January 18, 2024 • 46 mins
Daniel and Jorge talk about super complex numbers and how they might hold the secrets of the Universe.
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Speaker 1 (00:08):
Hey, Daniel, what are your favorite numbers in physics?
Speaker 2 (00:11):
Oh? There are so many good ones. I mean I
love planks 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.
Speaker 1 (00:26):
But have you ever wondered why numbers are so important
in the universe?
Speaker 2 (00:30):
I mean, numbers are like the currency of physics. We're
predicting things happening at times and places, and those are
just numbers.
Speaker 1 (00:38):
Wow. So does that mean you get paid in numbers?
Speaker 2 (00:42):
I get a number of dollars every year.
Speaker 1 (00:44):
I guess we all get paid in numbers if there's
just numbers in our bank account. If you're so lucky,
the whole universe is just numbers. I think you 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
(01:17):
Jorge Mack, cartoonists and the author of Oliver's Great Big Universe.
Speaker 2 (01:21):
Hi, I'm Daniel. I'm a particle physicist, and I've been
a professor at UC Irvine for a large number of years.
Speaker 1 (01:27):
Oh, too many to count? Or do you go into
some sort of like weird subspace when you do physics.
Speaker 2 (01:34):
No, I think I'm going to enter a puba phase
eventually and then emerge as an emeritus professor.
Speaker 1 (01:39):
That's a butterfly. That's a beautiful tenured emeritus.
Speaker 2 (01:43):
Butterfly, as a white haired moth.
Speaker 1 (01:45):
I think instead of wings, do you just have a
bunch of research papers taped together.
Speaker 2 (01:53):
I'm going to fly too close to the universe on
wings of research papers.
Speaker 1 (01:56):
Right, But you know, butterflies don't have mouths, right, so
you won't be to talk.
Speaker 2 (02:00):
Yeah, but they have those long newses, and so I'm
already said.
Speaker 1 (02:04):
There you go. You can sniff out science from there.
M hmm.
Speaker 2 (02:10):
I can snort up all the information in the universe.
Speaker 1 (02:12):
Oh boy, that sounds a little illegal.
Speaker 2 (02:15):
Are you going to arrest a butterfly? Is that what
has come to?
Speaker 1 (02:18):
Maybe a cocaine sniffing butterfly. Maybe maybe it will be
our informant for, you know, figuring out how the universe works.
Speaker 2 (02:26):
Maybe altering your mental state is important for understanding the universe.
Speaker 1 (02:30):
M I hear that's called the butterfly effect.
Speaker 2 (02:33):
I think that's something else.
Speaker 1 (02:35):
Meant something else. Did I get that wrong.
Speaker 2 (02:39):
I think it's called microducing.
Speaker 1 (02:40):
Yeah, I think this is called micro punning, which it
doesn't work as well. But anyways, welcome to our podcast
Daniel and Jorge Explain the Universe, a production of iHeartRadio.
Speaker 2 (02:49):
In which we give you a drip, drip, drip, little
doses of the incredible wonder we 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.
Speaker 1 (03:06):
That's right, We try to get you high on the
little dopamin head of wonder and amazement at how our
universe works and the surprising ways in which it still
puzzles scientists even today.
Speaker 2 (03:16):
And one thing that still puzzles scientists and philosophers is
at 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.
Speaker 1 (03:36):
Wait, wait, are you saying math is more important than physics.
Speaker 2 (03:41):
I mean, is English more important than Shakespeare. You can't
really compare the two things, you know.
Speaker 1 (03:47):
Wait with this analogy, Shakespeare's math in English is physics.
Speaker 2 (03:51):
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.
Speaker 1 (03:58):
I see. I see. So physics a comedy or a tragedy.
Speaker 2 (04:02):
It's a tragic comedy, for sure.
Speaker 1 (04:04):
It's a tragic comedy.
Speaker 2 (04:06):
We're all waiting for the final twist. Nobody knows how
it ends.
Speaker 1 (04:09):
That's right, it never ends well for Shakespeare.
Speaker 2 (04:12):
But I hope we laugh along the way.
Speaker 1 (04:14):
And you also have to wear tights when you do physics.
Speaker 3 (04:17):
Mmmm.
Speaker 2 (04:17):
And as always, the jester is the wisest one.
Speaker 1 (04:21):
I see. That's the engineer and the team.
Speaker 2 (04:24):
Did you just call engineers jokers? I mean, I'm just
gonna step slowly away from that.
Speaker 1 (04:29):
No. I know. We have a great sense of humor,
is what I'm saying. We're the smaltest person in the room.
Speaker 2 (04:35):
Often laughed at and unwisely ignored.
Speaker 1 (04:37):
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.
Speaker 2 (04:49):
Yeah, there are certain constants which seem to be important
to 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
(05:12):
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.
Speaker 1 (05:23):
And this is 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
understanding how things work at the particle level.
Speaker 2 (05:33):
Math 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.
Speaker 1 (05:46):
Okay, so it's at every level, but it seems like
in the Standard model there are especially some interesting numbers.
Speaker 2 (05:51):
Yes, absolutely, the theories of particle physics are very mathematical,
and not just in the sense that they're predicting numbers
and places in time. 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.
Speaker 1 (06:09):
Right, which sort of eraises 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.
Speaker 2 (06:21):
Yeah, and it makes you wonder, like what does that mean?
And our 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.
There's 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
(06:42):
all sort of mental games.
Speaker 1 (06:44):
Right, right, Like maybe the universe is not really physical,
maybe it's just sort of like conceptual theoretical.
Speaker 2 (06:50):
Yeah, but then you wonder, like what breeds fire into
all those concepts it makes us experience it. The other
side of that argument is that numbers are not universal,
they're not physics, 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
(07:10):
you can like describe the color orange using a bunch
of words, but none of those words are orange or
fully capture the oranginess of an orange.
Speaker 1 (07:19):
Mm and is one really the loneliest number? Is another
big physics question, right.
Speaker 2 (07:25):
Yes, yes, a very deep physics question. But we have
to 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.
(07:47):
There's a lot of hints there that the laws we
use to describe the universe are deeply mathematical.
Speaker 1 (07:52):
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 to be. On the podcast,
we'll be asking the question, what are octonions? Did I
say that right? Or is it actonions?
Speaker 2 (08:13):
I think it's oct onions, Like, give me eight onions
for that recipe, please.
Speaker 1 (08:17):
Isn't it like a blooming onion? Isn't that a dish
in a fried dish in one of these famous restaurants?
Is this like a fried onion that looks like an octopus?
Speaker 2 (08:28):
Are you shilling for Outback Steakhouse? Now?
Speaker 1 (08:30):
Oh? Is it out back?
Speaker 2 (08:31):
I don't know, never been.
Speaker 1 (08:32):
Maybe Applebee's is interested.
Speaker 2 (08:36):
I think it's maybe octonians or octanians, depending on where
you are in the world.
Speaker 1 (08:41):
So this is an interesting word. It sort of sounds
like octo, which is eight. 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.
Speaker 2 (08:55):
Mm hmm, yeah, ions exactly.
Speaker 1 (08:57):
Oh yeah, right right yeah ions. So is it denions?
Speaker 2 (09:00):
Then I think we should have eight different pronunciations of
this word.
Speaker 1 (09:05):
Now a trace you go, and then we'll spend eighty
eight minutes talking about it here on the podcast as.
Speaker 2 (09:11):
Part of our Cult of eight. There you go, send
eight dollars to join.
Speaker 1 (09:15):
There you go. We'll have eight dollars in our pockets.
In our eight pockets because I wear cargo pants exactly.
Speaker 2 (09:22):
Maybe these are the spiders of the universe.
Speaker 1 (09:23):
It's all an intricate web of eight. Well, anyways, as usually,
we were wondering how many people out there had thought
about or even heard of the word octonions and what
they think it might mean.
Speaker 2 (09:33):
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 two questions at Danielandjorge
dot com.
Speaker 1 (09:45):
So think about it for a second. What do you
think are octonions? And are they sold at Applebee's or
out back to steakhouse?
Speaker 2 (09:52):
And would they make you cry if you chop them?
Speaker 1 (09:54):
Here's what people had to say.
Speaker 4 (09:55):
What are onions, so octors, the prefix for eight onions
being something players, probably so eight layers stars.
Speaker 3 (10:05):
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 glue ons or any type of similar particle or
(10:27):
force carrier that is dependent on matrices for transformation.
Speaker 1 (10:37):
All right, well, pretty much the same as what I've
been guessing. Something to do with onions and eight and
or particles.
Speaker 2 (10:44):
Yeah, reasonable guess is absolutely.
Speaker 1 (10:46):
Is it like what happens when you smash together eight onions?
You get an ooked onion.
Speaker 2 (10:52):
We're working on the onion collider right now. We just
need a little bit more funding, eight dollars more and
we'll be there.
Speaker 1 (10:58):
You need a little bit more garlic, You're gonna say
a bit more seasoning. You need eighty eight billion dollars.
Speaker 2 (11:05):
I won't say no to eighty eight billion dollars, that's
for sure.
Speaker 1 (11:08):
Yeah, there you go. And then what happens at the
end when you collide the onions? Would you have to
eat them?
Speaker 2 (11:14):
I disappeared to my private island with all the cash.
Speaker 1 (11:18):
I think they can give you eighty years in prison
for that, Daniel, only if they catch me, only they
smell of the crime.
Speaker 2 (11:26):
I'll have an army of cocaine infused butterflies to protect me.
Speaker 1 (11:30):
Oh my gosh, you are a super villain there. I mean,
I've heard of sharks with lasers, but m drug butterflies
is a new level of villainy and they will never
see me coming. All right, Well, stick into it, Daniel.
What is an octonian and what do you think is
the right way to pronounce it.
Speaker 2 (11:47):
I pronounced it octonians, but it does make it sound
like creatures from the planet Octo or something.
Speaker 1 (11:53):
M I see you're going for the more tony pronunciation Octonians.
Speaker 2 (11:58):
Octonians exactly.
Speaker 1 (12:00):
So.
Speaker 2 (12:01):
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. They just have one component.
Octonians have eight components, one real and seven imaginary components.
Speaker 1 (12:24):
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.
Speaker 2 (12:32):
Imaginary numbers were invented in the sixteen hundreds, and they're
called imaginary because you don't see them in reality. They're
very useful in math. And there's basically one imaginary number,
which is I, which is the square root of minus one.
There's no real number which, if you multiplied by itself,
gives you minus one. So they invented a number. They
(12:52):
just call it I, and they say, if you multiply
I by itself, you get minus one. So it's a
new kind of number. It's different from any of the
numbers on the normal number line.
Speaker 1 (13:02):
Right, because like we had the number negative one, and
we have 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.
Speaker 2 (13:16):
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
(13:37):
what is the square root of negative one? So they
had to invent a new kind of number, sort of
imagine the real number line is like 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, and they have a number plane where every
number is two components, a real component and this imaginary component.
Speaker 1 (13:58):
Right, because you came up with us 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.
Speaker 2 (14:08):
Or negative two I. Right, the imaginary number line goes
both ways, right, right?
Speaker 1 (14:14):
And can you also have not I?
Speaker 2 (14:15):
You can have not it and like I'm not doing the.
Speaker 1 (14:17):
Dishes, Yes, that's what I mean, Like zero I, would
you call it not I?
Speaker 2 (14:23):
Oh? You can have zero? Yeah, the zero in the
number line is just zero Comma zero, right, zero real
numbers and zero imaginary.
Speaker 1 (14:30):
Right. So in physics, when you talk about an imaginary number,
you 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.
Speaker 2 (14:43):
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 or x and imaginary numbers are y.
Speaker 1 (14:56):
Right, And I think like all of most of quantum
particle physics is based on imaginary numbers, right, Like it's
a convenient mathematical space to do all of the math in.
Speaker 2 (15:05):
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 sixteen hundreds when they were invented,
before we had quantum mechanics, they were already useful. Italian
mathematicians came up with them as a way to 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,
(15:28):
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,
(15:51):
has a real component and an imaginary component. We couldn't
do quantum mechanics without complex numbers. They represent something real
about the universe.
Speaker 1 (16:00):
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.
Speaker 2 (16:09):
We don't know if the wave function is real, if
it 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
(16:32):
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 will 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.
Speaker 1 (16:53):
Well, the universe is definitely complicated, whether it's complex or
real or imaginary. Maybe it 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 to dig into that
and see whether or not it can help explain more
(17:14):
of the universe that we see around this and what
would that mean. But first, let's take a quick break.
All right, we're talking about Oktonians or Kanyans, which are
(17:35):
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.
Speaker 2 (17:50):
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.
Speaker 1 (18:08):
Well, was that like a big surprise, Like, wait a minute,
what does that mean about the universe that you need
imaginary numbers to describe it?
Speaker 2 (18:14):
Yeah? It has deep consequences 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
(18:37):
to quantum mechanics.
Speaker 1 (18:39):
Interesting, so there may be something point to something fundamental
about the nature of the universe.
Speaker 2 (18:43):
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, and 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.
So you have to come up with what mathematicians call
a division algebra, which is basically just like all the
(19:05):
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.
Speaker 1 (19:21):
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.
Speaker 2 (19:42):
Yeah, call it J. Right, J is like another square
root of negative one, a different a unique square root
of negative one. Now thing says you can't have more, right,
Maybe there's multiple square roots of negative one. And after
people discover complex numbers, mathematicians are 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
(20:04):
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, etc. People spend hundreds
of years on that.
Speaker 1 (20:15):
Let me think about that for a second. So now
we have a number plus a number I pleasent a
J number and you're saying, Jay's another square root of
a negative one. So if I'm multiplied J times J,
I get negative one. What if I'm look at IE
times J.
Speaker 2 (20:28):
So this is exactly what you have to do. You
have to come up with the rules of this triplet
system 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
(20:49):
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. Can't
do it with three.
Speaker 1 (21:01):
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.
Speaker 2 (21:11):
Yeah, you might imagine you should be able to do
it with any set of numbers, exactly the way you
can define spaces in any dimensions. You're gonna have a
one dimensional space or two dimensional space, so three dimensional space,
a nineteen 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
(21:31):
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
(21:54):
spent like decades on this, and he said once every morning,
coming down to breakfast, my kids ask me, well, pop up,
can you multiply triplets? And he always said nope, I
can always I can only add and subtract them. So
we spent decades like try to figure out how to
multiply and divide triplets of numbers and never succeeded.
Speaker 1 (22:11):
WHOA, but X, Y Z space works in three D
space all around this? Or are you saying that three
D space doesn't work mathematically or just triplet imaginary numbers
don't work mathematically?
Speaker 2 (22:22):
Triplet 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 two mathematical on
the podcast today. But this creates problems of having numbers
that don't have inverses, like one comma one coma zero
doesn't have an inverse, but it's non zero.
Speaker 1 (22:44):
Okay, I think I see what you're saying. I think
you're saying it's possible to just have like coordinates in
three D 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.
Speaker 2 (23:02):
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 together,
I always get a number that's not a zero if
(23:22):
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.
Speaker 1 (23:35):
Okay, so it doesn't work for three triple imaginary numbers,
but maybe for four imaginary numbers it does work.
Speaker 2 (23:41):
Exactly and the same. Irish mathematician William Hamilton he discovered
that if you put four numbers together, like four D
space one reel in 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 know need on quadruples of numbers, even though you
(24:02):
can't do it on triplets.
Speaker 1 (24:04):
Okay, So now that means that my number is not
just like eight, and it's not just like A plus
four I. It's like eight plus four I plus seven
J plus five K exactly.
Speaker 2 (24:16):
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
or other square roots of negative one, and that one
equation makes it all hang together. And when Hamilton had
(24:36):
this insight, he was like walking across this bridge in Dublin.
It 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.
Speaker 1 (24:48):
And you can still see it today.
Speaker 2 (24:51):
It's actually worn down, but they put a plaque on
that spot to commemorate it.
Speaker 1 (24:55):
And more important was this kid I pressed or because
the kids seem really interested in what was going on
with the math and his father.
Speaker 2 (25:03):
Yeah, I think he was just rooting for his dad,
you know, day his struggling with this crazy multiplication.
Speaker 1 (25:10):
And how much can a kid know about imaginary numbers.
Speaker 2 (25:14):
It's fascinating that the mathematics of it tells us what's allowed.
You can build a one D number, two D number,
a four D number, but not a three D number.
It's really interesting.
Speaker 1 (25:24):
Can you go five and six and seven?
Speaker 2 (25:26):
You can't. About one hundred years later, mathematicians proved that
the only sets of numbers you can do are one
real numbers, two complex numbers, four quaternions, and eight octonians.
Speaker 1 (25:39):
Ooh, and what can you keep going like sixteen thirty two?
Speaker 2 (25:43):
Nope, those are the only ones.
Speaker 1 (25:45):
What do you mean?
Speaker 2 (25:46):
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.
Speaker 1 (25:56):
That's it. That's nothing more in the hing more.
Speaker 2 (25:59):
In the universe. They prove this, and I try to
find a way to like explain this in intuitive sense,
but the proof is very, very complicated. But everybody's totally
convinced that one, two, four, and eight are the only
dimensions allowed for numbers that have consistent multiplication and division.
Speaker 1 (26:14):
You mean for like complex numbers, Yeah, complex numbers with
like extra imaginary dimensions exactly.
Speaker 2 (26:21):
You could have one extra imaginary dimension, which gives you
our familiar complex numbers are of two D. You can
have three imaginary dimensions, which gives you four D numbers quaternions.
Or you can have seven imaginary dimensions plus one reel
gives you eight dimensions for octonians.
Speaker 1 (26:38):
And still be consistent with our rules of math that
we know about in our universe. Could does be different
in another universe.
Speaker 2 (26:46):
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 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
(27:08):
basic rules.
Speaker 1 (27:09):
Okay, Now, I think the idea is that you know,
we had real numbers for a long time, and they
weren't great for regular physics. And then we found quantum
mechanics and particles, and we've figured out that one dimensional
imaginary numbers work really well to describe the math and
those theories where you have one extra imaginary dimension. But
you're saying, mathematician is also found out that you can
(27:31):
have three extra imaginary numbers or dimensions in a quaternion,
or seven extra imaginary dimensions in a Olktonian. 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.
Speaker 2 (27:49):
Exactly, Because obviously one D numbers are very relevant to physics,
two D numbers are very relevant to quantum mechanics. These
four D numbers you probably haven't heard of quaternions, but
these days we actually just call them four dimensional vectors,
and they're crucial for relativity. Like special relativity combines space
and time into a four dimensional structure where three of
(28:11):
them are similar to each other and one of them
is different. Right, doesn't that sound familiar?
Speaker 1 (28:15):
It sounds like space time.
Speaker 2 (28:17):
It sounds like space time exactly, And quaternions 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 quaternions the
rules of space time vectors. We call them four vectors.
We call them four vectors now instead of quaternions are
(28:37):
absolutely crucial to relativity.
Speaker 1 (28:40):
Meaning you need you can only do relativity math if
you're using quaternions, or it's just helpful to use them.
Speaker 2 (28:47):
No, they're fundamental to special relativity absolutely like.
Speaker 1 (28:49):
You can't you can't prove relativity without going into quaternions.
Speaker 2 (28:52):
Exactly if we didn't already have quaternions, we would have
needed to invent them or discover them, depending on your
philosophical take. When we build 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.
(29:15):
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.
Speaker 1 (29:22):
Mmmmm, Now you need four for relativity. Does it also
work if you call them formianss.
Speaker 2 (29:31):
They're a little crunchier, but they still were.
Speaker 1 (29:35):
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?
Speaker 2 (29:45):
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
(30:05):
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.
Speaker 1 (30:19):
But does it seem kind of random to you? Like
you need one extra 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 too? Here?
Speaker 2 (30:38):
Nobody understands it, right, then 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 two D numbers are
needed for quantum mechanics, four D numbers are needed for relativity. Well,
we've spent the laste hundred years looking for quantum mechanical relativity.
(31:01):
Maybe Octonians are the key to that, right, two times
four equals eight? After all, maybe the mathematical structure we
need to understand quantum gravity is based on Octonians.
Speaker 1 (31:12):
But why Goodtonians as anyone tried to apply quaternions to
like quantum gravity?
Speaker 2 (31:18):
Well, 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 mathematically, it does
not work. So it might just be that we're using
the wrong kind of number.
Speaker 1 (31:32):
Oh, I see, Like, maybe the key to it all
is just to go to Applebee's and order Sometonians.
Speaker 2 (31:40):
Absolutely, it might be well, I.
Speaker 1 (31:44):
Mean that is a joke, but like, I think that's
what you're saying. It's like, why why don't we order
an Actonian and see if it satisfies or it helps
us combine these theories of physics.
Speaker 2 (31:54):
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 and looking to see more of the pattern,
hoping that if you see more of it, you'll figure
(32:15):
out what's going on. Another approach is not to discover
anything new, but find new mathematical 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
(32:37):
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.
Speaker 1 (32:48):
Seem interesting, Like, that's a way to do science, Like
try to look for patterns and then come up with
ideas that might make sense of those patterns, and then
go back to the experiments to see if those theories are.
Speaker 2 (33:00):
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
(33:21):
we haven't yet figured out what they applied to, it's
worth thinking about what they might be applied to.
Speaker 3 (33:26):
Mmm.
Speaker 1 (33:27):
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 Atonium exactly.
Speaker 2 (33:40):
An Actonian could be the center of the onion eight
layers down.
Speaker 1 (33:44):
Yeah, there you go.
Speaker 2 (33:45):
I see where you're walking me to.
Speaker 1 (33:46):
Yes, all right, well, let's get into what these ideas
are about. How Altonians 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 stick into that. But first let's take another quick break.
(34:14):
All right, we're talking about Octonians, and are they the
imaginary numbers that scientists have been dreaming about to explain
everything in the universe. So, Daniel, you were saying that
the numbers is one imaginary number really helped describe quantum mechanics,
numbers with three imaginary numbers really help describe special relativity
and relativity, and because it lets you do math with
(34:36):
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 imaginary number, which might be the Octonian.
Speaker 2 (34:52):
Yeah, it might be exactly, And we're trying 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 the quaternions and complex numbers and real numbers
don't do. How are they different? And that's one fruitful
(35:12):
way to understand where they might be relevant.
Speaker 1 (35:15):
What do you mean, like, what can they do that's special?
Speaker 2 (35:17):
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
(35:39):
longer rank them necessarily. So as you add complex dimensions
you actually lose some capabilities.
Speaker 1 (35:44):
Things say crazier kind of right, like it's it's kind
of hard to order things in a plane or in
an X y plane exactly.
Speaker 2 (35:51):
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
(36:12):
don't commute. It like matters what order you do rotations in.
If you first turn left and then you turn up,
you get it to a different place than if you
do it in the other direction.
Speaker 1 (36:22):
For example, Wait, wait, I thought that you said that
quaternions and olktunians 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.
Speaker 2 (36:33):
These math rules that we require don't include that they commute.
They only include that you can multiply and divide an
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 order them quaternions,
you lose the ability to multiply them in any order
(36:53):
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, this is
really about space now, Octonians. Even weirder Octonians, which you
lose is the associative property. The associated property tells us basically,
you can distribute numbers within a parenthesis, like do you
do the multiplication within the parentheses first, or do you
(37:14):
distribute the number through the parentheses. It's 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.
(37:37):
The associated property doesn't hold for Octonians.
Speaker 1 (37:40):
M 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.
Speaker 2 (37:48):
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
(38:10):
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.
Speaker 1 (38:25):
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
(38:47):
them that means something special about how you can rotate
or not things in space m M exactly.
Speaker 2 (38:52):
So what does this mean about the universe that Octonians
don't respect the associative property. It's very strange, it's very unphysical,
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.
Speaker 1 (39:07):
Wait, doesn't it just mean you can't put on your
socks and your shoes at the same time.
Speaker 2 (39:11):
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.
Speaker 1 (39:18):
Yeah, it would make a difference, for sure. I'm sure
would work.
Speaker 2 (39:24):
I mean, I think some people put their underwear in
their pants on at the same time, don't they. I
don't know.
Speaker 1 (39:28):
Oh, well, well, why didn't you start with that analogy?
Speaker 2 (39:30):
I didn't want to talk about underwear? I guess.
Speaker 1 (39:35):
So you're saying that, uh, Quaternions, you can't do that.
You can't put your underwear inside of your pants and
then put them on both at the same time.
Speaker 2 (39:41):
You can for you can for quaternions. You can't for Octonians,
That's right.
Speaker 1 (39:45):
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?
Speaker 2 (39:55):
The underlying rule of the universe is somehow related to underpants.
We don't know.
Speaker 1 (39:59):
Hey, yeah, there you go, or d end up looking
like Superman with another word outside of your pants exactly.
Speaker 2 (40:06):
Maybe he's the original Octonian.
Speaker 1 (40:08):
That's right. Maybe maybe any Kryptonians to just really describe
the universe.
Speaker 2 (40:14):
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 pantalizing 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, ooh, well, that's
(40:37):
sort of like eight spatial dimensions and one time dimension
plus one dimension for like along the string, and so
this is kind of an eight there.
Speaker 1 (40:45):
But ten, you said didn't work? Why are they fixated
with ten if ten doesn't hold mathematically?
Speaker 2 (40:50):
Because string theory works in ten dimensions. There's a different
set of rules there. Mathematically, string theory works in ten.
Octonians work in eight. Can we somehow marry the two? Well,
maybe if.
Speaker 1 (41:00):
We take those ten dimensions the difference and make it nine.
Speaker 2 (41:03):
There's no compromising in math. It's not a negotiation with
the universe.
Speaker 1 (41:07):
You can't split the difference, well, I can't make quantum gravity. Worry.
How about uh, we'll split the difference.
Speaker 2 (41:15):
I'll give you points on the back end. All right?
How about that?
Speaker 1 (41:17):
That's right? Yes, we'll only make both of them half work.
Speaker 2 (41:21):
No, people are like, well, can we take eight of
those ten 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.
Speaker 1 (41:38):
Can't you just decrease the number of dimensions in string
theory or it wouldn't work anymore?
Speaker 2 (41:43):
No, it doesn't work. It only works in ten dimensions.
There's another version of it that works in twenty six,
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, number eight,
So maybe Octonians would be a better way to describe
(42:04):
the strong force. Maybe the whole reason we have trouble
doing calculations in quantum chromodynamics is because we're used in
the wrong kind of math, and it would all just
like click beautifully into place if we replaced it with octonians.
Speaker 1 (42:15):
Wait, what do mean there are only eight gluons? You're
just looking for things in nature that you know about
so far that somehow count it eight.
Speaker 2 (42:23):
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
(42:44):
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.
Speaker 1 (42:55):
Huh, 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.
Speaker 2 (43:07):
Time throwing some oranges and a couple of socks. Yeah, exactly.
I know people are working on that, right. There a
lot of people studying the nature of optunians and trying
to use them to build a physical theory, but nobody
succeeded so far. Just sort of like a direction some
people are sniffing in as we try to build mathematically
consistent theories of quantum gravity.
Speaker 1 (43:27):
Wow, you're just looking for the number eight.
Speaker 2 (43:29):
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 one hundred years.
So we're like digging deep in the mathematical toolbox to like, well,
what else we got in here? Let's see what else
could be useful.
Speaker 1 (43:43):
Well, I can't seem to get up before eight am?
Could that be a rule of the universe? I could
blame that.
Speaker 2 (43:49):
On Well, we could flip the blame. You could say
nobody's solved this problem because you won't get up before
eight am.
Speaker 1 (43:55):
Well, that would still make sense. That Still, that's still
an explanation for the universe.
Speaker 2 (44:00):
Or maybe it would take an announcement of quantum gravity
to get you out of it before eight am.
Speaker 1 (44:05):
Unlikely, unlikely in this universe. I don't think that was
mathematically hold. Yeah, 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 octogoggles on and look for things that
(44:25):
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.
Speaker 2 (44:35):
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.
Speaker 1 (44:52):
Right, And that's kind of what's happening with strength there.
Although that one seems to be following the rule.
Speaker 2 (44:56):
Of ten yeah, or the rule of twenty six.
Speaker 1 (44:58):
So it sounds like maybe physicists you need to sit
down and peel more onions.
Speaker 2 (45:05):
And stop crying.
Speaker 1 (45:07):
That's a right stop pretend. Pretend that tears come from
cutting these onions and not from being a physicist. And
maybe that will inspire some new theory. That may it'll
crack open, they'll peel away the truth of the universe.
Speaker 2 (45:20):
Or maybe we should just take a break in good
Applebee's and have a bloomin onion.
Speaker 1 (45:24):
Yeah, there you go. There are different ways. As long
as it's after eight am, I'll join you, all right, Well,
and 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.
Speaker 2 (45:42):
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.
Speaker 1 (45:52):
All right, Well, we hope you enjoyed that. Thanks for
joining us. See you next time.
Speaker 2 (46:01):
For more science and curiosity. Come find us on social media,
where we answer questions and post videos. We're on Twitter,
disc Org, Insta, and now TikTok. Thanks for listening, and
remember that Daniel and Jorge Explain the Universe is a
production of iHeartRadio. For more podcasts from iHeartRadio, visit the
iHeartRadio app, Apple Podcasts, or wherever you listen to your
(46:23):
favorite shows.
Hey, Daniel, what are your favorite numbers in physics?
Speaker 2 (00:11):
Oh? There are so many good ones. I mean I
love planks 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.
Speaker 1 (00:26):
But have you ever wondered why numbers are so important
in the universe?
Speaker 2 (00:30):
I mean, numbers are like the currency of physics. We're
predicting things happening at times and places, and those are
just numbers.
Speaker 1 (00:38):
Wow. So does that mean you get paid in numbers?
Speaker 2 (00:42):
I get a number of dollars every year.
Speaker 1 (00:44):
I guess we all get paid in numbers if there's
just numbers in our bank account. If you're so lucky,
the whole universe is just numbers. I think you 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
(01:17):
Jorge Mack, cartoonists and the author of Oliver's Great Big Universe.
Speaker 2 (01:21):
Hi, I'm Daniel. I'm a particle physicist, and I've been
a professor at UC Irvine for a large number of years.
Speaker 1 (01:27):
Oh, too many to count? Or do you go into
some sort of like weird subspace when you do physics.
Speaker 2 (01:34):
No, I think I'm going to enter a puba phase
eventually and then emerge as an emeritus professor.
Speaker 1 (01:39):
That's a butterfly. That's a beautiful tenured emeritus.
Speaker 2 (01:43):
Butterfly, as a white haired moth.
Speaker 1 (01:45):
I think instead of wings, do you just have a
bunch of research papers taped together.
Speaker 2 (01:53):
I'm going to fly too close to the universe on
wings of research papers.
Speaker 1 (01:56):
Right, But you know, butterflies don't have mouths, right, so
you won't be to talk.
Speaker 2 (02:00):
Yeah, but they have those long newses, and so I'm
already said.
Speaker 1 (02:04):
There you go. You can sniff out science from there.
M hmm.
Speaker 2 (02:10):
I can snort up all the information in the universe.
Speaker 1 (02:12):
Oh boy, that sounds a little illegal.
Speaker 2 (02:15):
Are you going to arrest a butterfly? Is that what
has come to?
Speaker 1 (02:18):
Maybe a cocaine sniffing butterfly. Maybe maybe it will be
our informant for, you know, figuring out how the universe works.
Speaker 2 (02:26):
Maybe altering your mental state is important for understanding the universe.
Speaker 1 (02:30):
M I hear that's called the butterfly effect.
Speaker 2 (02:33):
I think that's something else.
Speaker 1 (02:35):
Meant something else. Did I get that wrong.
Speaker 2 (02:39):
I think it's called microducing.
Speaker 1 (02:40):
Yeah, I think this is called micro punning, which it
doesn't work as well. But anyways, welcome to our podcast
Daniel and Jorge Explain the Universe, a production of iHeartRadio.
Speaker 2 (02:49):
In which we give you a drip, drip, drip, little
doses of the incredible wonder we 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.
Speaker 1 (03:06):
That's right, We try to get you high on the
little dopamin head of wonder and amazement at how our
universe works and the surprising ways in which it still
puzzles scientists even today.
Speaker 2 (03:16):
And one thing that still puzzles scientists and philosophers is
at 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.
Speaker 1 (03:36):
Wait, wait, are you saying math is more important than physics.
Speaker 2 (03:41):
I mean, is English more important than Shakespeare. You can't
really compare the two things, you know.
Speaker 1 (03:47):
Wait with this analogy, Shakespeare's math in English is physics.
Speaker 2 (03:51):
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.
Speaker 1 (03:58):
I see. I see. So physics a comedy or a tragedy.
Speaker 2 (04:02):
It's a tragic comedy, for sure.
Speaker 1 (04:04):
It's a tragic comedy.
Speaker 2 (04:06):
We're all waiting for the final twist. Nobody knows how
it ends.
Speaker 1 (04:09):
That's right, it never ends well for Shakespeare.
Speaker 2 (04:12):
But I hope we laugh along the way.
Speaker 1 (04:14):
And you also have to wear tights when you do physics.
Speaker 3 (04:17):
Mmmm.
Speaker 2 (04:17):
And as always, the jester is the wisest one.
Speaker 1 (04:21):
I see. That's the engineer and the team.
Speaker 2 (04:24):
Did you just call engineers jokers? I mean, I'm just
gonna step slowly away from that.
Speaker 1 (04:29):
No. I know. We have a great sense of humor,
is what I'm saying. We're the smaltest person in the room.
Speaker 2 (04:35):
Often laughed at and unwisely ignored.
Speaker 1 (04:37):
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.
Speaker 2 (04:49):
Yeah, there are certain constants which seem to be important
to 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
(05:12):
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.
Speaker 1 (05:23):
And this is 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
understanding how things work at the particle level.
Speaker 2 (05:33):
Math 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.
Speaker 1 (05:46):
Okay, so it's at every level, but it seems like
in the Standard model there are especially some interesting numbers.
Speaker 2 (05:51):
Yes, absolutely, the theories of particle physics are very mathematical,
and not just in the sense that they're predicting numbers
and places in time. 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.
Speaker 1 (06:09):
Right, which sort of eraises 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.
Speaker 2 (06:21):
Yeah, and it makes you wonder, like what does that mean?
And our 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.
There's 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
(06:42):
all sort of mental games.
Speaker 1 (06:44):
Right, right, Like maybe the universe is not really physical,
maybe it's just sort of like conceptual theoretical.
Speaker 2 (06:50):
Yeah, but then you wonder, like what breeds fire into
all those concepts it makes us experience it. The other
side of that argument is that numbers are not universal,
they're not physics, 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
(07:10):
you can like describe the color orange using a bunch
of words, but none of those words are orange or
fully capture the oranginess of an orange.
Speaker 1 (07:19):
Mm and is one really the loneliest number? Is another
big physics question, right.
Speaker 2 (07:25):
Yes, yes, a very deep physics question. But we have
to 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.
(07:47):
There's a lot of hints there that the laws we
use to describe the universe are deeply mathematical.
Speaker 1 (07:52):
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 to be. On the podcast,
we'll be asking the question, what are octonions? Did I
say that right? Or is it actonions?
Speaker 2 (08:13):
I think it's oct onions, Like, give me eight onions
for that recipe, please.
Speaker 1 (08:17):
Isn't it like a blooming onion? Isn't that a dish
in a fried dish in one of these famous restaurants?
Is this like a fried onion that looks like an octopus?
Speaker 2 (08:28):
Are you shilling for Outback Steakhouse? Now?
Speaker 1 (08:30):
Oh? Is it out back?
Speaker 2 (08:31):
I don't know, never been.
Speaker 1 (08:32):
Maybe Applebee's is interested.
Speaker 2 (08:36):
I think it's maybe octonians or octanians, depending on where
you are in the world.
Speaker 1 (08:41):
So this is an interesting word. It sort of sounds
like octo, which is eight. 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.
Speaker 2 (08:55):
Mm hmm, yeah, ions exactly.
Speaker 1 (08:57):
Oh yeah, right right yeah ions. So is it denions?
Speaker 2 (09:00):
Then I think we should have eight different pronunciations of
this word.
Speaker 1 (09:05):
Now a trace you go, and then we'll spend eighty
eight minutes talking about it here on the podcast as.
Speaker 2 (09:11):
Part of our Cult of eight. There you go, send
eight dollars to join.
Speaker 1 (09:15):
There you go. We'll have eight dollars in our pockets.
In our eight pockets because I wear cargo pants exactly.
Speaker 2 (09:22):
Maybe these are the spiders of the universe.
Speaker 1 (09:23):
It's all an intricate web of eight. Well, anyways, as usually,
we were wondering how many people out there had thought
about or even heard of the word octonions and what
they think it might mean.
Speaker 2 (09:33):
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 two questions at Danielandjorge
dot com.
Speaker 1 (09:45):
So think about it for a second. What do you
think are octonions? And are they sold at Applebee's or
out back to steakhouse?
Speaker 2 (09:52):
And would they make you cry if you chop them?
Speaker 1 (09:54):
Here's what people had to say.
Speaker 4 (09:55):
What are onions, so octors, the prefix for eight onions
being something players, probably so eight layers stars.
Speaker 3 (10:05):
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 glue ons or any type of similar particle or
(10:27):
force carrier that is dependent on matrices for transformation.
Speaker 1 (10:37):
All right, well, pretty much the same as what I've
been guessing. Something to do with onions and eight and
or particles.
Speaker 2 (10:44):
Yeah, reasonable guess is absolutely.
Speaker 1 (10:46):
Is it like what happens when you smash together eight onions?
You get an ooked onion.
Speaker 2 (10:52):
We're working on the onion collider right now. We just
need a little bit more funding, eight dollars more and
we'll be there.
Speaker 1 (10:58):
You need a little bit more garlic, You're gonna say
a bit more seasoning. You need eighty eight billion dollars.
Speaker 2 (11:05):
I won't say no to eighty eight billion dollars, that's
for sure.
Speaker 1 (11:08):
Yeah, there you go. And then what happens at the
end when you collide the onions? Would you have to
eat them?
Speaker 2 (11:14):
I disappeared to my private island with all the cash.
Speaker 1 (11:18):
I think they can give you eighty years in prison
for that, Daniel, only if they catch me, only they
smell of the crime.
Speaker 2 (11:26):
I'll have an army of cocaine infused butterflies to protect me.
Speaker 1 (11:30):
Oh my gosh, you are a super villain there. I mean,
I've heard of sharks with lasers, but m drug butterflies
is a new level of villainy and they will never
see me coming. All right, Well, stick into it, Daniel.
What is an octonian and what do you think is
the right way to pronounce it.
Speaker 2 (11:47):
I pronounced it octonians, but it does make it sound
like creatures from the planet Octo or something.
Speaker 1 (11:53):
M I see you're going for the more tony pronunciation Octonians.
Speaker 2 (11:58):
Octonians exactly.
Speaker 1 (12:00):
So.
Speaker 2 (12:01):
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. They just have one component.
Octonians have eight components, one real and seven imaginary components.
Speaker 1 (12:24):
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.
Speaker 2 (12:32):
Imaginary numbers were invented in the sixteen hundreds, and they're
called imaginary because you don't see them in reality. They're
very useful in math. And there's basically one imaginary number,
which is I, which is the square root of minus one.
There's no real number which, if you multiplied by itself,
gives you minus one. So they invented a number. They
(12:52):
just call it I, and they say, if you multiply
I by itself, you get minus one. So it's a
new kind of number. It's different from any of the
numbers on the normal number line.
Speaker 1 (13:02):
Right, because like we had the number negative one, and
we have 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.
Speaker 2 (13:16):
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
(13:37):
what is the square root of negative one? So they
had to invent a new kind of number, sort of
imagine the real number line is like 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, and they have a number plane where every
number is two components, a real component and this imaginary component.
Speaker 1 (13:58):
Right, because you came up with us 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.
Speaker 2 (14:08):
Or negative two I. Right, the imaginary number line goes
both ways, right, right?
Speaker 1 (14:14):
And can you also have not I?
Speaker 2 (14:15):
You can have not it and like I'm not doing the.
Speaker 1 (14:17):
Dishes, Yes, that's what I mean, Like zero I, would
you call it not I?
Speaker 2 (14:23):
Oh? You can have zero? Yeah, the zero in the
number line is just zero Comma zero, right, zero real
numbers and zero imaginary.
Speaker 1 (14:30):
Right. So in physics, when you talk about an imaginary number,
you 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.
Speaker 2 (14:43):
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 or x and imaginary numbers are y.
Speaker 1 (14:56):
Right, And I think like all of most of quantum
particle physics is based on imaginary numbers, right, Like it's
a convenient mathematical space to do all of the math in.
Speaker 2 (15:05):
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 sixteen hundreds when they were invented,
before we had quantum mechanics, they were already useful. Italian
mathematicians came up with them as a way to 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,
(15:28):
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,
(15:51):
has a real component and an imaginary component. We couldn't
do quantum mechanics without complex numbers. They represent something real
about the universe.
Speaker 1 (16:00):
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.
Speaker 2 (16:09):
We don't know if the wave function is real, if
it 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
(16:32):
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 will 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.
Speaker 1 (16:53):
Well, the universe is definitely complicated, whether it's complex or
real or imaginary. Maybe it 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 to dig into that
and see whether or not it can help explain more
(17:14):
of the universe that we see around this and what
would that mean. But first, let's take a quick break.
All right, we're talking about Oktonians or Kanyans, which are
(17:35):
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.
Speaker 2 (17:50):
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.
Speaker 1 (18:08):
Well, was that like a big surprise, Like, wait a minute,
what does that mean about the universe that you need
imaginary numbers to describe it?
Speaker 2 (18:14):
Yeah? It has deep consequences 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
(18:37):
to quantum mechanics.
Speaker 1 (18:39):
Interesting, so there may be something point to something fundamental
about the nature of the universe.
Speaker 2 (18:43):
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, and 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.
So you have to come up with what mathematicians call
a division algebra, which is basically just like all the
(19:05):
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.
Speaker 1 (19:21):
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.
Speaker 2 (19:42):
Yeah, call it J. Right, J is like another square
root of negative one, a different a unique square root
of negative one. Now thing says you can't have more, right,
Maybe there's multiple square roots of negative one. And after
people discover complex numbers, mathematicians are 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
(20:04):
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, etc. People spend hundreds
of years on that.
Speaker 1 (20:15):
Let me think about that for a second. So now
we have a number plus a number I pleasent a
J number and you're saying, Jay's another square root of
a negative one. So if I'm multiplied J times J,
I get negative one. What if I'm look at IE
times J.
Speaker 2 (20:28):
So this is exactly what you have to do. You
have to come up with the rules of this triplet
system 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
(20:49):
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. Can't
do it with three.
Speaker 1 (21:01):
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.
Speaker 2 (21:11):
Yeah, you might imagine you should be able to do
it with any set of numbers, exactly the way you
can define spaces in any dimensions. You're gonna have a
one dimensional space or two dimensional space, so three dimensional space,
a nineteen 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
(21:31):
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
(21:54):
spent like decades on this, and he said once every morning,
coming down to breakfast, my kids ask me, well, pop up,
can you multiply triplets? And he always said nope, I
can always I can only add and subtract them. So
we spent decades like try to figure out how to
multiply and divide triplets of numbers and never succeeded.
Speaker 1 (22:11):
WHOA, but X, Y Z space works in three D
space all around this? Or are you saying that three
D space doesn't work mathematically or just triplet imaginary numbers
don't work mathematically?
Speaker 2 (22:22):
Triplet 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 two mathematical on
the podcast today. But this creates problems of having numbers
that don't have inverses, like one comma one coma zero
doesn't have an inverse, but it's non zero.
Speaker 1 (22:44):
Okay, I think I see what you're saying. I think
you're saying it's possible to just have like coordinates in
three D 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.
Speaker 2 (23:02):
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 together,
I always get a number that's not a zero if
(23:22):
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.
Speaker 1 (23:35):
Okay, so it doesn't work for three triple imaginary numbers,
but maybe for four imaginary numbers it does work.
Speaker 2 (23:41):
Exactly and the same. Irish mathematician William Hamilton he discovered
that if you put four numbers together, like four D
space one reel in 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 know need on quadruples of numbers, even though you
(24:02):
can't do it on triplets.
Speaker 1 (24:04):
Okay, So now that means that my number is not
just like eight, and it's not just like A plus
four I. It's like eight plus four I plus seven
J plus five K exactly.
Speaker 2 (24:16):
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
or other square roots of negative one, and that one
equation makes it all hang together. And when Hamilton had
(24:36):
this insight, he was like walking across this bridge in Dublin.
It 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.
Speaker 1 (24:48):
And you can still see it today.
Speaker 2 (24:51):
It's actually worn down, but they put a plaque on
that spot to commemorate it.
Speaker 1 (24:55):
And more important was this kid I pressed or because
the kids seem really interested in what was going on
with the math and his father.
Speaker 2 (25:03):
Yeah, I think he was just rooting for his dad,
you know, day his struggling with this crazy multiplication.
Speaker 1 (25:10):
And how much can a kid know about imaginary numbers.
Speaker 2 (25:14):
It's fascinating that the mathematics of it tells us what's allowed.
You can build a one D number, two D number,
a four D number, but not a three D number.
It's really interesting.
Speaker 1 (25:24):
Can you go five and six and seven?
Speaker 2 (25:26):
You can't. About one hundred years later, mathematicians proved that
the only sets of numbers you can do are one
real numbers, two complex numbers, four quaternions, and eight octonians.
Speaker 1 (25:39):
Ooh, and what can you keep going like sixteen thirty two?
Speaker 2 (25:43):
Nope, those are the only ones.
Speaker 1 (25:45):
What do you mean?
Speaker 2 (25:46):
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.
Speaker 1 (25:56):
That's it. That's nothing more in the hing more.
Speaker 2 (25:59):
In the universe. They prove this, and I try to
find a way to like explain this in intuitive sense,
but the proof is very, very complicated. But everybody's totally
convinced that one, two, four, and eight are the only
dimensions allowed for numbers that have consistent multiplication and division.
Speaker 1 (26:14):
You mean for like complex numbers, Yeah, complex numbers with
like extra imaginary dimensions exactly.
Speaker 2 (26:21):
You could have one extra imaginary dimension, which gives you
our familiar complex numbers are of two D. You can
have three imaginary dimensions, which gives you four D numbers quaternions.
Or you can have seven imaginary dimensions plus one reel
gives you eight dimensions for octonians.
Speaker 1 (26:38):
And still be consistent with our rules of math that
we know about in our universe. Could does be different
in another universe.
Speaker 2 (26:46):
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 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
(27:08):
basic rules.
Speaker 1 (27:09):
Okay, Now, I think the idea is that you know,
we had real numbers for a long time, and they
weren't great for regular physics. And then we found quantum
mechanics and particles, and we've figured out that one dimensional
imaginary numbers work really well to describe the math and
those theories where you have one extra imaginary dimension. But
you're saying, mathematician is also found out that you can
(27:31):
have three extra imaginary numbers or dimensions in a quaternion,
or seven extra imaginary dimensions in a Olktonian. 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.
Speaker 2 (27:49):
Exactly, Because obviously one D numbers are very relevant to physics,
two D numbers are very relevant to quantum mechanics. These
four D numbers you probably haven't heard of quaternions, but
these days we actually just call them four dimensional vectors,
and they're crucial for relativity. Like special relativity combines space
and time into a four dimensional structure where three of
(28:11):
them are similar to each other and one of them
is different. Right, doesn't that sound familiar?
Speaker 1 (28:15):
It sounds like space time.
Speaker 2 (28:17):
It sounds like space time exactly, And quaternions 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 quaternions the
rules of space time vectors. We call them four vectors.
We call them four vectors now instead of quaternions are
(28:37):
absolutely crucial to relativity.
Speaker 1 (28:40):
Meaning you need you can only do relativity math if
you're using quaternions, or it's just helpful to use them.
Speaker 2 (28:47):
No, they're fundamental to special relativity absolutely like.
Speaker 1 (28:49):
You can't you can't prove relativity without going into quaternions.
Speaker 2 (28:52):
Exactly if we didn't already have quaternions, we would have
needed to invent them or discover them, depending on your
philosophical take. When we build 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.
(29:15):
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.
Speaker 1 (29:22):
Mmmmm, Now you need four for relativity. Does it also
work if you call them formianss.
Speaker 2 (29:31):
They're a little crunchier, but they still were.
Speaker 1 (29:35):
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?
Speaker 2 (29:45):
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
(30:05):
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.
Speaker 1 (30:19):
But does it seem kind of random to you? Like
you need one extra 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 too? Here?
Speaker 2 (30:38):
Nobody understands it, right, then 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 two D numbers are
needed for quantum mechanics, four D numbers are needed for relativity. Well,
we've spent the laste hundred years looking for quantum mechanical relativity.
(31:01):
Maybe Octonians are the key to that, right, two times
four equals eight? After all, maybe the mathematical structure we
need to understand quantum gravity is based on Octonians.
Speaker 1 (31:12):
But why Goodtonians as anyone tried to apply quaternions to
like quantum gravity?
Speaker 2 (31:18):
Well, 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 mathematically, it does
not work. So it might just be that we're using
the wrong kind of number.
Speaker 1 (31:32):
Oh, I see, Like, maybe the key to it all
is just to go to Applebee's and order Sometonians.
Speaker 2 (31:40):
Absolutely, it might be well, I.
Speaker 1 (31:44):
Mean that is a joke, but like, I think that's
what you're saying. It's like, why why don't we order
an Actonian and see if it satisfies or it helps
us combine these theories of physics.
Speaker 2 (31:54):
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 and looking to see more of the pattern,
hoping that if you see more of it, you'll figure
(32:15):
out what's going on. Another approach is not to discover
anything new, but find new mathematical 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
(32:37):
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.
Speaker 1 (32:48):
Seem interesting, Like, that's a way to do science, Like
try to look for patterns and then come up with
ideas that might make sense of those patterns, and then
go back to the experiments to see if those theories are.
Speaker 2 (33:00):
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
(33:21):
we haven't yet figured out what they applied to, it's
worth thinking about what they might be applied to.
Speaker 3 (33:26):
Mmm.
Speaker 1 (33:27):
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 Atonium exactly.
Speaker 2 (33:40):
An Actonian could be the center of the onion eight
layers down.
Speaker 1 (33:44):
Yeah, there you go.
Speaker 2 (33:45):
I see where you're walking me to.
Speaker 1 (33:46):
Yes, all right, well, let's get into what these ideas
are about. How Altonians 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 stick into that. But first let's take another quick break.
(34:14):
All right, we're talking about Octonians, and are they the
imaginary numbers that scientists have been dreaming about to explain
everything in the universe. So, Daniel, you were saying that
the numbers is one imaginary number really helped describe quantum mechanics,
numbers with three imaginary numbers really help describe special relativity
and relativity, and because it lets you do math with
(34:36):
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 imaginary number, which might be the Octonian.
Speaker 2 (34:52):
Yeah, it might be exactly, And we're trying 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 the quaternions and complex numbers and real numbers
don't do. How are they different? And that's one fruitful
(35:12):
way to understand where they might be relevant.
Speaker 1 (35:15):
What do you mean, like, what can they do that's special?
Speaker 2 (35:17):
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
(35:39):
longer rank them necessarily. So as you add complex dimensions
you actually lose some capabilities.
Speaker 1 (35:44):
Things say crazier kind of right, like it's it's kind
of hard to order things in a plane or in
an X y plane exactly.
Speaker 2 (35:51):
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
(36:12):
don't commute. It like matters what order you do rotations in.
If you first turn left and then you turn up,
you get it to a different place than if you
do it in the other direction.
Speaker 1 (36:22):
For example, Wait, wait, I thought that you said that
quaternions and olktunians 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.
Speaker 2 (36:33):
These math rules that we require don't include that they commute.
They only include that you can multiply and divide an
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 order them quaternions,
you lose the ability to multiply them in any order
(36:53):
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, this is
really about space now, Octonians. Even weirder Octonians, which you
lose is the associative property. The associated property tells us basically,
you can distribute numbers within a parenthesis, like do you
do the multiplication within the parentheses first, or do you
(37:14):
distribute the number through the parentheses. It's 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.
(37:37):
The associated property doesn't hold for Octonians.
Speaker 1 (37:40):
M 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.
Speaker 2 (37:48):
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
(38:10):
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.
Speaker 1 (38:25):
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
(38:47):
them that means something special about how you can rotate
or not things in space m M exactly.
Speaker 2 (38:52):
So what does this mean about the universe that Octonians
don't respect the associative property. It's very strange, it's very unphysical,
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.
Speaker 1 (39:07):
Wait, doesn't it just mean you can't put on your
socks and your shoes at the same time.
Speaker 2 (39:11):
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.
Speaker 1 (39:18):
Yeah, it would make a difference, for sure. I'm sure
would work.
Speaker 2 (39:24):
I mean, I think some people put their underwear in
their pants on at the same time, don't they. I
don't know.
Speaker 1 (39:28):
Oh, well, well, why didn't you start with that analogy?
Speaker 2 (39:30):
I didn't want to talk about underwear? I guess.
Speaker 1 (39:35):
So you're saying that, uh, Quaternions, you can't do that.
You can't put your underwear inside of your pants and
then put them on both at the same time.
Speaker 2 (39:41):
You can for you can for quaternions. You can't for Octonians,
That's right.
Speaker 1 (39:45):
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?
Speaker 2 (39:55):
The underlying rule of the universe is somehow related to underpants.
We don't know.
Speaker 1 (39:59):
Hey, yeah, there you go, or d end up looking
like Superman with another word outside of your pants exactly.
Speaker 2 (40:06):
Maybe he's the original Octonian.
Speaker 1 (40:08):
That's right. Maybe maybe any Kryptonians to just really describe
the universe.
Speaker 2 (40:14):
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 pantalizing 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, ooh, well, that's
(40:37):
sort of like eight spatial dimensions and one time dimension
plus one dimension for like along the string, and so
this is kind of an eight there.
Speaker 1 (40:45):
But ten, you said didn't work? Why are they fixated
with ten if ten doesn't hold mathematically?
Speaker 2 (40:50):
Because string theory works in ten dimensions. There's a different
set of rules there. Mathematically, string theory works in ten.
Octonians work in eight. Can we somehow marry the two? Well,
maybe if.
Speaker 1 (41:00):
We take those ten dimensions the difference and make it nine.
Speaker 2 (41:03):
There's no compromising in math. It's not a negotiation with
the universe.
Speaker 1 (41:07):
You can't split the difference, well, I can't make quantum gravity. Worry.
How about uh, we'll split the difference.
Speaker 2 (41:15):
I'll give you points on the back end. All right?
How about that?
Speaker 1 (41:17):
That's right? Yes, we'll only make both of them half work.
Speaker 2 (41:21):
No, people are like, well, can we take eight of
those ten 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.
Speaker 1 (41:38):
Can't you just decrease the number of dimensions in string
theory or it wouldn't work anymore?
Speaker 2 (41:43):
No, it doesn't work. It only works in ten dimensions.
There's another version of it that works in twenty six,
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, number eight,
So maybe Octonians would be a better way to describe
(42:04):
the strong force. Maybe the whole reason we have trouble
doing calculations in quantum chromodynamics is because we're used in
the wrong kind of math, and it would all just
like click beautifully into place if we replaced it with octonians.
Speaker 1 (42:15):
Wait, what do mean there are only eight gluons? You're
just looking for things in nature that you know about
so far that somehow count it eight.
Speaker 2 (42:23):
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
(42:44):
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.
Speaker 1 (42:55):
Huh, 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.
Speaker 2 (43:07):
Time throwing some oranges and a couple of socks. Yeah, exactly.
I know people are working on that, right. There a
lot of people studying the nature of optunians and trying
to use them to build a physical theory, but nobody
succeeded so far. Just sort of like a direction some
people are sniffing in as we try to build mathematically
consistent theories of quantum gravity.
Speaker 1 (43:27):
Wow, you're just looking for the number eight.
Speaker 2 (43:29):
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 one hundred years.
So we're like digging deep in the mathematical toolbox to like, well,
what else we got in here? Let's see what else
could be useful.
Speaker 1 (43:43):
Well, I can't seem to get up before eight am?
Could that be a rule of the universe? I could
blame that.
Speaker 2 (43:49):
On Well, we could flip the blame. You could say
nobody's solved this problem because you won't get up before
eight am.
Speaker 1 (43:55):
Well, that would still make sense. That Still, that's still
an explanation for the universe.
Speaker 2 (44:00):
Or maybe it would take an announcement of quantum gravity
to get you out of it before eight am.
Speaker 1 (44:05):
Unlikely, unlikely in this universe. I don't think that was
mathematically hold. Yeah, 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 octogoggles on and look for things that
(44:25):
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.
Speaker 2 (44:35):
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.
Speaker 1 (44:52):
Right, And that's kind of what's happening with strength there.
Although that one seems to be following the rule.
Speaker 2 (44:56):
Of ten yeah, or the rule of twenty six.
Speaker 1 (44:58):
So it sounds like maybe physicists you need to sit
down and peel more onions.
Speaker 2 (45:05):
And stop crying.
Speaker 1 (45:07):
That's a right stop pretend. Pretend that tears come from
cutting these onions and not from being a physicist. And
maybe that will inspire some new theory. That may it'll
crack open, they'll peel away the truth of the universe.
Speaker 2 (45:20):
Or maybe we should just take a break in good
Applebee's and have a bloomin onion.
Speaker 1 (45:24):
Yeah, there you go. There are different ways. As long
as it's after eight am, I'll join you, all right, Well,
and 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.
Speaker 2 (45:42):
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.
Speaker 1 (45:52):
All right, Well, we hope you enjoyed that. Thanks for
joining us. See you next time.
Speaker 2 (46:01):
For more science and curiosity. Come find us on social media,
where we answer questions and post videos. We're on Twitter,
disc Org, Insta, and now TikTok. Thanks for listening, and
remember that Daniel and Jorge Explain the Universe is a
production of iHeartRadio. For more podcasts from iHeartRadio, visit the
iHeartRadio app, Apple Podcasts, or wherever you listen to your
(46:23):
favorite shows.
Hosts And Creators
Daniel Whiteson
Kelly Weinersmith
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