So obviously
as a mathematician and a physicist
we're all very used to to sort of
playing around with Greek letters.
Omicron's a bit of an unusual one, right?
As you probably know the the all the
Covid variants are named after these
these Greek letters and they've been
going along in order. They got to the
Mu variant and then they skipped two and
went to Omicron. Obviously the first
thing you want to ask is why did they
skip skip two? Well the next letter after
after Mu was was Nu and they did they
thought that was too close to to 'new' as
in something big brand new.
One after that was Xi which was thought
to be too close to the name of the
Chinese president. So they thought that was
a bit bad, nobody wants to be associated
with the Covid variant so they they
went straight to Omicron or at least
that's the theory that's why they did it.
Omicron's bit of a weird Greek letter,
it's not one that we see that often in
as you know in maths but it does exist
and it does have a bit of history which
I think is is quite
interesting to talk about, and certainly
more pleasant to talk about I think than
uh the Covid virus. Well I think we
should go all the way back to the to the
second century with Ptolemy who was a,
you know, he wrote this Almagest which
was this big sort of article on
astronomy. And he was using Omicron
there. One thing we have to do is we have
to understand a little bit about how the
Greeks
wrote their numbers.
They had this kind of alphabetic
system; so for example Alpha represented
1, Beta represented 2, Gamma
represented 3 and so on. And they
went like that until they got up to up
to ten, so it went up one to nine. They
had special letters for 10, special
letters for 20, special letters for 30,
and so on; and then they would just pile it
all together. So I can give you an
example maybe just to illustrate it;
the number 666 - 666, devil's number. So a
Chi
and then I think this is a
Xi
and then a Zeta, and then they'd put a
line over the top. And what this means is-
so this
represents the number 6,
this represents the number 60 and this
represents 600,
okay, so 666, we put it all together.
Incidentally Omicron does appear in this,
okay?
It appears as the number 70. Okay so
Omicron which is just more like that-
(Brady: What's it look like?) - Well it's like- it's like an O basically.
But it also had another interesting use,
or people have theorised that it had
another interesting use, in Ptolemy's
system. So what Ptolemy did is he also
wanted to- he was trying to describe
astronomical systems and he was thinking
about you know sort of the angles in the
sky basically. So of course you have 360
degrees, you then have sort of 60
minutes and 60 seconds.
That that basically meant he was working
with kind of like a sexa-decimal
number system. So he needed a way to sort
of write those sort of 1/60th and 160th squared;
i.e either minutes and the and the
seconds. So suppose we wanted to do like 80
degrees 41 minutes
and three seconds, just for
you know pick a random one okay. How
would you write- how would Ptolemy have
written that? So he would have done the
80 okay, so the symbol for 80 is is pi,
so you'd have to put a pi in there. They'd
have had a new column. - (Why is there a)
(line above it by the way?)
That's to distinguish it from the
from the actual letters, okay, so they put
the lines above it for the numbers, okay.
80 is is is pi so that goes in the first
column because that's the number of
degrees. Second column, so now we need to
do 41 okay? So
40 is Mu
and there one is Alpha so that's written
like that. So that's that's 40 plus 1.
And then 3 is is Gamma so we put a
Gamma in there. Okay so this this is how
he would have written it, this
positional system. Then you have to ask
the question, right, well what if I didn't
have any minutes? What if I had zero
minutes?
Okay.
So didn't really have a symbol for that
well then so he said what what
Ptolemy used was- well he used this
symbol, which people have said is an
Omicron. Now there's a lot of debate
about this, because some people say, well
is it an Omicron? How could it be a
Omicron? Omicron's already 70.
But maybe it was an Omicron because you
couldn't put a 70 in any of these
columns anyway, because they only went up
to 60 right? You know 59. It didn't
really-
he could have got away with using an
Omicron there. - (It was like the first)
(symbol available to him that didn't have a use?)
Yeah yeah. Well it had a use - you
could you could put an Omicron in this
column right? Because you could have 70
degrees,
but it's but it wouldn't make sense to
have a 70 in either of these columns
right, because they only go up to 59.
There's a bit of arguing about this and
another reason people say he might have
used Omicron is that the Greek word for
nothing or none is called Ουδέν,
right, which begins with an Omicron in
Greek.
So so it kind of fitted in. So obviously
because of this then people start to
speculate, well this does look a little
bit like our 0 right? A lot actually, like a
lot. So people speculate, well are they
connected? But there's really no evidence
that they are connected, in fact almost
certainly not connected it's probably
just a massive coincidence because I
think obviously traditionally people
talk about that
our 0 came from much further east,
from India so it's almost certainly a
massive coincidence but that's this
ancient use of of Omicron.
So now I think we need to get a little
bit more modern. More recently, where has
Omicron popped up in the mathematical
literature? Okay so I know you've come
across Donald Knuth before, the great sort
of computer scientist from from Stanford.
Obviously we've seen him in the arrow
notation, the famous arrow notation which
we use for graham's number, so that's
pretty cool. He has this really
wonderfully cantankerous paper about
Omicron and its use in the mathematical literature-
I love this paper it's so
it's just so angry it's great.
I've got it here, so this is this is the
paper. Well I say it's modern but I mean
it was came out the year after I was
born so it's not that modern, 1976.
Here we are, so the title of the paper is
'Big Omicron, Big Omega and Big Theta'. The
reason he wrote this paper was because
he was really getting a bit fed up with
the way people were misusing Omicron in
the mathematical literature, and Omegas
as well actually. And he basically
decided that somebody needed to fix the
literature to sort out all the language
that people were using and normalise it and
that's the point of his cantankerous
paper which literally just sets
the sort of rules of what people should
be using. So what is Omicron, and I guess
Omega and Theta,
to Donald Knuth?
Well let me explain. It's to do with how
sort of fast things grow. If I say that
something is
Omicron, big Omicron, some people call it
Big O but
he calls it Omicron so we're going to go
with that. If something says
that
big Omicron n squared what does that
mean?
This means all the functions
which are bounded above; so they're
essentially smaller than things that
grow like n squared. So let me let me do
it graphically, so maybe that will
explain. So suppose
you've got-
we're going to send n to infinity okay,
so we're going to make n really really
big. So let me just draw n squared okay?
Grows like that okay? Or this could be
some multiple of n squared, this could be
some- you know, some number m times n
squared. Now all the functions that this
means, what does this actually mean, it
means all the functions which don't
quite grow as big as that. They don't get
they don't get as big as quickly as that.
So they might do this; so they might go
above in you know at small values of m
but eventually when you get big enough
they don't grow as fast, they always stay
below
some curve which is a multiple of n
squared.
So there's always a curve which is some
multiple of n squared which they stay
behind, and that's what this this
notation means
okay? Now it's important notation for
sort of thinking about how sort of
systems,
you know- maybe if you've got an
algorithm and you want to work out how
how sort of efficient it is; so we've got
to sort out n things and you want to
know, well,
how quickly does how many steps do I
need to use? Is it order n? Is it this
is it this this order n squared? Or is it
is it some other something else? It gives
you a feel for that, so it's really
important in that kind of thing. So
obviously you can put any function in
here right, okay, so you could do order n
cubed or the you know or any any
other function that- log n, anything right.
And you're just basically saying that
the the functions that you're talking
about don't grow as fast as some
multiple of that thing. What about
the other things that he's talked about
in the paper? Well they're kind of quite
simple so he has-
you can imagine something like Omega of
n squared, that's kind of the other way
around, they're the things that go faster
than it okay. - (The other side of the line.)
The other side of the line. So something
like this, so this would basically say
there was stuff which is also growing
like
you know some other n here. That they're
always bigger than some other number
times n squared, so they stay the other
side of the line. They can always find
something which is growing like n
squared, some multiple of n squared which-
and this guys will grow faster. Then
Theta which is the last one of the list,
that means you're talking about all the
functions which are
above something that grows like n
squared, some multiple of n squared, and
below
some other multiple of n squared. So
they're kind of bounded on either side.
These ones tell you all the functions
that are below,
these ones tell you all the functions
that are above,
and these ones tell you all the
functions that are in between
for some choice of m and n and
primed as I've written it. - (So with)
(Omicron and Omega we don't need two)
(lines do we, because they don't have to)
(be between, but with Theta there needs to)
(be two-)
(two lines to define our-) - Exactly, exactly.
So the Theta's kind of more precise, it
tells you it's really telling you an
awful lot of information that it's
really really growing like some multiple
of n squared so it's really trapped
between,
you know, that grow- two different growth
rates which are very alike. Whereas
the other two sort of just give you an
upper bound or a lower bound.
There's another
area where these are used, you don't just
have to take things to infinity to use
this notation, you can take things to to
constants or to zero; and that's where I
tended to come across this notation, this
this Omicron. And you could say, well what
is Omicron of x squared as x goes to
zero?
This basically means all the functions
which are smaller than x squared,
okay?
So then they're bounded above by x squared
so things like x- as x goes to zero, so
you can think things like x cubed or x
to the four. So as x goes to zero they're
always going- they're not growing as fast
as x squared, but it's the same kind of
idea. So for example let's-
let's draw the graph again, let's draw
something which goes like x squared
which is-
which can be 4 times x squared, or 5
times x squared, it's just a multiple of
x squared.
And we want all the functions which are-
we're going to x equals 0 now, so we're
interested in what's happening down here.
Okay, we're in the functions that are
like an x cubed which down here is is
growing more slowly,
okay. So all the functions which are
trapped down here but we're interested
in what's happening down down below okay,
near zero. - (Because eventually x cubed will)
(over-)
(in the other direction- ) - It'll overtake,
yeah absolutely absolutely. Yeah so you
couldn't have it- it's only a statement
about what happens to x equals zero. So
yes you see here, so here we're
interested in staying below as we go to
large values, here we're interested in
staying below as you go to small values
but you can use the same notation for
each, it just says you're bounded above
by by this thing by these this functions
which go like x squared or two times x
squared or three times x squared okay.
And of course you can apply the same; the
Omega and the Theta language there too.
And this is really common in physics
actually to use this. We use it in
perturbation theory, one thing we do in
physics is is perturbation theory which
is where you take
small stuff. So you- so to actually do
anything you take really small
quantities and you
expanded it, you sort of consider the the
first correction in that quantity.
Some things like- let me give you an
example: quantum electrodynamics. We
have a small number in quantum
electrodynamics which is the fine-
structure constant, which is 1 over 137,
that's a small number. So
what you can do is you can calculate
things to some sort of degree of
accuracy in the fine-structure constant.
So if we want like roughly a 1%
accuracy we only need to take the first
correction that's linear in the fine-
structure constant. And then we can go to
higher order, to things that go like the
square of the fine-structure constant
that would give us like 1% of
1% accuracy roughly. And you can
go higher and use this perturbation
theory. And that's how you do physics,
that's how you do physics, you can't read-
without perturbation theory physicists
are useless,
okay? Right so so we really need- and
obviously this kind of language which is
telling you how-
what are all the things that you're
sweeping under the carpet that you're
ignoring, what are all- how fast are they
growing? What are they bounded above by?
If we were talking about something like
the fine-structure constant we might say,
oh those terms are order Alpha squared,
you know, so we put an Omicron
Alpha squared or Omega Alpha cubed. They're
the things, so everything that we've
swept under the carpet that we're
ignoring, they're bounded above by this
thing, okay, functions which grow like
this thing.
And that's the idea of a-
so this is this is the use of Omicron in
in in maths and physics.
(When they named this new variant Omicron did you)
(straight away go, oh yeah I know that)
(symbol, I use it all the time in physics?)
Well I don't- see this is the thing,
funnily enough I don't think it does get
used. I never thought- I've used this all
the time right, this this notation is so
common right; I use it in my lectures, I use it
all the time when I'm writing stuff.
I never thought of it as an Omicron
really, I never- it never occurred
to me that it was an Omicron. Oh it's just an O,
right? You know, but, it's an Omicron
apparently according to Donald. That's what it is,
I should think of it as an Omicron
which I'm more than happy to do now.
I saw some debate on social media, people
say, oh we need- all need to start writing
papers with Omicrons now. Everyone's
like, everybody's already doing it, it's
like you just didn't realise so I think
that's quite cool.
Waiting for your chance to talk about it,
what book is this here? This is a book
called the Almagest by Ptolemy, and this
particular edition is very very old, 1515
is the publication date for this. It is
the first published edition of this book.
The book itself was actually written in
about 150 AD.
So I was asked on twitter to calculate the
number of SARS-Cov-2 particles, the
particles that caused Covid-19, how many
they were in the world? And so I did the
calculation and it turns out that there
are few enough to fit inside a regular
can of coke.