Latest / Daniel and Kelly’s Extraordinary Universe / What are octonions?

Transcript

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

• 0:08
  Speaker 1: Hey, Daniel, what are your favorite numbers in physics?
• 0:11
  Speaker 2: Oh? There are so many good ones. I mean I
• 0:14
  Speaker 2: love planks constant, which tells us about like when things
• 0:17
  Speaker 2: become quantum. I love the speed of light that tells
• 0:21
  Speaker 2: us all about relativity. Too many to choose from. There
• 0:24
  Speaker 2: are a number of favorites there.
• 0:26
  Speaker 1: But have you ever wondered why numbers are so important
• 0:30
  Speaker 1: in the universe?
• 0:30
  Speaker 2: I mean, numbers are like the currency of physics. We're
• 0:34
  Speaker 2: predicting things happening at times and places, and those are
• 0:37
  Speaker 2: just numbers.
• 0:38
  Speaker 1: Wow. So does that mean you get paid in numbers?
• 0:42
  Speaker 2: I get a number of dollars every year.
• 0:44
  Speaker 1: I guess we all get paid in numbers if there's
• 0:46
  Speaker 1: just numbers in our bank account. If you're so lucky,
• 0:50
  Speaker 1: the whole universe is just numbers. I think you said
• 0:52
  Speaker 1: that a number of times. I'm the number one fan
• 0:56
  Speaker 1: of numbers. I think I know a number of those
• 0:59
  Speaker 1: in physics and math. Too many to number. Hi. I'm
• 1:17
  Speaker 1: Jorge Mack, cartoonists and the author of Oliver's Great Big Universe.
• 1:21
  Speaker 2: Hi, I'm Daniel. I'm a particle physicist, and I've been
• 1:24
  Speaker 2: a professor at UC Irvine for a large number of years.
• 1:27
  Speaker 1: Oh, too many to count? Or do you go into
• 1:30
  Speaker 1: some sort of like weird subspace when you do physics.
• 1:34
  Speaker 2: No, I think I'm going to enter a puba phase
• 1:36
  Speaker 2: eventually and then emerge as an emeritus professor.
• 1:39
  Speaker 1: That's a butterfly. That's a beautiful tenured emeritus.
• 1:43
  Speaker 2: Butterfly, as a white haired moth.
• 1:45
  Speaker 1: I think instead of wings, do you just have a
• 1:49
  Speaker 1: bunch of research papers taped together.
• 1:53
  Speaker 2: I'm going to fly too close to the universe on
• 1:55
  Speaker 2: wings of research papers.
• 1:56
  Speaker 1: Right, But you know, butterflies don't have mouths, right, so
• 1:59
  Speaker 1: you won't be to talk.
• 2:00
  Speaker 2: Yeah, but they have those long newses, and so I'm
• 2:03
  Speaker 2: already said.
• 2:04
  Speaker 1: There you go. You can sniff out science from there.
• 2:09
  Speaker 1: M hmm.
• 2:10
  Speaker 2: I can snort up all the information in the universe.
• 2:12
  Speaker 1: Oh boy, that sounds a little illegal.
• 2:15
  Speaker 2: Are you going to arrest a butterfly? Is that what
• 2:17
  Speaker 2: has come to?
• 2:18
  Speaker 1: Maybe a cocaine sniffing butterfly. Maybe maybe it will be
• 2:22
  Speaker 1: our informant for, you know, figuring out how the universe works.
• 2:26
  Speaker 2: Maybe altering your mental state is important for understanding the universe.
• 2:30
  Speaker 1: M I hear that's called the butterfly effect.
• 2:33
  Speaker 2: I think that's something else.
• 2:35
  Speaker 1: Meant something else. Did I get that wrong.
• 2:39
  Speaker 2: I think it's called microducing.
• 2:40
  Speaker 1: Yeah, I think this is called micro punning, which it
• 2:43
  Speaker 1: doesn't work as well. But anyways, welcome to our podcast
• 2:46
  Speaker 1: Daniel and Jorge Explain the Universe, a production of iHeartRadio.
• 2:49
  Speaker 2: In which we give you a drip, drip, drip, little
• 2:52
  Speaker 2: doses of the incredible wonder we discovered about the universe,
• 2:55
  Speaker 2: everything that's out there that makes sense, and everything out
• 2:58
  Speaker 2: there that still puzzles us as we try to fit
• 3:00
  Speaker 2: the entire universe into a pattern that makes sense to
• 3:04
  Speaker 2: our little human minds.
• 3:06
  Speaker 1: That's right, We try to get you high on the
• 3:08
  Speaker 1: little dopamin head of wonder and amazement at how our
• 3:11
  Speaker 1: universe works and the surprising ways in which it still
• 3:15
  Speaker 1: puzzles scientists even today.
• 3:16
  Speaker 2: And one thing that still puzzles scientists and philosophers is
• 3:20
  Speaker 2: at any of it makes sense that these mathematical models
• 3:23
  Speaker 2: we build on our head can actually describe and even
• 3:26
  Speaker 2: predict what's going to happen out there in the universe.
• 3:30
  Speaker 2: That somehow mathematics seems to be not just the currency
• 3:33
  Speaker 2: of our physics, but the currency of the universe itself.
• 3:36
  Speaker 1: Wait, wait, are you saying math is more important than physics.
• 3:41
  Speaker 2: I mean, is English more important than Shakespeare. You can't
• 3:44
  Speaker 2: really compare the two things, you know.
• 3:47
  Speaker 1: Wait with this analogy, Shakespeare's math in English is physics.
• 3:51
  Speaker 2: No, Shakespeare's physics and English is math. Physics is speaking
• 3:54
  Speaker 2: in the language of mathematics. We are writing poetry in
• 3:57
  Speaker 2: the language of math.
• 3:58
  Speaker 1: I see. I see. So physics a comedy or a tragedy.
• 4:02
  Speaker 2: It's a tragic comedy, for sure.
• 4:04
  Speaker 1: It's a tragic comedy.
• 4:06
  Speaker 2: We're all waiting for the final twist. Nobody knows how
• 4:09
  Speaker 2: it ends.
• 4:09
  Speaker 1: That's right, it never ends well for Shakespeare.
• 4:12
  Speaker 2: But I hope we laugh along the way.
• 4:14
  Speaker 1: And you also have to wear tights when you do physics.
• 4:17
  Speaker 3: Mmmm.
• 4:17
  Speaker 2: And as always, the jester is the wisest one.
• 4:21
  Speaker 1: I see. That's the engineer and the team.
• 4:24
  Speaker 2: Did you just call engineers jokers? I mean, I'm just
• 4:27
  Speaker 2: gonna step slowly away from that.
• 4:29
  Speaker 1: No. I know. We have a great sense of humor,
• 4:30
  Speaker 1: is what I'm saying. We're the smaltest person in the room.
• 4:35
  Speaker 2: Often laughed at and unwisely ignored.
• 4:37
  Speaker 1: But yeah, it seems like there are a lot of
• 4:38
  Speaker 1: numbers in physics. There's an infinite number of numbers in math,
• 4:41
  Speaker 1: let's face it. But in physics it seems like there
• 4:43
  Speaker 1: are special numbers out there that have kind of a
• 4:45
  Speaker 1: special status because they're sort of significant in how the
• 4:48
  Speaker 1: universe works.
• 4:49
  Speaker 2: Yeah, there are certain constants which seem to be important
• 4:52
  Speaker 2: to tell us something about the universe, the speed of light,
• 4:55
  Speaker 2: planks constant, the gravitational constant. There are even numbers that
• 4:59
  Speaker 2: don't have units, you know, like the number of particles
• 5:02
  Speaker 2: we've discovered or ratios of masses that we think reveal
• 5:05
  Speaker 2: something deep about the universe. But even more than that,
• 5:08
  Speaker 2: there are systems of numbers. There are patterns of numbers
• 5:12
  Speaker 2: that reflect patterns we see in the universe, mathematical constructs
• 5:17
  Speaker 2: and all sorts of fancy mathematics that really are crucial
• 5:20
  Speaker 2: to understanding how the universe operates.
• 5:23
  Speaker 1: And this is kind of especially true at the microscopic level, right,
• 5:26
  Speaker 1: That's where a lot of these numbers and a lot
• 5:27
  Speaker 1: of this math comes from, and it comes from our
• 5:30
  Speaker 1: understanding how things work at the particle level.
• 5:33
  Speaker 2: Math an interesting point. I would say that math describes
• 5:36
  Speaker 2: the universe at every level. You know, we have mathematics
• 5:38
  Speaker 2: for fluid dynamics and for planetary evolution and for the
• 5:41
  Speaker 2: expansion of the whole universe. It's mathematics up and down.
• 5:46
  Speaker 1: Okay, so it's at every level, but it seems like
• 5:48
  Speaker 1: in the Standard model there are especially some interesting numbers.
• 5:51
  Speaker 2: Yes, absolutely, the theories of particle physics are very mathematical,
• 5:55
  Speaker 2: and not just in the sense that they're predicting numbers
• 5:58
  Speaker 2: and places in time. But the patterns we see in
• 6:02
  Speaker 2: the Standard Model use complex mathematical theories like group theory
• 6:06
  Speaker 2: and field theory and all sorts of like heavy hitting
• 6:08
  Speaker 2: mathematical apparatuses.
• 6:09
  Speaker 1: Right, which sort of eraises a question of whether the
• 6:12
  Speaker 1: universe itself is mathematical. Like maybe if you dig down
• 6:15
  Speaker 1: deep enough into the stuff we're all made out of
• 6:17
  Speaker 1: at the end, maybe we're just mathematical equations or formulas.
• 6:21
  Speaker 2: Yeah, and it makes you wonder, like what does that mean?
• 6:24
  Speaker 2: And our numbers real the way like stuff is real?
• 6:28
  Speaker 2: You know, like where is the number two? If two
• 6:30
  Speaker 2: is a physical thing, does it exist somewhere in the universe.
• 6:34
  Speaker 2: There's a whole fascinating branch of philosophy. But like how
• 6:37
  Speaker 2: we learn things about numbers because you can't like do
• 6:40
  Speaker 2: experiments with the number two and the number seven. It's
• 6:42
  Speaker 2: all sort of mental games.
• 6:44
  Speaker 1: Right, right, Like maybe the universe is not really physical,
• 6:47
  Speaker 1: maybe it's just sort of like conceptual theoretical.
• 6:50
  Speaker 2: Yeah, but then you wonder, like what breeds fire into
• 6:52
  Speaker 2: all those concepts it makes us experience it. The other
• 6:56
  Speaker 2: side of that argument is that numbers are not universal,
• 6:59
  Speaker 2: they're not physics, they're not natural. They're just sort of
• 7:02
  Speaker 2: the way that our human mind works. That you can
• 7:05
  Speaker 2: use numbers as a way to describe the universe, but
• 7:07
  Speaker 2: it doesn't mean they're part of the universe. The way
• 7:10
  Speaker 2: you can like describe the color orange using a bunch
• 7:13
  Speaker 2: of words, but none of those words are orange or
• 7:16
  Speaker 2: fully capture the oranginess of an orange.
• 7:19
  Speaker 1: Mm and is one really the loneliest number? Is another
• 7:23
  Speaker 1: big physics question, right.
• 7:25
  Speaker 2: Yes, yes, a very deep physics question. But we have
• 7:28
  Speaker 2: to see a lot of value in math. There's lots
• 7:31
  Speaker 2: of times when mathematicians have developed some cool little technique
• 7:35
  Speaker 2: just for fun, because they see cool patterns and they
• 7:37
  Speaker 2: like playing with them, and then later on physicists will
• 7:39
  Speaker 2: come along and be like, hey, that looks useful, and
• 7:42
  Speaker 2: just like pluck it out of their hands and go
• 7:44
  Speaker 2: insert it into our physics equations and get great insights.
• 7:47
  Speaker 2: There's a lot of hints there that the laws we
• 7:49
  Speaker 2: use to describe the universe are deeply mathematical.
• 7:52
  Speaker 1: Yeah, it seems like there are a lot of new
• 7:54
  Speaker 1: ideas coming up all the time about how to explain
• 7:56
  Speaker 1: the patterns that we see in the universe, maybe with
• 7:58
  Speaker 1: new kinds of math and so to be. On the podcast,
• 8:01
  Speaker 1: we'll be asking the question, what are octonions? Did I
• 8:11
  Speaker 1: say that right? Or is it actonions?
• 8:13
  Speaker 2: I think it's oct onions, Like, give me eight onions
• 8:16
  Speaker 2: for that recipe, please.
• 8:17
  Speaker 1: Isn't it like a blooming onion? Isn't that a dish
• 8:20
  Speaker 1: in a fried dish in one of these famous restaurants?
• 8:24
  Speaker 1: Is this like a fried onion that looks like an octopus?
• 8:28
  Speaker 2: Are you shilling for Outback Steakhouse? Now?
• 8:30
  Speaker 1: Oh? Is it out back?
• 8:31
  Speaker 2: I don't know, never been.
• 8:32
  Speaker 1: Maybe Applebee's is interested.
• 8:36
  Speaker 2: I think it's maybe octonians or octanians, depending on where
• 8:40
  Speaker 2: you are in the world.
• 8:41
  Speaker 1: So this is an interesting word. It sort of sounds
• 8:43
  Speaker 1: like octo, which is eight. Then you're ending it with onions,
• 8:47
  Speaker 1: which is a vegetable or a tuber. But it's also
• 8:52
  Speaker 1: sort of how some particle names ends, right.
• 8:55
  Speaker 2: Mm hmm, yeah, ions exactly.
• 8:57
  Speaker 1: Oh yeah, right right yeah ions. So is it denions?
• 9:00
  Speaker 2: Then I think we should have eight different pronunciations of
• 9:04
  Speaker 2: this word.
• 9:05
  Speaker 1: Now a trace you go, and then we'll spend eighty
• 9:07
  Speaker 1: eight minutes talking about it here on the podcast as.
• 9:11
  Speaker 2: Part of our Cult of eight. There you go, send
• 9:14
  Speaker 2: eight dollars to join.
• 9:15
  Speaker 1: There you go. We'll have eight dollars in our pockets.
• 9:18
  Speaker 1: In our eight pockets because I wear cargo pants exactly.
• 9:22
  Speaker 2: Maybe these are the spiders of the universe.
• 9:23
  Speaker 1: It's all an intricate web of eight. Well, anyways, as usually,
• 9:27
  Speaker 1: we were wondering how many people out there had thought
• 9:29
  Speaker 1: about or even heard of the word octonions and what
• 9:32
  Speaker 1: they think it might mean.
• 9:33
  Speaker 2: Thanks very much to everybody who participates in this audience
• 9:37
  Speaker 2: contribution segment of the podcast. We'd love if you joined
• 9:41
  Speaker 2: the crew. Please write to me two questions at Danielandjorge
• 9:44
  Speaker 2: dot com.
• 9:45
  Speaker 1: So think about it for a second. What do you
• 9:46
  Speaker 1: think are octonions? And are they sold at Applebee's or
• 9:51
  Speaker 1: out back to steakhouse?
• 9:52
  Speaker 2: And would they make you cry if you chop them?
• 9:54
  Speaker 1: Here's what people had to say.
• 9:55
  Speaker 4: What are onions, so octors, the prefix for eight onions
• 9:59
  Speaker 4: being something players, probably so eight layers stars.
• 10:05
  Speaker 3: I don't know, Honestly, I have no idea, but if
• 10:09
  Speaker 3: I were to have to guess, I would deconstruct the word.
• 10:14
  Speaker 3: So I think the prefix oct means it has to
• 10:19
  Speaker 3: do with eight and so maybe it has to do
• 10:21
  Speaker 3: with glue ons or any type of similar particle or
• 10:27
  Speaker 3: force carrier that is dependent on matrices for transformation.
• 10:37
  Speaker 1: All right, well, pretty much the same as what I've
• 10:40
  Speaker 1: been guessing. Something to do with onions and eight and
• 10:43
  Speaker 1: or particles.
• 10:44
  Speaker 2: Yeah, reasonable guess is absolutely.
• 10:46
  Speaker 1: Is it like what happens when you smash together eight onions?
• 10:50
  Speaker 1: You get an ooked onion.
• 10:52
  Speaker 2: We're working on the onion collider right now. We just
• 10:54
  Speaker 2: need a little bit more funding, eight dollars more and
• 10:56
  Speaker 2: we'll be there.
• 10:58
  Speaker 1: You need a little bit more garlic, You're gonna say
• 11:01
  Speaker 1: a bit more seasoning. You need eighty eight billion dollars.
• 11:05
  Speaker 2: I won't say no to eighty eight billion dollars, that's
• 11:07
  Speaker 2: for sure.
• 11:08
  Speaker 1: Yeah, there you go. And then what happens at the
• 11:11
  Speaker 1: end when you collide the onions? Would you have to
• 11:14
  Speaker 1: eat them?
• 11:14
  Speaker 2: I disappeared to my private island with all the cash.
• 11:18
  Speaker 1: I think they can give you eighty years in prison
• 11:20
  Speaker 1: for that, Daniel, only if they catch me, only they
• 11:24
  Speaker 1: smell of the crime.
• 11:26
  Speaker 2: I'll have an army of cocaine infused butterflies to protect me.
• 11:30
  Speaker 1: Oh my gosh, you are a super villain there. I mean,
• 11:34
  Speaker 1: I've heard of sharks with lasers, but m drug butterflies
• 11:38
  Speaker 1: is a new level of villainy and they will never
• 11:40
  Speaker 1: see me coming. All right, Well, stick into it, Daniel.
• 11:44
  Speaker 1: What is an octonian and what do you think is
• 11:46
  Speaker 1: the right way to pronounce it.
• 11:47
  Speaker 2: I pronounced it octonians, but it does make it sound
• 11:50
  Speaker 2: like creatures from the planet Octo or something.
• 11:53
  Speaker 1: M I see you're going for the more tony pronunciation Octonians.
• 11:58
  Speaker 2: Octonians exactly.
• 12:00
  Speaker 1: So.
• 12:01
  Speaker 2: Octonians are a kind of number, and they're sort of
• 12:04
  Speaker 2: in the category of complex numbers, but they're an extension
• 12:08
  Speaker 2: of complex numbers the way complex numbers have like two
• 12:11
  Speaker 2: components to them, the real and the imaginary part, which
• 12:14
  Speaker 2: are an extension of real numbers. They just have one component.
• 12:17
  Speaker 2: Octonians have eight components, one real and seven imaginary components.
• 12:24
  Speaker 1: Well, okay, wait, hold on, I think maybe some of
• 12:26
  Speaker 1: us might not be super familiar with our high school math.
• 12:30
  Speaker 1: What is an imaginary number in the first place.
• 12:32
  Speaker 2: Imaginary numbers were invented in the sixteen hundreds, and they're
• 12:34
  Speaker 2: called imaginary because you don't see them in reality. They're
• 12:38
  Speaker 2: very useful in math. And there's basically one imaginary number,
• 12:42
  Speaker 2: which is I, which is the square root of minus one.
• 12:46
  Speaker 2: There's no real number which, if you multiplied by itself,
• 12:49
  Speaker 2: gives you minus one. So they invented a number. They
• 12:52
  Speaker 2: just call it I, and they say, if you multiply
• 12:55
  Speaker 2: I by itself, you get minus one. So it's a
• 12:57
  Speaker 2: new kind of number. It's different from any of the
• 13:00
  Speaker 2: numbers on the normal number line.
• 13:02
  Speaker 1: Right, because like we had the number negative one, and
• 13:05
  Speaker 1: we have the function to take a square root of something,
• 13:08
  Speaker 1: but when you mix the two, it's like you get
• 13:10
  Speaker 1: something that's almost impossible, or that it's not on the
• 13:13
  Speaker 1: regular number line that we all use in our everyday.
• 13:16
  Speaker 2: Lives exactly, because any number on the regular number line
• 13:19
  Speaker 2: we call the real numbers, if you square it, you
• 13:22
  Speaker 2: get a positive number. Three squared is nine. Negative three
• 13:25
  Speaker 2: squared is also nine because the two negatives give you
• 13:28
  Speaker 2: a positive. So there's no number on the real number line,
• 13:31
  Speaker 2: which if you square it will give you a negative answer.
• 13:34
  Speaker 2: So there's no number that's an answer to the question
• 13:37
  Speaker 2: what is the square root of negative one? So they
• 13:39
  Speaker 2: had to invent a new kind of number, sort of
• 13:42
  Speaker 2: imagine the real number line is like the x axis.
• 13:45
  Speaker 2: They invented a new axis, like the y axis, which
• 13:48
  Speaker 2: is like the imaginary direction. So instead of having a
• 13:51
  Speaker 2: number line, and they have a number plane where every
• 13:54
  Speaker 2: number is two components, a real component and this imaginary component.
• 13:58
  Speaker 1: Right, because you came up with us or mathematicians came
• 14:01
  Speaker 1: up with I, and then you can have a whole
• 14:03
  Speaker 1: number line based on I. You can have like two
• 14:05
  Speaker 1: I and three I or four point seven I or
• 14:08
  Speaker 1: eight I.
• 14:08
  Speaker 2: Or negative two I. Right, the imaginary number line goes
• 14:11
  Speaker 2: both ways, right, right?
• 14:14
  Speaker 1: And can you also have not I?
• 14:15
  Speaker 2: You can have not it and like I'm not doing the.
• 14:17
  Speaker 1: Dishes, Yes, that's what I mean, Like zero I, would
• 14:21
  Speaker 1: you call it not I?
• 14:23
  Speaker 2: Oh? You can have zero? Yeah, the zero in the
• 14:24
  Speaker 2: number line is just zero Comma zero, right, zero real
• 14:28
  Speaker 2: numbers and zero imaginary.
• 14:30
  Speaker 1: Right. So in physics, when you talk about an imaginary number,
• 14:34
  Speaker 1: you talk about like a number that has both a
• 14:36
  Speaker 1: real component and an imaginary component, so you write it
• 14:39
  Speaker 1: as two numbers like seven plus eight I exactly.
• 14:43
  Speaker 2: That's what we call a complex number, something with a
• 14:46
  Speaker 2: real and an imaginary component, sort of like a coordinate
• 14:49
  Speaker 2: on the two dimensional complex plane. If you imagine real
• 14:53
  Speaker 2: numbers or x and imaginary numbers are y.
• 14:56
  Speaker 1: Right, And I think like all of most of quantum
• 14:58
  Speaker 1: particle physics is based on imaginary numbers, right, Like it's
• 15:01
  Speaker 1: a convenient mathematical space to do all of the math in.
• 15:05
  Speaker 2: Yeah, exactly. So there's a few important things to know
• 15:07
  Speaker 2: about complex numbers. Number one, they turned out to be
• 15:10
  Speaker 2: really useful. Back in the sixteen hundreds when they were invented,
• 15:13
  Speaker 2: before we had quantum mechanics, they were already useful. Italian
• 15:17
  Speaker 2: mathematicians came up with them as a way to solve problems,
• 15:20
  Speaker 2: and they found that they could get to the solutions
• 15:22
  Speaker 2: of tricky math problems much more rapidly if they used
• 15:25
  Speaker 2: these imaginary numbers, even though they didn't believe in them,
• 15:28
  Speaker 2: they didn't believe that they represented anything in the universe.
• 15:31
  Speaker 2: But now all of our quantum mechanics relies on these
• 15:34
  Speaker 2: complex numbers. Like the wave function that we talk about
• 15:37
  Speaker 2: all the time. In quantum mechanics, that's actually a complex
• 15:41
  Speaker 2: valued object. That's why you can't observe it directly. You
• 15:44
  Speaker 2: can only observe it when it's squared. That gives you
• 15:47
  Speaker 2: the probability. The wave function itself is a complex number,
• 15:51
  Speaker 2: has a real component and an imaginary component. We couldn't
• 15:54
  Speaker 2: do quantum mechanics without complex numbers. They represent something real
• 15:58
  Speaker 2: about the universe.
• 16:00
  Speaker 1: Wait, wait, what are you saying. Are you saying like
• 16:01
  Speaker 1: somehow the universe or particles are themselves imaginary, or that
• 16:06
  Speaker 1: they sort of operate in a two dimensional space.
• 16:09
  Speaker 2: We don't know if the wave function is real, if
• 16:11
  Speaker 2: it actually exists in the universe. What we do know
• 16:14
  Speaker 2: is that the mathematics needed to describe what particles do
• 16:18
  Speaker 2: in which direction they go has to have complex numbers
• 16:22
  Speaker 2: in it. Like that. Math just does not work without
• 16:24
  Speaker 2: complex numbers. So in our models, in our calculations, the
• 16:28
  Speaker 2: wave function that describes all these particles have complex values
• 16:32
  Speaker 2: in them. Now, when you predict the outcome of an experiment,
• 16:34
  Speaker 2: it always just gives you actual numbers or real numbers
• 16:37
  Speaker 2: that you can measure the particle will be at this
• 16:39
  Speaker 2: location at that time. But the intermediate states. The calculations
• 16:42
  Speaker 2: we do between the observations require complex numbers. Does that
• 16:46
  Speaker 2: mean that the universe is complex, that there are imaginary
• 16:49
  Speaker 2: components to it? We don't know. Deep question for philosophy.
• 16:53
  Speaker 1: Well, the universe is definitely complicated, whether it's complex or
• 16:57
  Speaker 1: real or imaginary. Maybe it is the topic for another podcast.
• 17:02
  Speaker 1: But here today we're talking about maybe a different kind
• 17:04
  Speaker 1: of imaginary number, one that is maybe eight times more
• 17:08
  Speaker 1: imaginary than an imaginary number. So to dig into that
• 17:11
  Speaker 1: and see whether or not it can help explain more
• 17:14
  Speaker 1: of the universe that we see around this and what
• 17:15
  Speaker 1: would that mean. But first, let's take a quick break.
• 17:31
  Speaker 1: All right, we're talking about Oktonians or Kanyans, which are
• 17:35
  Speaker 1: maybe extra imaginary numbers that might describe how the universe
• 17:39
  Speaker 1: works at the particle level. We're digging into that here today. Now, Daniel,
• 17:44
  Speaker 1: you're saying that imaginary numbers play a big part in
• 17:47
  Speaker 1: describing how things work at the quantum level.
• 17:50
  Speaker 2: Exactly, we need complex numbers in order to do quantum mechanics.
• 17:54
  Speaker 2: Like if they hadn't been invented already early this century
• 17:57
  Speaker 2: when we were developing quantum mechanics, we might have invented
• 18:00
  Speaker 2: them then, As it is, they already existed in the
• 18:03
  Speaker 2: sort of mathematical toolbox, and so we could just pluck
• 18:05
  Speaker 2: them out and say, oh, this is helpful, let's use it.
• 18:08
  Speaker 1: Well, was that like a big surprise, Like, wait a minute,
• 18:10
  Speaker 1: what does that mean about the universe that you need
• 18:12
  Speaker 1: imaginary numbers to describe it?
• 18:14
  Speaker 2: Yeah? It has deep consequences for mathematical philosophy. You know,
• 18:18
  Speaker 2: you can wonder if the wave function is real, does
• 18:20
  Speaker 2: that mean that imaginary numbers are real in some philosophical sense,
• 18:24
  Speaker 2: even if we don't call them real numbers in a
• 18:26
  Speaker 2: mathematical sense. We don't know what that means. But imaginary
• 18:30
  Speaker 2: numbers are also important in mathematics itself. It tells us
• 18:33
  Speaker 2: something about how numbers work even before they were applied
• 18:37
  Speaker 2: to quantum mechanics.
• 18:39
  Speaker 1: Interesting, so there may be something point to something fundamental
• 18:42
  Speaker 1: about the nature of the universe.
• 18:43
  Speaker 2: Yeah. Perhaps, because complex numbers are not just like something
• 18:47
  Speaker 2: mathematicians invented. You can't just say, well, I'm going to
• 18:49
  Speaker 2: take two numbers and stick them together, and now I've
• 18:51
  Speaker 2: got a new kind of number. When you do that,
• 18:53
  Speaker 2: you also have to decide, like, what are the rules
• 18:55
  Speaker 2: of those numbers, how do you add them? How do
• 18:57
  Speaker 2: you subtract them? What happens when you multiply the two numbers.
• 19:00
  Speaker 2: So you have to come up with what mathematicians call
• 19:02
  Speaker 2: a division algebra, which is basically just like all the
• 19:05
  Speaker 2: rules of how the math works for these numbers. And
• 19:07
  Speaker 2: it's not always easy to come up with that system.
• 19:10
  Speaker 2: For example, you can do it for complex numbers, but
• 19:12
  Speaker 2: you can't do it for triplets of numbers. So it
• 19:15
  Speaker 2: works for single numbers, it works for pairs of numbers,
• 19:18
  Speaker 2: it doesn't work for triplets of numbers.
• 19:21
  Speaker 1: Okay, wait, hold on, and I think you're talking about
• 19:22
  Speaker 1: now increasing the imaginariness of a number. So like a
• 19:26
  Speaker 1: regular imaginary number has a real component and then an
• 19:29
  Speaker 1: imaginary component which you get by multiplying another number by
• 19:34
  Speaker 1: the number I. So like an imaginary number is seven
• 19:37
  Speaker 1: plus eight I. Now you're talking about like adding a
• 19:40
  Speaker 1: third component.
• 19:42
  Speaker 2: Yeah, call it J. Right, J is like another square
• 19:45
  Speaker 2: root of negative one, a different a unique square root
• 19:48
  Speaker 2: of negative one. Now thing says you can't have more, right,
• 19:51
  Speaker 2: Maybe there's multiple square roots of negative one. And after
• 19:55
  Speaker 2: people discover complex numbers, mathematicians are like, oh, that's cool.
• 19:58
  Speaker 2: Can you do the same thing with three numbers. If
• 20:01
  Speaker 2: you put three numbers together like a real number, and
• 20:04
  Speaker 2: then some number of I and some number of J,
• 20:07
  Speaker 2: can you also then make like consistent mathematics so you
• 20:11
  Speaker 2: know how to multiply, divide, subtract, etc. People spend hundreds
• 20:14
  Speaker 2: of years on that.
• 20:15
  Speaker 1: Let me think about that for a second. So now
• 20:17
  Speaker 1: we have a number plus a number I pleasent a
• 20:20
  Speaker 1: J number and you're saying, Jay's another square root of
• 20:23
  Speaker 1: a negative one. So if I'm multiplied J times J,
• 20:25
  Speaker 1: I get negative one. What if I'm look at IE
• 20:28
  Speaker 1: times J.
• 20:28
  Speaker 2: So this is exactly what you have to do. You
• 20:30
  Speaker 2: have to come up with the rules of this triplet
• 20:32
  Speaker 2: system to figure out what happens when you multiply different
• 20:35
  Speaker 2: numbers together, what happens when you divide them. And it
• 20:38
  Speaker 2: turns out there's no consistent way to define multiplication and
• 20:43
  Speaker 2: division so that it all makes sense and keeps the
• 20:45
  Speaker 2: mathematicians happy. If you have I and J in addition
• 20:49
  Speaker 2: to the real numbers, like, you cannot build a mathematical
• 20:52
  Speaker 2: system based on numbers with three components. You can do
• 20:55
  Speaker 2: it with one that's just the normal numbers. You can
• 20:57
  Speaker 2: do it with two that's just the complex numbers. Can't
• 21:00
  Speaker 2: do it with three.
• 21:01
  Speaker 1: But I feel like when you said before, like having
• 21:05
  Speaker 1: two numbers, or like an real and an imaginary. It's
• 21:07
  Speaker 1: sort of like an X and a Y. Isn't this
• 21:09
  Speaker 1: just like having X, Y and Z.
• 21:11
  Speaker 2: Yeah, you might imagine you should be able to do
• 21:12
  Speaker 2: it with any set of numbers, exactly the way you
• 21:14
  Speaker 2: can define spaces in any dimensions. You're gonna have a
• 21:17
  Speaker 2: one dimensional space or two dimensional space, so three dimensional space,
• 21:20
  Speaker 2: a nineteen dimensional space, an infinite dimensional space. Geometry has
• 21:24
  Speaker 2: no restrictions, right, But the rules of mathematics, for some
• 21:28
  Speaker 2: reason constrain the number of numbers we can pack in
• 21:31
  Speaker 2: together and still have everything makes sense. Mathematicians worry about
• 21:35
  Speaker 2: like does every number have an inverse? If I take
• 21:38
  Speaker 2: a number, is there always some other number I can
• 21:40
  Speaker 2: multiply it by to get one? Are there unique zeros
• 21:44
  Speaker 2: or not? And it turns out that mathematics based on
• 21:46
  Speaker 2: triplets just doesn't work. There's no way to put it together.
• 21:50
  Speaker 2: This is famous Irish mathematician named William Rowan Hamilton who
• 21:54
  Speaker 2: spent like decades on this, and he said once every morning,
• 21:57
  Speaker 2: coming down to breakfast, my kids ask me, well, pop up,
• 22:00
  Speaker 2: can you multiply triplets? And he always said nope, I
• 22:02
  Speaker 2: can always I can only add and subtract them. So
• 22:05
  Speaker 2: we spent decades like try to figure out how to
• 22:07
  Speaker 2: multiply and divide triplets of numbers and never succeeded.
• 22:11
  Speaker 1: WHOA, but X, Y Z space works in three D
• 22:14
  Speaker 1: space all around this? Or are you saying that three
• 22:17
  Speaker 1: D space doesn't work mathematically or just triplet imaginary numbers
• 22:20
  Speaker 1: don't work mathematically?
• 22:22
  Speaker 2: Triplet imaginary numbers don't work mathematically. Like you might think, well,
• 22:25
  Speaker 2: why can't you just put three numbers together and then say,
• 22:28
  Speaker 2: you know, multiplying two triplets means multiplying the components, right.
• 22:32
  Speaker 2: I don't think we want to get two mathematical on
• 22:34
  Speaker 2: the podcast today. But this creates problems of having numbers
• 22:38
  Speaker 2: that don't have inverses, like one comma one coma zero
• 22:42
  Speaker 2: doesn't have an inverse, but it's non zero.
• 22:44
  Speaker 1: Okay, I think I see what you're saying. I think
• 22:46
  Speaker 1: you're saying it's possible to just have like coordinates in
• 22:48
  Speaker 1: three D space, like x, Y and z, But if
• 22:51
  Speaker 1: you want to call it like X plus y, I
• 22:54
  Speaker 1: plus z J, that doesn't work like if you want
• 22:56
  Speaker 1: to put them all together as one number with an
• 22:58
  Speaker 1: addition sign between the different coordinates, that doesn't work.
• 23:02
  Speaker 2: It doesn't work if you also require that you can
• 23:04
  Speaker 2: do things with these numbers, like can you take any
• 23:07
  Speaker 2: of these two numbers and multiply them together and still
• 23:09
  Speaker 2: get a number. If you add them together, do you
• 23:11
  Speaker 2: still get a number? We rely on that for the
• 23:13
  Speaker 2: normal numbers. Right, any number you give me, I can
• 23:15
  Speaker 2: always find another number to multiply it by to get one.
• 23:18
  Speaker 2: Or if you give me two numbers I multiply together,
• 23:20
  Speaker 2: I always get a number that's not a zero if
• 23:22
  Speaker 2: both of the numbers you gave me were not zero.
• 23:25
  Speaker 2: You can't build rules like that, which you need to
• 23:27
  Speaker 2: do any interesting math or any interesting physics. You can't
• 23:30
  Speaker 2: build rules like that for triplets. You can do it
• 23:32
  Speaker 2: for singles and for pairs, but not for triplets.
• 23:35
  Speaker 1: Okay, so it doesn't work for three triple imaginary numbers,
• 23:38
  Speaker 1: but maybe for four imaginary numbers it does work.
• 23:41
  Speaker 2: Exactly and the same. Irish mathematician William Hamilton he discovered
• 23:45
  Speaker 2: that if you put four numbers together, like four D
• 23:48
  Speaker 2: space one reel in three imaginary numbers, you call these quaternions.
• 23:53
  Speaker 2: This actually works that the mathematics hangs together. You can
• 23:56
  Speaker 2: build multiplication and division, you can do all the mathematics
• 23:59
  Speaker 2: you know need on quadruples of numbers, even though you
• 24:02
  Speaker 2: can't do it on triplets.
• 24:04
  Speaker 1: Okay, So now that means that my number is not
• 24:06
  Speaker 1: just like eight, and it's not just like A plus
• 24:09
  Speaker 1: four I. It's like eight plus four I plus seven
• 24:12
  Speaker 1: J plus five K exactly.
• 24:16
  Speaker 2: And in that system, I squared is negative one, J
• 24:20
  Speaker 2: squared is negative one. K squared is negative one. And
• 24:24
  Speaker 2: if you multiply I times J times k, you also
• 24:27
  Speaker 2: get negative one. And that's the key J and K
• 24:31
  Speaker 2: or other square roots of negative one, and that one
• 24:33
  Speaker 2: equation makes it all hang together. And when Hamilton had
• 24:36
  Speaker 2: this insight, he was like walking across this bridge in Dublin.
• 24:40
  Speaker 2: It came to him in a flash, and he actually
• 24:43
  Speaker 2: chiseled the formula onto the bridge like mathematical graffiti in
• 24:47
  Speaker 2: the moment because he didn't want to forget it.
• 24:48
  Speaker 1: And you can still see it today.
• 24:51
  Speaker 2: It's actually worn down, but they put a plaque on
• 24:53
  Speaker 2: that spot to commemorate it.
• 24:55
  Speaker 1: And more important was this kid I pressed or because
• 24:59
  Speaker 1: the kids seem really interested in what was going on
• 25:01
  Speaker 1: with the math and his father.
• 25:03
  Speaker 2: Yeah, I think he was just rooting for his dad,
• 25:05
  Speaker 2: you know, day his struggling with this crazy multiplication.
• 25:10
  Speaker 1: And how much can a kid know about imaginary numbers.
• 25:14
  Speaker 2: It's fascinating that the mathematics of it tells us what's allowed.
• 25:17
  Speaker 2: You can build a one D number, two D number,
• 25:20
  Speaker 2: a four D number, but not a three D number.
• 25:22
  Speaker 2: It's really interesting.
• 25:24
  Speaker 1: Can you go five and six and seven?
• 25:26
  Speaker 2: You can't. About one hundred years later, mathematicians proved that
• 25:30
  Speaker 2: the only sets of numbers you can do are one
• 25:32
  Speaker 2: real numbers, two complex numbers, four quaternions, and eight octonians.
• 25:39
  Speaker 1: Ooh, and what can you keep going like sixteen thirty two?
• 25:43
  Speaker 2: Nope, those are the only ones.
• 25:45
  Speaker 1: What do you mean?
• 25:46
  Speaker 2: Everything else runs into mathematical problems. You can't have consistent
• 25:49
  Speaker 2: multiplication and division without running into ugly problems with the zeros.
• 25:53
  Speaker 2: It only works for one, two, four, and eight.
• 25:56
  Speaker 1: That's it. That's nothing more in the hing more.
• 25:59
  Speaker 2: In the universe. They prove this, and I try to
• 26:01
  Speaker 2: find a way to like explain this in intuitive sense,
• 26:04
  Speaker 2: but the proof is very, very complicated. But everybody's totally
• 26:07
  Speaker 2: convinced that one, two, four, and eight are the only
• 26:10
  Speaker 2: dimensions allowed for numbers that have consistent multiplication and division.
• 26:14
  Speaker 1: You mean for like complex numbers, Yeah, complex numbers with
• 26:18
  Speaker 1: like extra imaginary dimensions exactly.
• 26:21
  Speaker 2: You could have one extra imaginary dimension, which gives you
• 26:24
  Speaker 2: our familiar complex numbers are of two D. You can
• 26:27
  Speaker 2: have three imaginary dimensions, which gives you four D numbers quaternions.
• 26:31
  Speaker 2: Or you can have seven imaginary dimensions plus one reel
• 26:35
  Speaker 2: gives you eight dimensions for octonians.
• 26:38
  Speaker 1: And still be consistent with our rules of math that
• 26:41
  Speaker 1: we know about in our universe. Could does be different
• 26:45
  Speaker 1: in another universe.
• 26:46
  Speaker 2: You can come up with whatever rules of math you want,
• 26:48
  Speaker 2: but if you want to do physics with it, you
• 26:50
  Speaker 2: need to know how to multiply the numbers, how to
• 26:52
  Speaker 2: divide them. You need some confidence that you're not going
• 26:54
  Speaker 2: to run into zeros all the time. So these are
• 26:57
  Speaker 2: like pretty basic requirements for mathematical system that's going to
• 27:00
  Speaker 2: underlie physics. You can invent math however you like and
• 27:03
  Speaker 2: have it be useless or useful or whatever, but if
• 27:06
  Speaker 2: you wanted to do physics, you have to follow these
• 27:08
  Speaker 2: basic rules.
• 27:09
  Speaker 1: Okay, Now, I think the idea is that you know,
• 27:11
  Speaker 1: we had real numbers for a long time, and they
• 27:13
  Speaker 1: weren't great for regular physics. And then we found quantum
• 27:16
  Speaker 1: mechanics and particles, and we've figured out that one dimensional
• 27:20
  Speaker 1: imaginary numbers work really well to describe the math and
• 27:24
  Speaker 1: those theories where you have one extra imaginary dimension. But
• 27:28
  Speaker 1: you're saying, mathematician is also found out that you can
• 27:31
  Speaker 1: have three extra imaginary numbers or dimensions in a quaternion,
• 27:37
  Speaker 1: or seven extra imaginary dimensions in a Olktonian. And now
• 27:41
  Speaker 1: I think maybe what you're saying is that scientists are wondering,
• 27:45
  Speaker 1: do these extra super numbers also maybe describe something about
• 27:49
  Speaker 1: the universe.
• 27:49
  Speaker 2: Exactly, Because obviously one D numbers are very relevant to physics,
• 27:53
  Speaker 2: two D numbers are very relevant to quantum mechanics. These
• 27:56
  Speaker 2: four D numbers you probably haven't heard of quaternions, but
• 28:00
  Speaker 2: these days we actually just call them four dimensional vectors,
• 28:03
  Speaker 2: and they're crucial for relativity. Like special relativity combines space
• 28:08
  Speaker 2: and time into a four dimensional structure where three of
• 28:11
  Speaker 2: them are similar to each other and one of them
• 28:13
  Speaker 2: is different. Right, doesn't that sound familiar?
• 28:15
  Speaker 1: It sounds like space time.
• 28:17
  Speaker 2: It sounds like space time exactly, And quaternions have exactly
• 28:20
  Speaker 2: that structure, three imaginary numbers and one real. So the
• 28:25
  Speaker 2: real numbers like time and the three imaginary numbers are
• 28:27
  Speaker 2: like three dimensions of space. And so these quaternions the
• 28:31
  Speaker 2: rules of space time vectors. We call them four vectors.
• 28:34
  Speaker 2: We call them four vectors now instead of quaternions are
• 28:37
  Speaker 2: absolutely crucial to relativity.
• 28:40
  Speaker 1: Meaning you need you can only do relativity math if
• 28:43
  Speaker 1: you're using quaternions, or it's just helpful to use them.
• 28:47
  Speaker 2: No, they're fundamental to special relativity absolutely like.
• 28:49
  Speaker 1: You can't you can't prove relativity without going into quaternions.
• 28:52
  Speaker 2: Exactly if we didn't already have quaternions, we would have
• 28:55
  Speaker 2: needed to invent them or discover them, depending on your
• 28:58
  Speaker 2: philosophical take. When we build relativity the same way, we
• 29:02
  Speaker 2: absolutely need complex numbers to describe quantum mechanics. So all
• 29:05
  Speaker 2: the self consistent ways we know how to build complex
• 29:08
  Speaker 2: numbers so far are very very useful for physics. And
• 29:12
  Speaker 2: it turns out there's a limited number of these sets.
• 29:15
  Speaker 2: You can't just like make up any number. So if one, two,
• 29:18
  Speaker 2: and four are super valuable for physics, then maybe eight
• 29:21
  Speaker 2: is also.
• 29:22
  Speaker 1: Mmmmm, Now you need four for relativity. Does it also
• 29:26
  Speaker 1: work if you call them formianss.
• 29:31
  Speaker 2: They're a little crunchier, but they still were.
• 29:35
  Speaker 1: All right, I'm just kidding. So you're saying, now the
• 29:38
  Speaker 1: question is, like, do numbers with seven extra imaginary dimensions
• 29:42
  Speaker 1: also maybe could they be used for something in physics?
• 29:45
  Speaker 2: That's exactly the question, and it's worth thinking about because
• 29:48
  Speaker 2: we've made progress so many times when we've developed some
• 29:51
  Speaker 2: mathematical tool group theory, field theory, complex numbers and then
• 29:56
  Speaker 2: later found it applicable to physics. So let's like get
• 29:59
  Speaker 2: ahead of the game. Let's say, oh, here's a kind
• 30:02
  Speaker 2: of mathematical tool which has been useful complex numbers of
• 30:05
  Speaker 2: dimensions one, two, and four. If there's only one more
• 30:08
  Speaker 2: kind of complex number out there, let's see if maybe
• 30:11
  Speaker 2: it tells us something about physics. Maybe it's the mathematical
• 30:14
  Speaker 2: structure to understand some other patterns that are out there
• 30:17
  Speaker 2: that we didn't have explanations for.
• 30:19
  Speaker 1: But does it seem kind of random to you? Like
• 30:21
  Speaker 1: you need one extra imaginary dimension to explain quantum physics
• 30:24
  Speaker 1: that's at the microscopic level, but then you need three
• 30:28
  Speaker 1: extra imaginary dimensions to describe relativity, which is usually at
• 30:32
  Speaker 1: a macro level, isn't it kind of random? Like you
• 30:35
  Speaker 1: need four here eight there too? Here?
• 30:38
  Speaker 2: Nobody understands it, right, then nobody has an intuitive understanding
• 30:41
  Speaker 2: for why one, two, four, and eight? Like, there is
• 30:44
  Speaker 2: this proof that shows that these are definitely the only
• 30:46
  Speaker 2: ones you can do. Nobody really knows philosophically what it means,
• 30:49
  Speaker 2: but it's very suggestive. Right, If two D numbers are
• 30:53
  Speaker 2: needed for quantum mechanics, four D numbers are needed for relativity. Well,
• 30:56
  Speaker 2: we've spent the laste hundred years looking for quantum mechanical relativity.
• 31:01
  Speaker 2: Maybe Octonians are the key to that, right, two times
• 31:04
  Speaker 2: four equals eight? After all, maybe the mathematical structure we
• 31:08
  Speaker 2: need to understand quantum gravity is based on Octonians.
• 31:12
  Speaker 1: But why Goodtonians as anyone tried to apply quaternions to
• 31:17
  Speaker 1: like quantum gravity?
• 31:18
  Speaker 2: Well, absolutely nobody has a consistent theory of quantum gravity
• 31:21
  Speaker 2: that works. And every time we try to write down
• 31:24
  Speaker 2: the theory of quantum gravity, it breaks mathematically, it does
• 31:27
  Speaker 2: not work. So it might just be that we're using
• 31:30
  Speaker 2: the wrong kind of number.
• 31:32
  Speaker 1: Oh, I see, Like, maybe the key to it all
• 31:34
  Speaker 1: is just to go to Applebee's and order Sometonians.
• 31:40
  Speaker 2: Absolutely, it might be well, I.
• 31:44
  Speaker 1: Mean that is a joke, but like, I think that's
• 31:46
  Speaker 1: what you're saying. It's like, why why don't we order
• 31:49
  Speaker 1: an Actonian and see if it satisfies or it helps
• 31:52
  Speaker 1: us combine these theories of physics.
• 31:54
  Speaker 2: Yeah. Absolutely, because remember that there's lots of ways to
• 31:57
  Speaker 2: make progress in physics when you don't understand the patterns
• 31:59
  Speaker 2: you're looking at. One way is to see more of
• 32:02
  Speaker 2: the pattern. You're like, well, what are all these particles?
• 32:04
  Speaker 2: Why do we have these particles not other particles. One
• 32:07
  Speaker 2: strategy is discover more particles. That's what I'm doing, building colliders,
• 32:10
  Speaker 2: smashing stuff together and looking to see more of the pattern,
• 32:13
  Speaker 2: hoping that if you see more of it, you'll figure
• 32:15
  Speaker 2: out what's going on. Another approach is not to discover
• 32:18
  Speaker 2: anything new, but find new mathematical structures that naturally describe
• 32:22
  Speaker 2: the patterns you're seeing. For example, that's how we thought
• 32:26
  Speaker 2: up the Higgs boson. It wasn't just discovered. It was
• 32:28
  Speaker 2: thought up by looking at the patterns that exist already
• 32:31
  Speaker 2: in the particles that we had, coming up with a
• 32:34
  Speaker 2: new mathematical structure to describe that group theory and the
• 32:37
  Speaker 2: unification of electromagnetism and the weak force and realizing that
• 32:41
  Speaker 2: that needed a Higgs boson. And so there's a lot
• 32:43
  Speaker 2: of progress to be made in just coming up with
• 32:46
  Speaker 2: a new mathematical way to look at the patterns we already.
• 32:48
  Speaker 1: Seem interesting, Like, that's a way to do science, Like
• 32:52
  Speaker 1: try to look for patterns and then come up with
• 32:54
  Speaker 1: ideas that might make sense of those patterns, and then
• 32:57
  Speaker 1: go back to the experiments to see if those theories are.
• 33:00
  Speaker 2: Exactly Because mathematics has its own rules, right, you can't
• 33:03
  Speaker 2: just invent whatever mathematics you want. Mathematics is telling us
• 33:07
  Speaker 2: very clearly, Hey guys, there's only four kind of numbers
• 33:09
  Speaker 2: you can use, by the way, if you want to
• 33:10
  Speaker 2: describe physics. And that seems like a pretty big clue.
• 33:13
  Speaker 2: It's like, well, if we only have four tools, let's
• 33:15
  Speaker 2: make sure to use all of them, right, especially if
• 33:18
  Speaker 2: the first three were so absolutely useful, So even if
• 33:21
  Speaker 2: we haven't yet figured out what they applied to, it's
• 33:24
  Speaker 2: worth thinking about what they might be applied to.
• 33:26
  Speaker 3: Mmm.
• 33:27
  Speaker 1: Interesting, Okay, So I think what you're saying is that maybe,
• 33:29
  Speaker 1: like if we take the universe and we peel back
• 33:32
  Speaker 1: its layers one by one and try to get to
• 33:35
  Speaker 1: the core of it. Maybe at the center of it
• 33:37
  Speaker 1: is an Atonium exactly.
• 33:40
  Speaker 2: An Actonian could be the center of the onion eight
• 33:42
  Speaker 2: layers down.
• 33:44
  Speaker 1: Yeah, there you go.
• 33:45
  Speaker 2: I see where you're walking me to.
• 33:46
  Speaker 1: Yes, all right, well, let's get into what these ideas
• 33:51
  Speaker 1: are about. How Altonians might describe what's inside of the universe,
• 33:54
  Speaker 1: what's at the core of it. How does it all work,
• 33:57
  Speaker 1: How does it make sense mathematically? Will these theories work?
• 34:00
  Speaker 1: Let's stick into that. But first let's take another quick break.
• 34:14
  Speaker 1: All right, we're talking about Octonians, and are they the
• 34:17
  Speaker 1: imaginary numbers that scientists have been dreaming about to explain
• 34:21
  Speaker 1: everything in the universe. So, Daniel, you were saying that
• 34:24
  Speaker 1: the numbers is one imaginary number really helped describe quantum mechanics,
• 34:29
  Speaker 1: numbers with three imaginary numbers really help describe special relativity
• 34:33
  Speaker 1: and relativity, and because it lets you do math with
• 34:36
  Speaker 1: space time, three dimensions of space and one of time.
• 34:40
  Speaker 1: Now you're saying that maybe if you want to combine
• 34:43
  Speaker 1: these two theories, quantum mechanics and special relativity, maybe you
• 34:48
  Speaker 1: need an extra imaginary number, which might be the Octonian.
• 34:52
  Speaker 2: Yeah, it might be exactly, And we're trying to figure
• 34:55
  Speaker 2: out where this might be relevant to the universe. And
• 34:58
  Speaker 2: one thing we can do to figure that out is
• 35:00
  Speaker 2: to look at the properties of octonians, like what do
• 35:03
  Speaker 2: octonians do the quaternions and complex numbers and real numbers
• 35:08
  Speaker 2: don't do. How are they different? And that's one fruitful
• 35:12
  Speaker 2: way to understand where they might be relevant.
• 35:15
  Speaker 1: What do you mean, like, what can they do that's special?
• 35:17
  Speaker 2: So it's more about what they can't do. Every time
• 35:20
  Speaker 2: you add more complex numbers, you sort of lose some ability.
• 35:24
  Speaker 2: Like within real numbers, you can order them, right, you
• 35:26
  Speaker 2: can say this number is bigger than that number, is
• 35:28
  Speaker 2: bigger than that number. They have a very well defined order.
• 35:30
  Speaker 2: But once you go to complex numbers, you can't necessarily
• 35:33
  Speaker 2: say that, like which number is bigger one or I?
• 35:36
  Speaker 2: They have like the same magnitude, but you can no
• 35:39
  Speaker 2: longer rank them necessarily. So as you add complex dimensions
• 35:42
  Speaker 2: you actually lose some capabilities.
• 35:44
  Speaker 1: Things say crazier kind of right, like it's it's kind
• 35:47
  Speaker 1: of hard to order things in a plane or in
• 35:50
  Speaker 1: an X y plane exactly.
• 35:51
  Speaker 2: And then when you go to quaternions, basically space time vectors.
• 35:55
  Speaker 2: What you lose is commutation, the fact that like A
• 35:59
  Speaker 2: times b is usually equal to B times a. But
• 36:02
  Speaker 2: that's not true for four dimensional vectors, because multiplying numbers
• 36:05
  Speaker 2: in four dimensions is like rotation. It's like taking a
• 36:08
  Speaker 2: vector and turning it, and in four dimensional space, rotations
• 36:12
  Speaker 2: don't commute. It like matters what order you do rotations in.
• 36:16
  Speaker 2: If you first turn left and then you turn up,
• 36:19
  Speaker 2: you get it to a different place than if you
• 36:21
  Speaker 2: do it in the other direction.
• 36:22
  Speaker 1: For example, Wait, wait, I thought that you said that
• 36:24
  Speaker 1: quaternions and olktunians work because they follow all the math rules.
• 36:28
  Speaker 1: You're saying now that maybe they sort of work, but
• 36:30
  Speaker 1: you kind of have to give up some math rules exactly.
• 36:33
  Speaker 2: These math rules that we require don't include that they commute.
• 36:35
  Speaker 2: They only include that you can multiply and divide an
• 36:37
  Speaker 2: add and subtract. Commutation is not a requirement. It turns
• 36:41
  Speaker 2: out that as the numbers get more complex, you lose
• 36:43
  Speaker 2: some of the properties we're familiar with of real numbers.
• 36:46
  Speaker 2: So complex numbers you lose the ability to order them quaternions,
• 36:50
  Speaker 2: you lose the ability to multiply them in any order
• 36:53
  Speaker 2: you want. A times B is not the same as
• 36:55
  Speaker 2: B times A. So that's really interesting. It tells us
• 36:57
  Speaker 2: something about the structure of space time, right, this is
• 37:00
  Speaker 2: really about space now, Octonians. Even weirder Octonians, which you
• 37:05
  Speaker 2: lose is the associative property. The associated property tells us basically,
• 37:09
  Speaker 2: you can distribute numbers within a parenthesis, like do you
• 37:12
  Speaker 2: do the multiplication within the parentheses first, or do you
• 37:14
  Speaker 2: distribute the number through the parentheses. It's sort of hard
• 37:16
  Speaker 2: to think about the associated property of math intuitively. In
• 37:19
  Speaker 2: a physical sense, it would be sort of like, you know,
• 37:21
  Speaker 2: you're used to putting your socks on and then your shoes.
• 37:24
  Speaker 2: What if you first put your socks in your shoes
• 37:26
  Speaker 2: and then put your feet into the socks, you'd end
• 37:29
  Speaker 2: up in the same place. Right, It doesn't really matter
• 37:31
  Speaker 2: if you put the socks on first or put the
• 37:33
  Speaker 2: socks in the shoes, But for Octonians it does matter.
• 37:37
  Speaker 2: The associated property doesn't hold for Octonians.
• 37:40
  Speaker 1: M I see, you lose another math rule, and you
• 37:44
  Speaker 1: also lose the ability to order them, and you also
• 37:46
  Speaker 1: lose the ability to like commute them.
• 37:48
  Speaker 2: You said exactly, So that's a clue, right. It tells
• 37:52
  Speaker 2: us that maybe the fundamental nature of the universe doesn't
• 37:55
  Speaker 2: respect this associative property. Maybe parentheses actually do matter to
• 38:00
  Speaker 2: whatever mathematics really rules the universe. That's just like a clue,
• 38:04
  Speaker 2: and then we go off hunting in the physical world
• 38:06
  Speaker 2: for maybe something that's like that. Is there some theory
• 38:10
  Speaker 2: of physics that we can build that doesn't have this
• 38:13
  Speaker 2: requirement of associativity of our numbers. Maybe that's the right direction.
• 38:17
  Speaker 2: It's a very very vague clue, but that's the kind
• 38:20
  Speaker 2: of clue we can get from looking at the structure
• 38:22
  Speaker 2: of the mathematics of Octonians.
• 38:25
  Speaker 1: Oh, I see, because the pattern is that every time
• 38:27
  Speaker 1: you have one of these imaginary number sets, their limitations
• 38:31
  Speaker 1: somehow correspond to a rule or a feature of the universe.
• 38:35
  Speaker 1: Like when you had one extra dimension and you can't
• 38:38
  Speaker 1: order those numbers, that corresponds to something special about quantum particles,
• 38:43
  Speaker 1: or when you have a quaternion and you can't commute
• 38:47
  Speaker 1: them that means something special about how you can rotate
• 38:50
  Speaker 1: or not things in space m M exactly.
• 38:52
  Speaker 2: So what does this mean about the universe that Octonians
• 38:56
  Speaker 2: don't respect the associative property. It's very strange, it's very unphysical,
• 39:00
  Speaker 2: like no intuition for it. What it tells us is
• 39:02
  Speaker 2: that a theory of the universe built on Octonians is
• 39:05
  Speaker 2: going to be very counterintuitive.
• 39:07
  Speaker 1: Wait, doesn't it just mean you can't put on your
• 39:08
  Speaker 1: socks and your shoes at the same time.
• 39:11
  Speaker 2: It means it makes a difference whether you put your
• 39:13
  Speaker 2: socks in your shoes and then put them both on,
• 39:15
  Speaker 2: or put your socks on and then your shoes.
• 39:18
  Speaker 1: Yeah, it would make a difference, for sure. I'm sure
• 39:22
  Speaker 1: would work.
• 39:24
  Speaker 2: I mean, I think some people put their underwear in
• 39:26
  Speaker 2: their pants on at the same time, don't they. I
• 39:27
  Speaker 2: don't know.
• 39:28
  Speaker 1: Oh, well, well, why didn't you start with that analogy?
• 39:30
  Speaker 2: I didn't want to talk about underwear? I guess.
• 39:35
  Speaker 1: So you're saying that, uh, Quaternions, you can't do that.
• 39:38
  Speaker 1: You can't put your underwear inside of your pants and
• 39:40
  Speaker 1: then put them on both at the same time.
• 39:41
  Speaker 2: You can for you can for quaternions. You can't for Octonians,
• 39:45
  Speaker 2: That's right.
• 39:45
  Speaker 1: I'm sorry you can't for Octonians. You're asking, like, is
• 39:48
  Speaker 1: there a rule in the universe that somehow corresponds to
• 39:51
  Speaker 1: that limitation with your pants exactly?
• 39:55
  Speaker 2: The underlying rule of the universe is somehow related to underpants.
• 39:58
  Speaker 2: We don't know.
• 39:59
  Speaker 1: Hey, yeah, there you go, or d end up looking
• 40:02
  Speaker 1: like Superman with another word outside of your pants exactly.
• 40:06
  Speaker 2: Maybe he's the original Octonian.
• 40:08
  Speaker 1: That's right. Maybe maybe any Kryptonians to just really describe
• 40:13
  Speaker 1: the universe.
• 40:14
  Speaker 2: We don't know. But in the meantime, physicists are on
• 40:16
  Speaker 2: the hunt for eights. We're looking for things out there
• 40:20
  Speaker 2: in nature patterns which include the number eight. And there
• 40:23
  Speaker 2: are some pantalizing hints, you know. For example, string theory
• 40:26
  Speaker 2: we know likes to exist in ten spatial dimensions because
• 40:31
  Speaker 2: the mathematics of how those things compactify works best in
• 40:34
  Speaker 2: ten dimensions. So some people have said, ooh, well, that's
• 40:37
  Speaker 2: sort of like eight spatial dimensions and one time dimension
• 40:40
  Speaker 2: plus one dimension for like along the string, and so
• 40:43
  Speaker 2: this is kind of an eight there.
• 40:45
  Speaker 1: But ten, you said didn't work? Why are they fixated
• 40:48
  Speaker 1: with ten if ten doesn't hold mathematically?
• 40:50
  Speaker 2: Because string theory works in ten dimensions. There's a different
• 40:53
  Speaker 2: set of rules there. Mathematically, string theory works in ten.
• 40:56
  Speaker 2: Octonians work in eight. Can we somehow marry the two? Well,
• 41:00
  Speaker 2: maybe if.
• 41:00
  Speaker 1: We take those ten dimensions the difference and make it nine.
• 41:03
  Speaker 2: There's no compromising in math. It's not a negotiation with
• 41:07
  Speaker 2: the universe.
• 41:07
  Speaker 1: You can't split the difference, well, I can't make quantum gravity. Worry.
• 41:13
  Speaker 1: How about uh, we'll split the difference.
• 41:15
  Speaker 2: I'll give you points on the back end. All right?
• 41:16
  Speaker 2: How about that?
• 41:17
  Speaker 1: That's right? Yes, we'll only make both of them half work.
• 41:21
  Speaker 2: No, people are like, well, can we take eight of
• 41:22
  Speaker 2: those ten string dimensions and say those are described by
• 41:25
  Speaker 2: Octonians and then another one of them is time and
• 41:28
  Speaker 2: another one is a string dimension. It's a real stretch.
• 41:31
  Speaker 2: But people are looking for eightishness somewhere out there in
• 41:34
  Speaker 2: the patterns and wondering if it can be described by Octonians.
• 41:38
  Speaker 1: Can't you just decrease the number of dimensions in string
• 41:41
  Speaker 1: theory or it wouldn't work anymore?
• 41:43
  Speaker 2: No, it doesn't work. It only works in ten dimensions.
• 41:45
  Speaker 2: There's another version of it that works in twenty six,
• 41:48
  Speaker 2: but string theory doesn't work in eight dimensions. Another direction
• 41:51
  Speaker 2: people are going in is to try to describe the
• 41:53
  Speaker 2: strong force. The strong force is something we really don't
• 41:56
  Speaker 2: understand very well. But it has eight gluons, right, number eight,
• 42:01
  Speaker 2: So maybe Octonians would be a better way to describe
• 42:04
  Speaker 2: the strong force. Maybe the whole reason we have trouble
• 42:07
  Speaker 2: doing calculations in quantum chromodynamics is because we're used in
• 42:10
  Speaker 2: the wrong kind of math, and it would all just
• 42:11
  Speaker 2: like click beautifully into place if we replaced it with octonians.
• 42:15
  Speaker 1: Wait, what do mean there are only eight gluons? You're
• 42:18
  Speaker 1: just looking for things in nature that you know about
• 42:21
  Speaker 1: so far that somehow count it eight.
• 42:23
  Speaker 2: Yeah, exactly, And there are eight kinds of gluons because
• 42:25
  Speaker 2: remember that there are three different colors in the strong force, red, green,
• 42:30
  Speaker 2: and blue, and each gluon carries two colors, And so
• 42:33
  Speaker 2: the way the mathematics works is that it's three squared
• 42:36
  Speaker 2: minus one. You get eight different kinds of gluons that
• 42:39
  Speaker 2: you can have. So, yeah, we're just looking for things
• 42:42
  Speaker 2: in nature that have eight in them. Like, what about
• 42:44
  Speaker 2: the universe is eightish? Can we describe it with octonians?
• 42:48
  Speaker 2: So far nobody's made it work. Octonians mathematically beautiful, mathematically consistent,
• 42:53
  Speaker 2: so far totally physically useless.
• 42:55
  Speaker 1: Huh, Like, what did you decide to try to make
• 42:57
  Speaker 1: quantum gravity work with octonians? Where does that put us?
• 43:00
  Speaker 1: Aren't there like eight dimensions or things like that? In
• 43:04
  Speaker 1: between the two of them, like the wave function plus
• 43:06
  Speaker 1: the space.
• 43:07
  Speaker 2: Time throwing some oranges and a couple of socks. Yeah, exactly.
• 43:12
  Speaker 2: I know people are working on that, right. There a
• 43:14
  Speaker 2: lot of people studying the nature of optunians and trying
• 43:16
  Speaker 2: to use them to build a physical theory, but nobody
• 43:19
  Speaker 2: succeeded so far. Just sort of like a direction some
• 43:22
  Speaker 2: people are sniffing in as we try to build mathematically
• 43:25
  Speaker 2: consistent theories of quantum gravity.
• 43:27
  Speaker 1: Wow, you're just looking for the number eight.
• 43:29
  Speaker 2: Yes, exactly. We're desperate for clues because this is the
• 43:32
  Speaker 2: biggest unsolved problem in physics. How to marry quantum mechanics
• 43:35
  Speaker 2: and relativity. We haven't figured out in one hundred years.
• 43:38
  Speaker 2: So we're like digging deep in the mathematical toolbox to like, well,
• 43:40
  Speaker 2: what else we got in here? Let's see what else
• 43:42
  Speaker 2: could be useful.
• 43:43
  Speaker 1: Well, I can't seem to get up before eight am?
• 43:46
  Speaker 1: Could that be a rule of the universe? I could
• 43:48
  Speaker 1: blame that.
• 43:49
  Speaker 2: On Well, we could flip the blame. You could say
• 43:51
  Speaker 2: nobody's solved this problem because you won't get up before
• 43:54
  Speaker 2: eight am.
• 43:55
  Speaker 1: Well, that would still make sense. That Still, that's still
• 43:58
  Speaker 1: an explanation for the universe.
• 44:00
  Speaker 2: Or maybe it would take an announcement of quantum gravity
• 44:02
  Speaker 2: to get you out of it before eight am.
• 44:05
  Speaker 1: Unlikely, unlikely in this universe. I don't think that was
• 44:08
  Speaker 1: mathematically hold. Yeah, I think there's YouTube for that. I'll
• 44:13
  Speaker 1: just catch it on YouTube later, all right, Well, but
• 44:16
  Speaker 1: it's sort of an interesting idea or direction in which
• 44:19
  Speaker 1: to look for new theories about the universe. Like maybe
• 44:22
  Speaker 1: you put your octogoggles on and look for things that
• 44:25
  Speaker 1: work or that seem to manifest themselves in sets of eight.
• 44:30
  Speaker 1: Maybe that could be a way to get to a
• 44:33
  Speaker 1: theory of everything exactly.
• 44:35
  Speaker 2: If mathematics really does reflect something deep about the universe,
• 44:39
  Speaker 2: then mathematicians building tools can actually construct sort of like
• 44:42
  Speaker 2: proto bits of future physics theories that we could just
• 44:45
  Speaker 2: like click into place. We've done it before with group
• 44:48
  Speaker 2: theory and field theory and differential geometry, so we hope
• 44:51
  Speaker 2: it happens again.
• 44:52
  Speaker 1: Right, And that's kind of what's happening with strength there.
• 44:54
  Speaker 1: Although that one seems to be following the rule.
• 44:56
  Speaker 2: Of ten yeah, or the rule of twenty six.
• 44:58
  Speaker 1: So it sounds like maybe physicists you need to sit
• 45:02
  Speaker 1: down and peel more onions.
• 45:05
  Speaker 2: And stop crying.
• 45:07
  Speaker 1: That's a right stop pretend. Pretend that tears come from
• 45:10
  Speaker 1: cutting these onions and not from being a physicist. And
• 45:14
  Speaker 1: maybe that will inspire some new theory. That may it'll
• 45:17
  Speaker 1: crack open, they'll peel away the truth of the universe.
• 45:20
  Speaker 2: Or maybe we should just take a break in good
• 45:22
  Speaker 2: Applebee's and have a bloomin onion.
• 45:24
  Speaker 1: Yeah, there you go. There are different ways. As long
• 45:27
  Speaker 1: as it's after eight am, I'll join you, all right, Well,
• 45:31
  Speaker 1: and interesting dive into mathematics and how it matches up
• 45:34
  Speaker 1: with physics and how it's helping us maybe understand how
• 45:37
  Speaker 1: it all works, or at least how it has helped
• 45:40
  Speaker 1: us and maybe could help us in the future.
• 45:42
  Speaker 2: And future mathematicians and physicists might look back and say, oh,
• 45:45
  Speaker 2: it was so obvious how it all clicked into place.
• 45:47
  Speaker 2: But here we are at the forefront of human ignorance,
• 45:50
  Speaker 2: just not seeing it come together.
• 45:52
  Speaker 1: All right, Well, we hope you enjoyed that. Thanks for
• 45:54
  Speaker 1: joining us. See you next time.
• 46:01
  Speaker 2: For more science and curiosity. Come find us on social media,
• 46:04
  Speaker 2: where we answer questions and post videos. We're on Twitter,
• 46:08
  Speaker 2: disc Org, Insta, and now TikTok. Thanks for listening, and
• 46:11
  Speaker 2: remember that Daniel and Jorge Explain the Universe is a
• 46:14
  Speaker 2: production of iHeartRadio. For more podcasts from iHeartRadio, visit the
• 46:18
  Speaker 2: iHeartRadio app, Apple Podcasts, or wherever you listen to your
• 46:23
  Speaker 2: favorite shows.