looking for patterns in nature that met these criteria. And from that, they were very successful in a lot of degrees.
Speaker: Yeah, but at the same time, what they had is I'm going to call proto axioms. They weren't actual axioms anywhere near the sense, but they were rules that they used for developing mathematics. And they were 10 opposites that they used in one of the traditions of Pythagoreanism.
Speaker: Yeah, I'll be, I very much look forward to hearing this. You know what I was saying though, like with respect to how you design a learning algorithm or, or machine learning, like, I wonder if you could do it in a way where, where it's, I'll just say, you know, in quotes, highly motivated to, to find certain things. You know what I mean? Like.
Speaker: Oh, yeah. Yeah. And I think that this is an example of basically if you view human beings as being a gigantic neural network simulation, this is us closing in on axioms, but not necessarily getting there yet. And I just think it's fascinating these 10 opposites and the way that they kind of sound like axioms and they bring up, as we'll see, very important mathematical and physical questions.
Speaker: So are these ten axioms going to sound like some kind of Joel Osteen examples for a good life? You know what I mean? Where it doesn't actually say anything, but it seems like it does.
Speaker: It's more like guidelines for doing math or guidelines for looking at the universe. So instead of being rules for looking at the universe, it's guidelines. Okay, let's hear them. Let's hear them. Yeah, they'll clear things up. So the first opposite pair of opposites, and I think that's called a dyad, but I'm not sure, is the finite versus the infinite. And that's still something that we talk about to this day. Okay.
Speaker: And so that was the first thing that they considered. The second is quiescent versus moving. Okay, hold on, quiescent. Can we break that down for 8th grade terminology? Oh yeah, sure, not moving.
Speaker: Oh, you could have guessed. It's opposites. What do you think it means? Still versus moving. Oh, yeah. I guess the thing that I was referencing used the word quiet. And so I assumed it was root word quiet. It's like Joel Osteen. One of those things that sounds like it means something. Oh, yeah. OK.
Speaker: Then we have odd versus even, which isn't very, I mean, I guess still in number theory, we still kind of talk about odd versus even, but. And symmetry breaking. Yeah, definitely. Then the next one is straight versus crooked, which I guess is the fastest path between two things versus not the fastest path. Sorry. OK. Yeah. OK. I mean, I don't think of those as opposites. I just think of one of them is like wrong. Yeah.
Speaker: Yeah, like, yeah, like these aren't like these were considered to be the reason why they had 10 is because they worship the number 10. But so so you can see they weren't very scientific about it. They're very like they assigned random stimulus meaning. OK, then to rattle off the other ones, one versus many light versus darkness, right versus left, which has to do a lot with chirality.
Speaker: Good versus evil square versus parallel parallel square versus parallelogram and male versus female Wow, okay. Okay interesting. So so there's this idea in speech and debate We've got a false dichotomy where basically you got two things that are falsely opposed to each other, you know
Speaker: Oh yeah, and also, any two opposites is gonna have to, I mean, when two people are on a seesaw, there has to be a hinge. So, I mean, every opposite has to be within the context of something, and they didn't see that. And just a quick note about sacred geometry, as it's called. This is something definitely used by the Pythagoreans, but they're nothing more than aesthetic ratios and the human tendency to assign qualities to arbitrary things.
Speaker: Sacred Geometry has no real value to mathematics. It has value to aesthetics. And I think that it's necessary to bring that up because, in my opinion, it's dangerous to assign qualities to arbitrary things when you have a better system available. This has been bursting your bubble with Jonathan Baca since 1992.
Speaker: However, it's still interesting. I've also seen YouTube videos that involve sacred geometry. There is some beauty in that because I think that... Oh, there's absolutely beauty in it. The value is extremely aesthetic. But to say that because something is aesthetic, it does something for your life that it doesn't do.
Speaker: Yeah. And I think there's examples of this and things like, uh, I actually saw one, one video on sacred geometry where they, uh, uh, they started off with geometry and they ended up with step by step healing crystals. But at some point I was like, wow, if I didn't, you know, if I wasn't a critical, like like a train, a critical thinker, I would have fallen for this and I would have brought crystals for all my relatives who are suffering some ailment because it's convincing. It's, it's, it's, yeah, we, we are, we are able to be, uh, suede.
Speaker: Yeah, I mean, things that sound like reasons sometimes aren't filtered out by emotion. That's why you have to have a strong emotional and logical mind to do mathematics with. You know, not to jump around too much, but I will say Isaac Newton was not immune to this. He spent a lot of his time studying alchemy, as well as codes in the Bible. He thought that he could get codes from the Bible that would give him special knowledge about things like the end times.
Speaker: Oh yeah, and I mean, I myself have like a pen that I draw with, like if I want to draw something well, it's kind of a lucky pen, but I realize that that's just Pavlovian conditioning at its base. Yeah, and what's also interesting is I think that when we talk about the developments of certain beliefs, especially
Speaker: I'm wondering if that has something to do with our own human reward system. You know what I mean? Like you do something really well and you're wearing a certain pair of boxers or something and then you always want to wear that pair, you know? Well, humans have a thing called, I mean,
Speaker: It's a survival strategy, first of all, because if you do something right and you get something good out of it, then it's better for you to repeat doing that thing than take the chance of not doing that thing just in general. I mean, our mind allows us to refine that significantly.
Speaker: but second of all humans and crows I think we've talked about this before on the podcast have a an inherent attraction to new things So if you have it so if you combine those two you get a perfect storm. Oh wow fascinating. That's just fascinating
Speaker: You know, it'd be interesting to do a survey of superstition and beliefs among professional athletes, but also among maybe stockbrokers or other people who have intense jobs.
Speaker: I mean, some people consider economists themselves to be modern numerologists in a lot of regards. If you look at many of the main indices that have been used throughout the 20th century, we don't know what's going on in economics. We know what's going on fiscally, but we don't know economically.
Speaker: So back when I was teaching one of the things that I love to do I absolutely love to do it was thought experiments with my students where I would start them off on a concept and then I'd let them explore that concept and come back to me with what they've what they've Discovered in these discussions one of the things we talked about is is if you have two circles of a given radius Let's just call it a radius of one unit and
Speaker: If you have two circles, these two circles, and I'm talking about a flat two -dimensional circles, they touch at a single point. However, they overlap at two points. You will always have this. They overlap at two points. You can draw a straight line through those two points. You've got a straight line. However, here's a question. How many points do they overlap at? Do they perfectly overlap at?
Speaker: They coincided an infinite amount of different differently identifiable. Yes. Yes So this is one of the first ways that we brought up infinity with my sixth graders and and It's amazing because because you know you think that you can't imagine infinity But you just did when you when you put two circles of equal radii one on top of the other they overlap at infinite points now extend this this whole
Speaker: this conversation into higher dimensions. Let's talk about spheres. So just like with circles that touch, two spheres will touch at a single point. However, things get a little bit different when you begin to overlap. How many points do two spheres overlap at?
Speaker: They overlap in a circle. Yes. And that's, there you go. And that's infinite. However, however, how many points do two spheres of, um, identical radii overlap at that's also, it's like an infinity of infinities. And what's interesting about that infinity of infinities is that it's provable. That is the same infinity as a circle, just in a different dimension. Yeah, yeah, absolutely. So, so these were some discussions that we had, and I just thought it was the coolest thing in the world.
Speaker: And one of the problems that you brought up was the circle where the circles coincide with one another, the infinite amount of points versus that's actually that goes back to a Pythagorean concept, the one versus the many, which is still a problem to this day, the problem with the one and the many, it's called.
Speaker: Okay. Yeah, yeah. So again, there's nothing new under the sun. These ideas have been explored before. Having said that to me and to my kids, it was the first time we explored them. So I think what this shows is either some kind of a cultural bias towards thinking in this way or a very natural way of thinking in this way. And I think that
Speaker: having people discover that kind of stuff on their own is essential. If you read through Euclid's elements, don't just read the proofs. Anybody could read a proof. Try to do the proofs. I've got one more for you. We're talking about straight lines as well. So let's say that you pick any two lines on a circle, any two lines at all. And there is a curve between them. And that curve corresponds to an angle, of course.
Speaker: It's sort of like on the earth. Jonathan, if you and I are standing next to each other and we're a mile apart because the earth, you know, approximates a globe or a sphere, there's going to be a curve between where you and I stand. So essentially, a straight line could be a circle with a radius of infinity, right?
Speaker: Oh, I mean, definitely once in one sense. I mean, that was something that I used to obsess about. Oh, yeah, I just I just love that. I mean, you know, like the relative to two points, the closer they are on a circle, the straighter that that line gets. But here's the funny thing. If you had two people on a on a on a circle.
Speaker: The only time that they are on the same level is when they're in the same spot. Is that right? Yeah, like there's no curvature between them whatsoever. On a circle, the only possible way that two people are on
Speaker: the same spot like with with no curvature between them I need to work on a way of explaining this better the only way is if they are in the exact same spot but what happens when you have two people who are you know who are standing on a circle and they are in different spots therefore you there is a curvature and you extend the size of the circle to infinity then would they be on the same level the same plane
Speaker: Well, it depends if you want to consider every increment up to infinity, then no, but if you consider the limit, yes. Okay. Yeah. Yeah. Fun. So fun thought experiments. I just love those. This episode is all about geometry and how geometry is used.
Speaker: As many of you know, our partner Brilliant .org has many courses to choose from in the fields of math, science, and computer science. But did you know that it has an entire section just for practicing fundamentals? One of these practice courses is called Outside the Box Geometry. I especially like the way the course was structured and that it went into detail about things like tessellations and 3D geometry. As a tutor, I can say that working through this course would greatly strengthen and even augment one's understanding of geometry.
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Speaker: And now, back to the episode. The human mind is an intricate web of connections that tries to reduce everything to the simplest possible explanation that still fits the data that is presented. The tendency to reduce and refine led eventually to the development of axioms. Sets of concrete rules that describe or govern systems most likely date back to time and memorial. Think about children's games and how much they are governed naturally by sets of rules.
Speaker: but sense of concrete rules that describe or govern mathematical systems are much younger, and date back to about 300 BCE with Euclid. These were, naturally, a way of describing what we now call Euclidean geometry. That is to say, the geometry that we encounter day to day. Of course we can encounter non -Euclidean geometry in forms like gravity, but these concepts are not naturally apparent to humans. The beauty of Euclid's geometry was not in its complexity, but its simplicity, as we shall see.
Speaker: So as we're talking about the development of the field of geometry, Euclid of course came after Pythagoreans. Do you know when? Um, I think it was a couple hundred years after. Okay. Okay. And do you know the, the, the state of the Pythagorean cult? I'll just call it. I think he was pretty much wiped out by then. Okay. Okay. Fascinating. So then with Euclid, do we know, do we right now with knowledge that we have of that time, do we know what may have been his influences or, you know, how he arrived at what he arrived at?
Speaker: This is a time in history where the facts were starting to be considered important to be written down. In ancient Sumeria, they would say that a king ruled for 144 ,000 years.
Speaker: they would exaggerate that. So in that sense, there was journalistic integrity. However, when it came to attributing things to people or even doing biographies of people, there was not really developed for, it would not be developed for over a thousand years. I mean, look at Beowulf, he probably didn't exist.
Speaker: Yeah, I see. So it could be that Euclid was entirely the genius. I have no doubt he was a genius. Oh, he's absolutely a genius. It could be that he was entirely the genius behind the book, Euclid's Element, or it could be he was his roommate and he ripped him off. I'm just kidding. But you're good. Yeah, so that sometimes happens.
Speaker: But it does seem like he certainly is noted as the person who codified this in the book, Euclid's Element. And you know, I have a friend of mine who thinks that that book is even more beautiful than Einstein's Relativity. Okay, Special Relativity is an application of Euclidean geometry in four dimensions.
Speaker: And general relativity is non -Euclidean geometry, but it's already using geometry. So mathematically, I would agree that Euclid's book is... I mean, I haven't got through it. I've read a lot of it, but I'm probably only a fifth of the way through. I have the book at my house and it's a big old book. You could hit somebody with it and do damage.
Speaker: Yeah, so I mean we've talked about his postulates before on this podcast. And are you familiar with these postulates, Adam? Is the postulate like the 180 degrees and the inside of a triangle?
Speaker: Well, that's a proof that comes from the postulates. The postulates are the axioms. These are assumptions that are made without any kind of proof. And I think one of the reasons why it's difficult to remember these, because I have difficulty remembering them as well, because they are just so basic that you forget. For example, it's possible to draw a straight line between two points as one of the postulates.
Speaker: And that's obvious, right? Yeah. Would you argue with me that you could draw a straight line between any two points? I wouldn't argue that. Yeah, and that's why it's a postulate. You could continue a straight line indefinitely. Describe a circle with any center and radius. All right angles are equal, so all that makes is pretty self -explanatory, right?
Speaker: All right, angles are equal. Isn't that how is that? That seems like like that's like saying all circles have the same shape or something. Yeah, I know. Yeah, there's kind of a node aspect. Yeah, no. These are the five no does. These are the five toys. Boy, if you just taught it as no does in geometry, students would learn it probably much quicker than postulates. Yeah.
Speaker: Well, the fifth one is less no duh. And that's if a straight line crosses two other straight lines and the interior angles sum up to less than two right angles, then if you continue these lines indefinitely, those two original straight lines will intersect. Basically what that's saying is that Willa will not intersect.
Speaker: No, they will intersect. Because think about a pair of chopsticks. If you look at the line, if you're holding a pair of chopsticks and you click them together, any line that passes them, it's going to have two angles on the inside of the chopsticks right where your index finger and thumb hold them. And that's going to be less than two right angles. So that's the intuition behind that.
Speaker: But it's such an ugly postulate that people tried until the 1800s to prove it. What's interesting about these postulates, too, is that they rest upon common notions, which in our modern day would also be axioms. And we've talked about - Is there a difference between an axiom and a postulate? Not really. A postulate is just a name for an axi - It's like an old version. Actually, I'm curious about that. If we have anybody who's really technical, I wonder if there is a definition. And obviously there's a computer. We could just Google it, but we'll leave that to our listeners. We'll give them homework. Here's your homework.
Speaker: Yeah, when we don't research things, it's called homework. You just lost 2 ,000 subscribers by giving them homework. No. No, no, no. But yeah, so the common notions are they sound kind of the same things that are equal to the same thing are equal to one another. If equals are added to equals, then their sums are equal.
Speaker: It's things like that. Yeah. And again, it's interesting because, you know, if we're talking about mathematics and if we talk about education, when you are talking about these to somebody, it does seem like obvious points. And it's almost like in order to get profound, you have to talk about the really, really, really obvious things, you know, fundamentals. This shows kind of a link of what you just said between geometry and analysis, because
Speaker: You reminded me of one time several years ago, I decided to come up with an axiomatic system for geometries. You could represent geometric problems just using strings of algebraic notation. And I actually needed to add axioms to the original ones because even these aren't doing all the assumptions when it comes to axioms.
Speaker: So when you translate geometry into algebraic notation, you have to actually add a lot of extra little assumptions and things like that. If you don't even include the logical assumptions, it requires a definition of equality itself if you go deep into it. And I know we've talked about how axiomatic systems aren't perfect, but just in my own attempt, I needed to add things. Maybe somebody could do better than I did.
Speaker: An example of Euclidean geometry is constructing an equilateral triangle on a given finite straight line. So what that means is you have a straight line just floating around in space and you want to build a triangle on top of it. How do you do that? Okay. So obviously what you'd have to do is you have to, you know, if you can be specific here and I might, I might be redundant in here, but you have to place a point somewhere that is not on that line.
Speaker: Yeah, and actually what's interesting about this proof is we shall see it requires a lot of assumptions because I remember translating this into my axiomatic geometric system and the proof was a lot longer than Euclid's proof. Oh, wow. And Euclid's proof is pretty much as follows. Let's call the left dot of the finite line L and the right side R. So you got left side and right side.
Speaker: So you draw a circle with L being the center and R being the radius. OK, got it. So you have a straight line, 2 .LR. Yeah. OK. So now what you do is you revolve the straight line around L. OK. And then you do the same thing but around R. So now you have two circles that are intersecting one another. OK.
Speaker: and where the two circles intersect, those are, because a circle is by definition the locus of points that are equidistant from one central point, right?
Speaker: Yeah, what's the radius of the two circles that we just drew? The little line. That's the radius. And so the distance between L and any point on the left circle is the same as R and any point on the right circle. So when the two circles intersect, that's going to be the same distance from L and R. So it's going to be an equilateral triangle. And that's the very first proof in Euclid's book.
Speaker: And one cool thing about Euclinian geometry that we're just going to mention real quick is that it didn't end with Euclid in 300 BC. In fact, it was continued for like two millennia and continues to the lesser extent to this day, but definitely had fervor for at least two millennia.
Speaker: For example, in 1837, there was a proof that you could not trisect an angle. And in 1882, when Einstein was in school, they proved that you cannot square the circle using Euclidean geometry. What does that mean? Oh, using a compass and a straight edge, you can't construct a square and a circle that have the same area. Oh, interesting.
Speaker: Yeah, and you can't trisect an angle either. What does that mean? You can't divide an angle into three smaller angles that are equal. That's crazy. Now what if, can I just propose an idea? Sure. What if you had a triangle piece of paper and you folded it such that the edge fell halfway between the other edge and the fold?
Speaker: Well, actually, the thing is with that is that you're adjusting two things at the same time. So you could do that. But what's cool about paper folding is it turns out that it's more powerful than compass and straight edge. You could trisect an angle easily using a sheet of paper.
Speaker: Oh, but if it's just a compass and a straight edge, then you cannot. Yeah. I mean, it's possible to square the circle. If you like using a record player and a and a compass and a straight edge, you could you could square the circle like Archimedes proved that actually. Oh, OK. But yeah, just so he was down with the record. He was down vinyls. Archimedes was down with the record player.
Speaker: Yeah, he said that it had a warmer tone. In the 19th century, many geometers tried and failed to prove Euclid's parallel postulate. However, a revolution was well on the way. Beginning with Janos Bolyai and Nikolai Ivanovich's publication of results, wherein the parallel postulate was not valid, a brand new tradition of non -Euclidean geometry arose. Today, we know that the universe itself is non -Euclidean.
Speaker: and that the fact is a result of general relativity, which is covered in episode 21, Einstein's biggest idea.
Speaker: Lines on spheres can be considered to be lines in the traditional geometric sense, what is called as the great circle. So like the equator is a great circle because it has the same radius as the sphere. So you could draw a triangle on a sphere by just drawing three great circles and having them intersect at three points. I can do this example of a big triangle. So let's say I start out
Speaker: in brazil at the equator okay is um are you guys with me i'm following i'm following so your ship is pointed north at this point right or just assuming it is okay sure get it but it's like a weird hovercraft so it can go other directions but it's uh but it's just we start going in the same direction that it's facing so you start going north
Speaker: Okay, so we're all going north on our hovercraft ship. So you go all the way to the North Pole. It's kind of cold up here. And then you turn 90 degrees and you start going down. And so now you're traveling right, right?
Speaker: okay so you said night to your right okay sure yeah and you go all the way down to africa okay and so now your ship if you if you if you realize if you think about this the ship was pointing north at verse but as soon as you got to the north pole and started going down the ship is now pointing to the uh to the east because we didn't turn at the north pole we just started going a different direction we started we kept our heading
Speaker: Oh, okay. And then when we go backwards on the equator to our original point, our ship has now turned by 90 degrees, even though there's no point at which we actually turned. And that's only possible in non -euclidean geometry.
Speaker: Okay, that's because the internal angles that we turned on were more than, there were more than, the total sum of the angles were more than 180. Yeah, 90 degrees each. And it's actually possible to show that if you do that with any path on a sphere, any closed loop, so if you start at one point and go take a journey and go back to the original point and you don't turn at any time, just like we did in our example,
Speaker: the angle that you turn is proportional to the area that you enclose on the surface of the sphere. So you might think, well, this sounds a lot like Euclidean geometry because we're just talking about things that can be embedded within Euclidean geometry, but that's a key term, embedded. And I want to do an analogy. You mentioned earlier, there's this book called Flatlands, A Romance in Two Dimensions. That book is all about living in 2D.
Speaker: And so you have to imagine that when you live on the surface of a sphere, there's people now that believe that the world is flat. That is the essence of non -Euclidean geometry. And in fact, space itself is curved around, in four dimensions, the Earth. It's why we have gravity, because space is curved.
Speaker: And, uh, yeah, so non -nucleating geometry was essential for Einstein's theories. And it's a very real site. People don't think of it as a real form of mathematics a lot of times because it's not plain geometry, but it really is. Yeah. It's amazing that even on, on a globe, I think it's not always the case. Um, you know, if you have two locations on a globe, it's not always a case that what you would consider the straight line approach is the shortest.
Speaker: I think that has to do with a great circle distance versus going, uh, large doodle. Um, or if you were to punch through the earth, you know, like in a straight line, then obviously that, that was certainly would be the straightest, but.
Speaker: Oh yeah, but you can't punch to the earth in non Euclidean geometry. Exactly. Exactly. So we're going to talk about the future of geometry a lot more, but things got complicated and we realized that we could do a whole episode on the topic of the recent advancements in the future of geometry itself. This took me by complete surprise. I honestly was almost of the opinion that the geometry was neat, but kind of outdated and we don't really do a whole lot with it anymore. So I was surprised. Learn something new every day.
Speaker: Yeah, I mean, if you want to look up the kind of things that geometry does now, if you're mathematically inclined, I would only do this. But there's inter -universal tech Mueller theory, which only developed in 2012. Fiber bundles, which are a way of like making derivatives extend to manifold. And I keep using the word manifold. I know everybody who knows what I'm talking about thinks that I'm
Speaker: don't know what I'm talking about. Manifold is just like a connected, a thing that, a series of points that are connected enough to call them something like a plane. And to say that inter -universal tech Mueller 3 has to do with Manifold is like saying that math has to do with numbers. It's not saying much. You know what? You try, dude. Yeah, yeah. In order to really get a flavor for this and to distill the concepts down, it takes an effort. So points for trying.
Speaker: One recent thing, and I would look this up on YouTube if I were you, and this is very easily understood, and it's a very recent discovery from the 1970s. I'll set up the problem right now. Let's say you had a sphere, and the sphere has the property that
Speaker: Let's say you crunched it up, the sphere could poke through the sphere itself. So it's made out of a cloth that can go through itself. And this cloth has the property that if you pinch it, if you pinch the cloth at all, it will light on fire and destroy the entire sphere.
Speaker: So the question is, how do you turn a sphere inside out without making it pinch? If you just pull opposite ends of it, then you're going to have a ring that's basically essentially pinched because it'll try to go through itself and you can't do that. And the way that you have to turn sphere inside out is very interesting. And just look that up and you'll see what kind of things are still being done in geometry. Keywords. What are the keywords if you're to YouTube that?
Speaker: just search on YouTube how to turn a sphere inside out. Yeah. And that is utterly fascinating. It is a really fascinating proof. It almost reminds me of those old, um, what do you call it? The horseshoe puzzles, the, the, the iron puzzles, like, like the, like the blacksmith puzzles where you have to, Oh yeah. It's totally that kind of thinking. Yeah. Yeah, absolutely. Where it's totally possible. You just have to see things the right way. What now, again, for our listeners who are not familiar with blacksmith puzzles, how do you explain those?
Speaker: Those are kind of like if you've ever tangled up like a keyring just like that, but more complicated Yeah, so imagine if you've got like a bunch of circles some iron circles that have small Slits in them, but but the slit is too small for the girth of the of the circles to fit through but there's just certain ways that you can line them up where they slip right through and
Speaker: It's sort of like, I think of if you've ever been in a really crowded concert and you don't have a direct path to the stage but you can slide through where people are standing somehow, that's some analogy. Or like when you have to move into a new place and you try to get your couch through the door?
Speaker: Yeah, yeah where it's still very possible. It just requires some I'm sorry about this cliche term the out -of -the -box thinking we need to replace Let's do since it's 2018. It's a new year. Let's replace the term out -of -the -box thinking I think that that term is run its course non Euclidean thinking. Thank you non Euclidean thinking, you know, I don't know What do you think out of the sphere thinking out of the sphere? Yeah, I'm trying to think of a different
Speaker: Out of the box. Yeah, I don't know. We'll have to think of a new term that means non -traditional thinking. Well, we had to think out of the box to come up with that. So one other thing that I want to talk about as we're discussing the future of geometry is the topic of computer science. When I think of computer science, I think of solving problems algebraically, but I seldom think about
Speaker: geometric proofs in computer science. Jonathan, can you tell us a little bit about what you researched about using computer science in geometry problems?
Speaker: Yeah, actually, automated proofs have been the topic of research since the development of artificial intelligence in the 1950s and 60s. And in the 1970s, a program was written, I wish I had the name of the program and the researchers to do them justice, such that it did geometric proofs. And there's a proof in Euclid's Elements called the Pons Esonoreum.
Speaker: And basically what it is, is it's saying that if you have a triangle with two equal sides, prove that the angles on the bottom of the sides are equal to one another. And it's a difficult proof because it requires more abstract thinking than the proofs before it.
Speaker: And this computer, back in the 1970s, this program, this very specific program, not this general form of programming, came up with a brand new proof of the Ponzasinorium problem. And it didn't do anything else novel. It was pretty much a dud besides that. But it came up with something beautiful and new. Wow. So a computer program created a proof that no human had thought of before. That's fascinating.
Speaker: And a lot of the questions having to do with this are, was it proof in the brains of the computer programmers and did it come out in a weird way? Or was it an essentially computerized thing that a human wouldn't be able to come up with? But those are philosophical questions that are best left to another episode.
Speaker: OK, OK, wow. So in other words, you said it's been a dud. So once it came up with this novel proof, then there's been no other novel proof with geometry. It didn't really come up with any good other proofs. It came up with very simple proofs that were very obvious, and it wasn't able to prove anything very complicated. That was the only complicated thing it could prove. OK, wow. And did it in a novel way. Geometry brought mathematics into the modern age with the development of axioms and continues to be a fruitful area of discovery.
Speaker: Looking into the future, we can see that by exploiting our human hardware in the best way, we will continue to improve our understanding of geometry. I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. On this episode today, we had... Adam. And Adam, is there anything that you'd like to say to the audience or plug? Nope. You could email us again at breakingmathpodcast at gmail .com. Visit our Facebook at facebook .com slash breakingmathpodcast. Visit our website at breakingmathpodcast .com.
Speaker: or patreon at patreon .com says breaking math podcast we're just like all over the place there so we got all of our bases covered we're also on twitter in fact we've gotten some correspondence from twitter too so breaking math pod is our twitter handle yeah today we know that the universe itself is non euclidean and that the fact is
Speaker: the next section your mom's none Euclidean geometry you can edit that out
