Daniel and Kelly’s Extraordinary Universe - What are octonions?

Episode Date: January 18, 2024

Daniel and Jorge talk about super complex numbers and how they might hold the secrets of the Universe. See omnystudio.com/listener for privacy information.

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I mean, I love Planck's Constant, which tells us about like when things become quantum. I love the speed of light that tells us all about relativity. Too many to choose from. There are a number of favorites there. But have you ever wonder why numbers are so important in the universe? I mean, numbers are like the currency of physics. We're predicting things happening at times and places, and those are just numbers. Wow.

So does that mean you get paid in numbers? I get a number of dollars every year I guess we all get paid in numbers There's just numbers in our bank account If you're so lucky The whole universe is just numbers I think you've said that a number of times I'm the number one fan of numbers

I think I know a number of those in physics And math, too many to number Hi, I'm Jorge McCartunist and the author of Oliver's Great Big Universe. Hi, I'm Daniel. I'm a particle physicist, and I've been a professor at UC Irvine for a large number of years. Oh, too many to count. or do you go into some sort of like weird subspace when you do physics? No, I think I'm going to enter a pupa phase eventually and then emerge as an emeritus professor.

As a butterfly. That's a beautiful tenured emeritus butterfly. As a white-haired moth, I think. Instead of wings, you just have a bunch of research papers taped together. I'm going to fly too close to the universe on wings of research papers. Right. But you know butterflies don't have mouths. right? So you want people to talk.

Yeah, but they have those long noses and so I'm already said. Oh, there you go. You can sniff out science from there. I can snort up all the information in the universe. Oh, boy, that sounds a little illegal. Are you going to arrest a butterfly? Is that what this is coming to? Maybe a cocaine sniffing butterfly. Maybe. Maybe it'll be our informant for, you know, figuring out how the universe works. Maybe altering your mental state is important for understanding the universe.

I hear that's called the butterfly effect. I think that's something else, man. Did I get that wrong? I think it's called microducing. Yeah, I think this is called micropunning, which doesn't work as well. But anyways, welcome to our podcast, Daniel and Jorge, Explain the Universe, a production of IHeart Radio. In which we give you a drip, drip, drip, little doses of the incredible wonder

we've discovered about the universe. Everything that's out there that makes sense and everything out there that still puzzles us as we try to fit the entire universe into a pattern that makes sense to our little human minds. That's right. We try to get you high on the little dopamine hit

of wonder and amazement at how our universe works and the surprising ways in which it still puzzles scientists even today. And one thing that still puzzles scientists and philosophers is that any of it makes sense that these mathematical models we build on our head can actually describe and even predict what's going to happen out there in the universe.

That somehow mathematics seems to be not just the currency of our physics, but the currency of the universe itself. Wait, wait. Are you saying math is more important than physics? I mean, is English more important than Shakespeare? You can't really compare the two things, you know? Wait, wait, in this analogy, Shakespeare's math and English is physics? No, Shakespeare's physics and English is math.

Physics is speaking in the language of mathematics. We are writing poetry in the language of math. I see. I see. So is physics a comedy or a tragedy? It's a tragic comedy for sure. It's a tragic comedy. We're all waiting for the final twist.

Nobody knows how it ends. That's right. It never ends well for Shakespeare. But I hope we laugh along the way. And you also have to wear tight when you do physics? And as always, the jester is the wisest one. Oh, I see. That's the engineer in the team.

Did you just call engineers, jokers? I mean, I'm just going to step slowly away from that. No, I know. We have a great sense of humor is what I'm saying. We're the smartest person in the room. Often laughed at and unwisely ignored. But yeah, it seems like there are a lot of numbers in physics. There's an infinite number of numbers in math, let's face it.

But in physics, it seems like there are special numbers out there that have kind of a special status because they're sort of significant in how the universe works. Yeah, there are certain constants which seem to be important that tell us something about the universe, the speed of light, planks constant, the gravitational constant. There are even numbers that don't have units, you know, like the number of particles we've discovered or ratios of masses that we think reveal something deep about the universe. But even more than that, there are systems of numbers. There are patterns of numbers that reflect patterns we see in the universe, mathematical constructs and all sorts of fancy mathematics that really are crucial to

understanding how the universe operates. kind of especially true at the microscopic level, right? That's where a lot of these numbers and a lot of this math comes from, and it comes from our attempt at understanding how things work at the particle level. That's an interesting point. I would say that math describes the universe at every level. You know, we have mathematics for fluid dynamics and for planetary evolution and for the expansion of the whole universe. It's mathematics up and down. Okay, so it's at every level, but it seems like in the standard model, there are especially some interesting numbers. Yes, absolutely. The theories of particle physics are very mathematical and not just in the sense that they're predicting

numbers and places and times, but the patterns we see in the standard model use complex mathematical theories like group theory and field theory and all sorts of like heavy hitting mathematical apparatuses. Right, which sort of raises a question of whether the universe itself is mathematical. Like maybe if you dig down deep enough into the stuff we're all made out of at the end, maybe we're just mathematical equations or formulas. Yeah, and it makes you wonder like what does that mean and are numbers real the way like stuff is real you know like where is the number two if two is a physical thing does it exist somewhere in the universe is a whole fascinating branch of philosophy but like how we learn things about numbers because you can't like do experiments with the number

two and the number seven it's all sort of mental games right right like maybe the universe is not really physical maybe it's just sort of like conceptual theoretical yeah but then you wonder like what breathes fire into all those concepts. It makes us experience it. You know, the other side of that argument is that numbers are not universal. They're not physical. They're not natural. They're just sort of the way that our human mind works, that you can use numbers as a way to describe the universe, but it doesn't mean they're part of the universe. The way you can like describe the color orange using a bunch of words, but none of those words are orangy or fully capture the oranginess of an orange.

And is one really the loniest number? It's another big physics question, right? Yes. Yes, a very deep physics question. But we don't see a lot of value in math. There's lots of times when mathematicians have developed some cool little technique just for fun because they see cool patterns and they like playing with them. And then later on, physicists will come along and be like, hey, that looks useful and

just like pluck it out of their hands and go insert it into our physics equations and get great insights. There's a lot of hints there that the laws we use to describe the universe are deeply mathematical. Yeah, it seems like there are a lot of new ideas coming up all the time about how to explain the patterns that we see in the universe, maybe with new kinds of math. And so today on the podcast, we'll be asking the question, What are octonians? Did I say that right? Or is it octonians? I think it's octonions. Like, give me eight onions for that. recipe please. Isn't it like a blooming onion? Isn't that a dish, a fried dish in one of these

famous restaurants? Is this like a fried onion that looks like an octopus? Are you shilling for Outback Steakhouse now? Oh, is it Outback? I don't know, never been. Maybe Applebee's is interested. I think it's maybe Octonians or Octonians, depending on where you are in the world. So this is an interesting word. It sort of sounds like Octo, which is eight, but then you're ending it with onions, which is a vegetable or a tuber, but it's also sort of how some particle names ends, right? Mm-hmm.

Yeah, ions, exactly. Oh, yeah, right, right, yeah, ions. So is it octanions then? I think we should have eight different pronunciations of this word. And then we'll spend 88 minutes talking about it here on the podcast. As part of our cult of eight. There you go. Send $8 to join.

There you go. have $8 in our pockets, in our eight pockets because I wear cargo pants. Exactly. Maybe these are the spiders of the universe. It's all an intricate web of eight. Well, anyways, as usual, we were wondering how many people at there had thought about or even heard of the word Octonians and what they think it might mean. Thanks very much to everybody who participates in this audience contribution segment of the podcast. We'd love if you joined the crew, please write to me to questions at Danielanhorpe.com. So think about it for a second.

What do you think are octonians and are they sold at Applebee's or Outbacked Steakhouse? And would they make you cry if you chop them? Here's what people had to say. What are oct onions? So act is the prefix for eight onions being something with layers probably. So eight layers, stars, I don't know. Honestly, I have no idea. But if I were to have to guess, I would deconstruct the word.

So I think the prefix oct means it has to do with eight. And so maybe it has to do with gluons or any type of similar particle or force carrier that is dependent on matrices for transformation. All right. Well, pretty much the same as what I've been guessing, something to do with onions and eight and or particles. Yeah, reasonable guesses, absolutely. Is it like what happens when you smash together eight onions? You get an uctonion? We're working on the onion collider right now.

We just need a little bit more funding, $8 more, and we'll be there. You need a little bit more garlic. I thought you're going to say a little more seasoning. You need $88 billion. I won't say no to $88 billion, that's for sure. Yeah, there you go. And then what happens at the end when you collide the onions? Would you have to eat them?

I disappeared in my private island with all the cash. I think they can give you eight years in prison for that, Daniel. Only if they catch me. Only they smell out the crime. I'll have an army of cocaine-infused butterflies to protect me. Oh, my gosh. You are a super villain there. I mean, I've heard of sharks with lasers, but drug butterflies is a new level of villainy.

They will never see me coming. All right, well, let's stick into it, Daniel. is an octonian and what do you think is the right way to pronounce it i pronounce it octonians but it does make it sound like creatures from the planet octo or something i see you're going for the more tony pronunciation octonians octonians exactly so octonians are a kind of number and they're sort of in the category of complex numbers but they're an extension of complex numbers the way complex numbers have like two components to them, the real and the imaginary part, which are an extension of real numbers that just have one component.

Octonians have eight components, one real and seven imaginary components. Well, okay, wait, hold on. I think maybe some of us might not be super familiar with our high school math. What is an imaginary number in the first place? Imaginary numbers were invented in the 1600s, and they're called imaginary because you don't see them in reality, but they're very useful in math. And there's basically one imaginary number, which is I, which is the square root of minus 1. There's no real number, which if you multiply it by itself, gives you minus 1.

So they invented a number. They just call it I. And they say, if you multiply I by itself, you get minus 1. So it's a new kind of number. It's different from any of the numbers on the normal number line. Right, because like we had the number negative 1 and we had the function to take a square root of something. But when you mix the two, it's like you get something that's almost impossible or that it's not on the regular number line that we all use in our everyday lives. Exactly, because any number on the regular number line, we call the real numbers, if you square it, you get a positive number.

Three squared is nine. Negative three squared is also nine because the two negatives give you a positive. So there's no number on the real number line, which if you square it will give you a negative answer. So there's no number that's an answer to the question, what is the square? root of negative 1. So they had to invent a new kind of number. So imagine the real number line is like at the x-axis. They invented a new axis, like the y-axis, which is like the imaginary direction. So instead of having a number line, now you have a number plane where every number has two components, a real component and this imaginary component. Right, because you came up with

I, or mathematicians came up with I, and then you can have a whole number line based on i you can have like two i and three i or four point seven i or eight i or negative two i right the imaginary number line goes both ways right right and can you also have not i you can have not it like i'm not doing the dishes yeah that's what i mean like zero i would you call it not i oh you can have zero yeah the zero and the number line is just zero comma zero right zero real numbers and zero imaginary right so in physics when you talk about an imaginary number uh you talk about an imaginary number talk about like a number that has both a real component and an imaginary component. So you write it as two numbers, like seven plus eight I. Exactly. That's what we call a complex number,

something with a real and an imaginary component. Sort of like a coordinate on the two-dimensional complex plane. If you imagine real numbers are X and imaginary numbers are Y. Right. And I think like all of most of quantum particle physics is based on imaginary numbers, right? Like it's a convenient, you know, mathematical space to do all of the math in. Yeah, exactly. So there's a few important things to know about complex numbers. Number one, they turned out to be really useful. Back in the 1600s, when they were invented before we had quantum mechanics, they were

already useful. Italian mathematicians came up with them as a way to like solve problems and they found that they could get to the solutions of tricky math problems much more rapidly if they used these imaginary numbers, even though they didn't believe in them. They didn't believe that they represented anything in the universe. But now all of our quantum mechanics relies on these complex numbers. Like the wave function that we talk about all the time in quantum mechanics, that's actually a complex valued object. That's why you can't observe it directly.

You can only observe it when it's squared that gives you the probability. The wave function itself is a complex number. It has a real component and an imaginary component. We couldn't do quantum mechanics without complex numbers. They represent something real about the universe. Now, wait, wait, wait, what are you saying? Are you saying, like, somehow the universe or particles are themselves imaginary or that they sort of operate in a two-dimensional space?

We don't know if the wave function is real, if it, like, actually exists in the universe. What we do know is that the mathematics needed to describe what particles do in which direction they go has to have complex numbers in it. Like, that math just does not work without complex numbers. So in our models, in our calculations, the wave function, that describes all these particles have complex values in them. Now, when you predict the outcome of an experiment, it always just gives you actual numbers or real numbers that you can measure.

The particle would be at this location at that time. But the intermediate states, the calculations we do between the observations require complex numbers. Does that mean that the universe is complex, that there are imaginary components to it? We don't know. Deep question for philosophy.

Well, the universe is definitely complicated, whether it's complex or real or imaginary. maybe is the topic for another podcast. But here today we're talking about maybe a different kind of imaginary number, one that is maybe eight times more imaginary than an imaginary number. So let's dig into that and see whether or not it can help explain more of the universe that we see around this and what would that mean? But first, let's take a quick break.

LaGuardia Airport. The holiday rush, parents hauling luggage, kids gripping their new Christmas toys. Then, at 6.33 p.m., everything changed. There's been a bombing at the TWA terminal. Apparently, the explosion actually impelled metal glass. The injured were being loaded into ambulances. Just a chaotic, chaotic scene. In its wake, a new kind of enemy emerged. and it was here to stay.

Terrorism. Law and order criminal justice system is back. In season two, we're turning our focus to a threat that hides in plain sight. That's harder to predict and even harder to stop. Listen to the new season of Law and Order Criminal Justice System on the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts. My boyfriend's professor is way too friendly, and now I'm seriously suspicious. Well, wait a minute, Sam, maybe her boyfriend's just looking for extra credit.

Well, Dakota, it's back to school week on the OK Storytime podcast, so we'll find out soon. This person writes, my boyfriend has been hanging out with his young professor a lot. He doesn't think it's a problem, but I don't trust her. Now, he's insisting we get to know each other, but I just want her gone. Now, hold up. Isn't that against school policy? That sounds totally inappropriate. Well, according to this person, this is her boyfriend's former professor, and they're the same age. And it's even more likely that they're cheating. He insists there's nothing between them.

I mean, do you believe him? Well, he's certainly trying to get this person to believe him because he now wants them both to meet. So, do we find out if this person's boyfriend really cheated with his professor or not? To hear the explosive finale, listen to the OK Storytime podcast on the IHeart Radio app, Apple Podcasts, or wherever you get your podcast. Hey, sis, what if I could promise you you never had to listen to a condescending finance, bro, tell you how to manage your money again. Welcome to Brown Ambition. This is the hard part when you pay down those credit cards. If you haven't gotten to the bottom of why you were racking up credit or turning to credit cards, you may just recreate the same problem a year from now. When you do feel like you are bleeding from these high interest rates, I would start shopping for a debt consolidation loan, starting with your local credit union, shopping around online, looking for some online, online lenders because they tend to have fewer fees and be more affordable. Listen, I am not here to judge. It is so expensive in these streets.

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which are maybe extra imaginary numbers that might describe how the universe works at the particle level. We're digging into that here today. Now, Daniel, you're saying that imaginary numbers play a big part in describing how things work at the quantum level. Exactly. We need complex numbers in order to do quantum mechanics. Like if they hadn't been invented already early this century when we were developing quantum mechanics, we might have invented them then. As it is, they already existed in the sort of mathematical toolbox and

so we could just pluck them out and say, oh, this is helpful. Let's use it. Well, was that a big surprise? Like, wait a minute. What does that mean about the universe that you need imaginary numbers to describe it? Yeah, it has deep consequences. is for mathematical philosophy. You know, you can wonder if the wave function is real, does that mean that imaginary numbers are real in some philosophical sense, even if we don't call them real numbers in a mathematical sense? We don't know what that means.

But imaginary numbers are also important in mathematics itself. It tells us something about how numbers work, even before they were applied to quantum mechanics. Oh, interesting. So there may be something, point to something fundamental about the nature of the universe. Yeah, perhaps. because complex numbers are not just like something mathematicians invented.

You can't just say, well, I'm going to take two numbers and stick them together. Now I've got a new kind of number. When you do that, you also have to decide, like, what are the rules of those numbers? How do you add them? How do you subtract them? What happens when you multiply the two numbers, et cetera? So you have to come up with what mathematicians call a division algebra, which is basically just like all the rules of how the math works for these numbers.

And it's not always easy to come up with that system. For example, you can do it for complex numbers, but you can't do it for triplets of numbers. So it works for single numbers. It works for pairs of numbers. It doesn't work for triplets of numbers. Okay, wait, hold on. And I think you're talking about now increasing the imaginariness of a number.

So like a regular imaginary number has a real component. And then an imaginary component, which you get by multiplying another number by the number I. So like an imaginary number is seven plus eight I. Now you're talking about like adding a third component. Yeah, call it J. Right. J is like another square root of negative one, a different, a unique square root of negative one. Nothing says you can't have more, right? Maybe there's multiple square roots of negative one. And after people discover complex numbers, mathematicians like, oh, that's cool. Can you do the same thing with three numbers? If you put three numbers together,

like a real number and then some number of I and some number of J, can you also then make like consistent mathematics so you know how to multiply, divide, subtract, et cetera. People spend hundreds of years on that. Let me think about that for a second. So now we have a number plus a number I plus a J number. And you're saying J is another square root of a negative one. So if I multiply J times J, I get negative one. What if I multiply I times J?

So this is exactly what you have to do. You have to come up with the rules of this triplet system. You get to figure out what happens when you multiply different numbers together. What happens when you divide them? and it turns out there's no consistent way to define multiplication and division so that it all makes sense and keeps the mathematicians happy if you have i and j in addition to the real numbers like you cannot build a mathematical system based on numbers with three components you can do it with one that's just the normal numbers you can do it with two that's just the complex numbers you can't do it with three but i feel like when you said before like having two numbers or like an real and an imaginary it's sort of like an X and a Y. Isn't this just like having X, Y, and Z? Yeah. You might imagine you should be able to do with any set of numbers exactly the way you can define spaces in any dimensions. You can have a one dimensional space, a two dimensional space, a three dimensional space, a 19th

dimensional space, an infinite dimensional space. Geometry has no restrictions, right? But the rules of mathematics, for some reason, constrain the number of numbers we can pack in together and still have everything makes sense. Mathematicians worry about like, does every number have an inverse If I take a number, is there always some other number I can multiply it by to get one? Are there unique zeros or not? And it turns out that mathematics based on triplets just doesn't work. There's no way to put it together. This is famous Irish mathematician named William Rowan Hamilton who spent like decades on this.

And he said once every morning coming down to breakfast, my kids ask me, well, Papa, can you multiply triplets? And he always said, nope, I can only add and subtract them. So we spent decades like trying to figure out how to multiply and divide triplets of numbers and never succeeded. Whoa. But XYZ space works in 3D space all around this. Are you saying that 3D space doesn't work mathematically or just triplet imaginary numbers don't work mathematically? Triplit imaginary numbers don't work mathematically. Like you might think, well, why can't you just put three numbers together and then say, you know, multiplying two triplets means multiplying the components, right?

I don't think we want to get too mathematical on the podcast today, but this creates problems of having numbers that don't have inverses. Like 1 comma 1, 0 doesn't have an inverse, but it's non-zero. Okay, I think I see what you're saying. I think you're saying it's possible to just have like coordinates in 3D space, like X, Y, and Z. But if you want to call it like X plus Y, I, plus Z, J, that doesn't work. Like if you want to put them all together as one number with an addition sign between the different coordinates, that doesn't work. It doesn't work if you also require that you can do things with these numbers.

Like, can you take any of these two numbers and multiply them together and still get a number? If you add them together, do you still get a number? We rely on that for the normal numbers, right? Any number you give me, I can always find another number to multiply it by to get one. Or if you give me two numbers, I multiply them together, I always get a number that's not a zero. If both of the numbers you gave me were not zero. You can't build rules like that, which you need to do any interesting math or any interesting physics. You can't build rules like that for triplets.

You can do it for singles and for pairs, but not for triplets. Okay, so it doesn't work for three triple imaginary numbers, but maybe for four imaginary numbers, it does work. Exactly. And the same Irish mathematician, William Hamilton, he discovered that if you put four numbers together, like 4D space, one reel and three imaginary numbers, you call these quaternions. This actually works that the mathematics hangs together. You can build multiplication and division. You can do all the mathematics you need on quadruples of numbers, even though. you can't do it on triplets.

Okay, so now that means that my number is not just like eight, and it's not just like eight plus four I, it's like eight plus four I plus seven J plus five K. Exactly. And in that system, I squared is negative one, J squared is negative one, K squared is negative one, and if you multiply I times J times K,

you also get negative one. And that's the key. J and K are other square roots of negative one. negative one. And that one equation makes it all hang together. And when Hamilton had this insight, he was like walking across this bridge in Dublin. It like came to him in a flash. And he actually chiseled the formula onto the bridge like mathematical graffiti in the moment because he didn't want to forget it. And you can still see it today? It's actually worn down, but they put a plaque on that spot to commemorate it. And more important, was this kid impressed? Because the kid seemed

really interested in what was going on with the math and his father. Yeah, I think he was just rooting for his dad, you know, wondering why his dad was struggling with this crazy multiplication. And how much can a kid know about imaginary numbers? It's fascinating that the mathematics of it tells us what's allowed. You can build a 1D number, a 2D number, a 4D number, but not a 3D number. It's really interesting. You go 5 and 6 and 7?

You can't. About 100 years later, mathematicians pretty. that the only sets of numbers you can do are one, real numbers, two complex numbers, four quaternians, and eight octonians. And can you keep going, like 16, 32? Nope. Those are the only ones. What do you mean?

Everything else runs into mathematical problems. You can't have consistent multiplication and division without running into ugly problems with the zeros. It only works for one, two, four, and eight. That's it. That's it. Nothing more in the universe. they prove this.

And I try to find a way to explain this in intuitive sense, but the proof is very, very complicated. But everybody's totally convinced that 1, 2, 4, and 8 are the only dimensions allowed for numbers that have consistent multiplication and division. You mean for like complex numbers? Yeah, complex numbers. With like extra imaginary dimensions. Exactly.

You could have one extra imaginary dimension, which gives you our familiar complex numbers of 2D. You can have three imaginary dimensions, which gives you four. 40 numbers, quaternians, or you can have seven imaginary dimensions, plus one real gives you eight dimensions for octonians. And still be consistent with our rules of math that we know about in our universe. Could those be different in another universe?

You can come up with whatever rules of math you want, but if you want to do physics with it, you need to know how to multiply the numbers, how to divide them. You need some confidence that you're not going to like run into zeros all the time. So these are like pretty basic requirements for mathematical. system that's going to underlie physics. You can invent math however you like and have it be useless or useful or whatever, but if you wanted to do physics, you have to follow these basic rules. Okay, now, I think the idea is that, you know, we had real numbers for a long time

and they worked great for regular physics, and then we found quantum mechanics and particles, and we figured out that one-dimensional imaginary numbers worked really well to describe the math and those theories where you have one extra imaginary dimension. But you're saying mathematicians also found out that you can have three extra imaginary numbers or dimensions in a quaternian or seven extra imaginary dimensions in an octonian. And now I think maybe what you're saying is that scientists are wondering, do these extra super numbers also maybe describe something about the universe? Exactly. Because obviously, 1D numbers are very relevant to physics. 2D numbers are very relevant to quantum mechanics.

These 4D numbers, you probably haven't heard of Quaternians, but these days we actually just call them four-dimensional vectors, and they're crucial for relativity, like special relativity combined space and time into a four-dimensional structure where three of them are similar to each other and one of them is different, right? Doesn't that sound familiar? Sounds like space-time. It sounds like space-time, exactly. And Quaternians have exactly that structure, three imaginary numbers and one real. So the real numbers like time. and the three imaginary numbers are like three dimensions of space.

And so these quaternians, the rules of space time vectors, we call them four vectors. We call them four vectors now instead of quaternians are absolutely crucial to relativity. Meaning you can only do relativity math if you're using quaternions or it's just helpful to use them. No, they're fundamental to special relativity. Like you can't prove relativity without going into quaternians. Exactly. If we didn't already have quaternians, we would have needed to an. invent them or discover them, depending on your philosophical take, when we built relativity.

The same way we absolutely need complex numbers to describe quantum mechanics. So all the self-consistent ways we know how to build complex numbers so far are very, very useful for physics. And it turns out there's a limited number of these sets. You can't just like make up any number. So if one, two, and four are super valuable for physics, then maybe eight is also. Now you need four for relativity. Does that also work if you call them phornians?

Or funnians? They're a little crunchier, but they still work. All right, I'm just kidding. So you're saying now the question is like, do numbers with seven extra imaginary dimensions also maybe could they be used for something in physics? That's exactly the question. And it's worth thinking about because we've made progress so many times when we've developed some mathematical tool, group theory, field theory, complex numbers. and then later found it applicable to physics. So let's like get ahead of the game.

Let's say, oh, here's a kind of mathematical tool which has been useful, complex numbers of dimensions one, two, and four. If there's only one more kind of complex number out there, let's see if maybe it tells us something about physics. Maybe it's the mathematical structure to understand some other patterns that are out there that we didn't have explanations for. But does it seem kind of random to you? Like, you need one X-ray imaginary dimension to explain quantum physics.

that's at the microscopic level. But then you need three extra imaginary dimensions to describe relativity, which is usually at a macro level. Isn't it kind of random? Like you need four here, eight there, two here? Nobody understands it. Nobody has an intuitive understanding for why,

one, two, four, and eight. Like there is this proof that shows that these are definitely the only ones you can do. Nobody really knows philosophically what it means. But it's very suggestive, right? If 2D numbers are needed for quantum mechanics, 4D numbers are needed for relativity, well, we've spent the last 100 years looking for quantum mechanical relativity. Maybe octonians are the key to that, right?

2 times 4 equals 8, after all, maybe the mathematical structure we need to understand quantum gravity is based on octonians. But why go to octonians? Has anyone tried to apply quaternians to like quantum gravity? Oh, absolutely. Nobody has a consistent theory of quantum gravity. that works. And every time we try to write down

the theory of quantum gravity, it breaks. Mathematicallyly, it does not work. So it might just be that we're using the wrong kind of number. Oh, I see. Like maybe the key to it all is just to go to Applebee's

and order some octonians. Absolutely. It might be. Well, I mean, that is a joke. Like, I think that's what you're saying. It's like, why don't we order a Nectonian and see if it satisfies or it helps us combine these theories of physics.

Yeah, absolutely. Because remember that there's lots of ways to make progress in physics. When you don't understand the patterns you're looking at, one way is to see more of the pattern. You're like, well, what are all these particles? Why do we have these particles, not other particles? One strategy is discover more particles. That's what I'm doing.

Building colliders, smashing stuff together, looking to see more of the pattern, hoping that if you see more of it, you'll figure out what's going on. Another approach is not to discover anything new, but find new mathematical structure. structures that naturally describe the patterns you're seeing. For example, that's how we thought up the Higgs boson.

It wasn't just discovered. It was thought up by looking at the patterns that exist already in the particles that we had, coming up with a new mathematical structure to describe that group theory and the unification of electromagnetism and the weak force and realizing that that needed a Higgs boson. And so there's a lot of progress to be made in just coming up with a new mathematical way to look at the patterns we already see. Interesting. Like, that's a way to do science, like try to look for patterns and then come up with ideas that might, you know, make sense of those patterns and then go back to the experiments to see if those theories are true.

Exactly. Because mathematics has its own rules, right? You can't just invent whatever mathematics you want. Mathematics is telling us very clearly, hey, guys, there's only four kind of numbers you can use, by the way, if you want to describe physics. And that seems like a pretty big clue. It's like, well, if we only have four tools, let's make sure to use all of them, right? especially if the first three were so absolutely useful. So even if we haven't yet figured out what they applied to, it's worth thinking about what they might be applied to. Interesting. Okay, so I think what you're saying is that maybe like if we take the universe

and we peel back its layers one by one and try to get to the core of it, maybe at the center of it is an octonian. Exactly. An octonian could be the center of the onion, eight layers down. The onion, yeah, there you go. I see where you're walking me to, yes, absolutely. All right. Well, let's get into what these ideas are about how Oktunians might describe what's inside of the universe, what's at the core of it. How does it all work? How does it make sense mathematically? Will these theories work? Let's dig into that. But first, let's take another quick break. parents hauling luggage, kids gripping their new Christmas toys.

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Welcome to Season 2 of the Good Stuff. Listen to the Good Stuff podcast on the IHeart Radio app, Apple Podcasts, or wherever you get your podcast. All right, we're talking about Octanians, and are they the imaginary numbers that scientists have been dream? dreaming about to explain everything in the universe. So then you were saying that the number of the one imaginary number really help describe quantum mechanics. Numbers with three imaginary numbers really help describe special relativity and relativity.

And because it lets you do math with space time, three dimensions of space and one of time. Now you're saying that maybe if you want to combine these two theories, quantum mechanics and special relativity, maybe you need an extra. for imaginary number, which might be the octonian. Yeah, it might be exactly. And we try to figure out where this might be relevant to the universe. And one thing we can do to figure that out is to look at the properties of octonians. Like, what do octonians do that quaternians and complex numbers and real numbers don't do? How are they different?

And that's one fruitful way to understand where they might be relevant. What do you mean? Like what can they do? That's special. So it's more about what they can't do. Every time you add more complex numbers, you sort of lose some ability. Like within real numbers, you can order them, right? You can say this number is bigger than that number is bigger than that number.

They have a very well-defined order. But once you go to complex numbers, you can't necessarily say that. Like, which number is bigger? One or I, they have like the same magnitude, but you can no longer rank them necessarily. So as you add complex dimensions, you actually lose some capabilities. Things get crazier kind of, right? Like it's kind of hard to order things in a, in a, plane or in an xy plane. Exactly. And then when you go to quaternions, basically space time vectors,

what you lose is commutation. The fact that like A times B is usually equal to B times A. But that's not true for four dimensional vectors because multiplying numbers in four dimensions is like rotation. It's like taking a vector and turning it. And in four dimensional space, rotations don't commute. It like matters what order you do rotations in. If you first turn left and then you turn up, get to a different place than if you do it in the other direction, for example. Wait, wait. I thought that you said that Quaternians and Elktonians work because they follow all the math rules. You're saying now that maybe they sort of work, but you kind of have to give up some math rules. Exactly. These math rules that we require don't include that they commute.

They only include that you can multiply and divide and add and subtract. Commutation is not a requirement. It turns out that as the numbers get more complex, you lose some of the properties we're familiar with of real numbers. So complex numbers, you lose the ability to organize. them. Quaternians, you lose the ability to multiply them in any order you want. A times B is not the same as B times A. So that's really interesting. It tells us something about the structure of space time, right? That this is really about space. Now, octonians, even weirder. Octonians, what you lose is the associative property. The associated property tells us basically you can distribute numbers within a parentheses like, do you do the multiplication within the parentheses first or do you distribute

the number through the parentheses? Sort of hard to think about the associated property of math, intuitively, in a physical sense, it would be sort of like, you know, you're used to putting your socks on and then your shoes. What if you first put your socks in your shoes and then put your feet into the socks? You'd end up in the same place, right? It doesn't really matter if you put the socks on first or put the socks in the shoes. But for Octonians, it does matter. The associated property doesn't hold for Octonians. I see.

You lose another math rule. And you also lose the ability to order them and you also lose the ability to like commute them. you said. Exactly. So that's a clue, right? It tells us that maybe the fundamental nature of the universe doesn't respect this associative property. Maybe parentheses actually do matter to whatever mathematics really rules the universe. That's just like a clue. And then we go off hunting in the physical world for maybe something that's like that. Is there some theory of physics that we can build that doesn't have this requirement of associativity of our numbers? Maybe that's the right direction. It's a very, very vague clue, but that's the kind of clue we can get from looking at

the structure of the mathematics of octonians. Oh, I see. Because the pattern is that every time you have one of these imaginary number sets, their limitations somehow correspond to a rule or a feature of the universe. Like when you had one extra dimension and you can't order those numbers, that corresponds to something special about quantum particles. Or when you have a quaternion and you can't commute them. That means something special about how you can rotate or not things in space. Exactly. So what does this mean about the universe that Octonians don't respect the associative property? It's very strange. It's very unphysical. We have like no intuition for it. What it tells us is that a theory of the universe built on Octonians is going to be very counterintuitive.

Wait, doesn't it just mean you can't put on your socks and your shoes at the same time? It means it makes a difference whether you put your socks in your shoes and then put them both. on or put your socks on and then your shoes. It would make a difference for sure. I'm not even sure how that would work. I mean, I think some people put their underwear and their pants on at the same time, don't they? I don't know. Oh, well, why didn't you start with that analogy?

I didn't want to talk about underwear, I guess. So you're saying that Quaternians, you can't do that. You can put your underwear inside of your pants and then put them on both at the same time. You can for Quaternians. You can't for Octonians. That's right. I'm sorry. You can't for octonians. You're asking, like, is there a rule in the universe that somehow corresponds to that limitation with your pants? Exactly. The underlying rule of the universe is somehow related to underpants. We don't know. Yeah, there you go. Or do you end up looking like Superman with an underwear outside of your pants?

Exactly. Maybe he's the original octonian. That's right. Maybe you need cryptonians to really describe the universe. We don't know, but in the meantime, physicists are on the hunt for eights. We're looking for things out there in nature, patterns which include the number eight. And there are some tantalizing hints. You know, for example, string theory, we know, likes to exist in ten spatial dimensions because the mathematics of how those things compactify works best in ten dimensions. So some people have said, oh, well, that's sort of like eight spatial dimensions and one-time dimension

plus one dimension for like along the string. And so there's kind of an eight there. But 10, you said, didn't work. Why are they fixated with 10 if 10 doesn't hold mathematically? Because string theory works in 10 dimensions. There's a different set of rules there mathematically. String theory works in 10. Octonians work in 8.

Can we somehow marry the two? Well, maybe if we take those 10 dimensions. You split the difference and make it nine? There's no compromising in math. It is not a negotiation with the universe. He can't split the difference. Well, I can't make quantum gravity. work? How about we'll split the difference?

I'll give you points on the back end, all right? How about that? That's right. Yes. We'll only make both them half work. No, people are like, well, can we take eight of those 10 string dimensions and say those are described by octonians? And then another one of them is time. And another one is a string dimension. It's a real stretch. But people are looking for eightishness somewhere out there in the patterns and wondering if it can be described by octonians. Can you just decrease the number of dimensions in string theory? Or it wouldn't work anymore. No, it doesn't work. It only works in 10 dimensions. There's another version of it that

works in 26, but string theory doesn't work in eight dimensions. Another direction people are going in is to try to describe the strong force. The strong force is something we really don't understand very well, but it has eight gluons, right? There's the number eight. So maybe octonians would be a better way to describe the strong force. Maybe the whole reason we have trouble doing calculations in quantum chromodynamics is because we're using the wrong kind of math. And it would all just like click beautifully into place if we replaced it with octonians. Wait, what do you mean? There are only eight gluons.

You're just looking for things in nature that you know about so far that somehow counted eight. Yeah, exactly. And there are eight kinds of gluons because remember that there are three different colors in the strong force, red, green, and blue. And each gluon carries two colors. And so the way the mathematics works is that it's three squared minus one. You get eight different kinds of gluons that you can have. So yeah, we're just looking for things in nature that have eight in them. Like what about the universe is eightish?

Can we describe it with octonians? So far, nobody's made it work. Octonians mathematically beautiful, mathematically consistent. So far, totally physically useless. Like what did you decide to try to make quantum gravity work with octonians? Where does that put us? Aren't there like eight dimensions or things like that in between the two of them? Like the wave function plus the space time?

Throw in some oranges and a couple of socks. Yeah, exactly. And no, people are working on that, right? A lot of people studying the nature of Octonians and trying to use them to build a physical theory, but nobody's succeeded so far. It's just sort of like a direction some people are sniffing in as we try to build mathematically consistent theories of quantum gravity. Wow. You're just looking for the number eight. Yes, exactly.

We're desperate for clues because this is the biggest unsolved problem in physics, how to marry quantum mechanics and relativity. We haven't figured out in a hundred years. So we're like digging deep in the mathematical toolbox to be like, well, what else we got in here? Let's see what else could be useful. Well, I can't seem to get up before 8 a.m. Could that be a rule of the universe? I could blame that. Well, we could flip the blame.

We could say nobody's solved this problem because you won't get up before 8 a.m. Well, that would still make sense. That's still an explanation for the universe. Or maybe it would take an announcement of quantum gravity to get you out of bed before 8 a.m. unlikely in this universe I don't think that was mathematically hold I think there's YouTube for that I'll just catch it on YouTube later

all right well but it's sort of an interesting idea or direction in which to look for new theories about the universe like maybe you put your octogogles on and look for things that work or that seem to manifest themselves in sets of eight maybe that could be a way to

get to a theory of everything. Exactly. If mathematics really does reflect something deep about the universe, then mathematicians building tools can actually construct sort of like proto bits of future physics theories that we could just like click into place. We've done it before with group theory and field theory and differential geometry. So we hope it happens again. Right. And that's kind of what's happening with strength theory, although that one seems to be following the rule of 10. Yeah, or the rule of 26. So it sounds like maybe physicists just need to sit down and peel more onions. And stop crying.

That's right. Pretend the tears come from cutting these onions and not from being a physicist, and maybe that will inspire some new theory. Maybe it'll crack open. They'll peel away the true truth of the universe. Or maybe we should just take a break and go to Applebee's and have a blooming onion. Yeah, there you go. There are different ways.

As long as it's after 8 a.m., I'll join you. All right. Well, an interesting dive into mathematics and how it, matches up with physics and how it's helping us maybe understand how it all works or at least how it has helped us and maybe could help us in the future. And future mathematicians and physicists might look back and say, oh, it was so obvious how it all clicked into place. But here we are at the forefront of human ignorance, just not seeing it come together. All right. Well, we hope you enjoyed that. Thanks for joining us. See you next time.

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criminal justice system on the iHeart radio app apple podcasts or wherever you get your podcasts my boyfriend's professor is way too friendly and now i'm seriously suspicious wait a minute sam maybe her boyfriend's just looking for extra credit well dakota luckily it's back to school week on the okay story time podcast so we'll find out soon this person writes my boyfriend's been hanging out with his young professor a lot he doesn't think it's a problem but i don't trust her now he's insisting we get to know each other but i just want her gone Hold up. Isn't that against school policy? That seems inappropriate. Maybe find out how it ends by listening to the OK Storytime podcast and the IHeart Radio app, Apple Podcasts, or wherever you get your podcasts.

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