("Carmen Ohio")
(audience chattering)
- Good afternoon.
My name is Marty Golubitsky.
It's my pleasure to welcome you
to this month's Science Sundays talk.
I'm a professor in the
Mathematics Department
and also associated with
the Mathematical Biosciences Institute,
the center that's
sponsoring today's event.
Today's talk gives those of us
who organize the Science Sundays series
the opportunity to show off yet another
of the many talented
faculty who can be found
at Ohio State in the College
of Arts and Sciences.
As some of you may know,
several years ago, Ohio State decided
to grow its research strength
in several discovery-themed areas.
One of these areas was
translational data analytics,
which intersects MBI's interest.
Indeed, a growth area in the
bio and biomedical sciences
is the analysis of big data,
such as it appears in
the studies of the genome
and personalized medicine.
In his day job, Professor Kahle
uses a perhaps unlikely combination
of pure maths, topology, combinatorics,
and probability theory
to discover structure
and shape in big data.
Today, however, Doctor Kahle will explore
accomplishments of Greek mathematics,
many of which are still relevant today.
Join me in welcoming Matt Kahle,
who will talk about Archimedes,
mathematical superhero
of the ancient world.
(audience applause)
- Thanks for the kind introduction,
and thanks for the opportunity
to give a public talk.
It's unusual, I think, to get to talk
about our, say, mathematical interests
(laughs) in public.
(audience laughing)
They usually don't let us do that.
(audience laughing)
We have conferences that
are sort of for specialists,
and we just sort of,
there's 10 other people
in the world who are
interested in the same thing
that you are. (audience laughing)
But I feel like everyone should
be interested in Archimedes.
And the more I've learned about him,
the more amazed I've been.
So I think that everybody
has heard of Pythagoras,
the Pythagorean theorem.
And people have also heard of Euclid
and Euclid's "Elements" is probably
the most impactful geometry
text in human history.
But I think Archimedes was
the greatest of them all,
of the ancient Greek mathematicians.
And there's a strong case to be made
for greatest mathematician of all time.
But, I mean, it is very subjective.
And so we can come back
to debating that later.
So here's a nice painting
of Archimedes by Fetti.
It's "Archimedes Thoughtful,"
is the title of the painting.
And there's a few things
that are maybe noteworthy
in here.
He has some kind of
mathematical instruments,
a protractor, and it looks like maybe
he's got a drawing of
a sphere on this paper.
We will come back to this.
So he's sometimes called
Archimedes of Syracuse.
So this is where Sicily is.
Sicily is the largest
island in the Mediterranean.
And this was part of the
Greek Empire at the time.
So we think of him as a
great Greek mathematician,
although now Sicily is part of Italy.
And Syracuse is, we can
zoom in a little bit.
It's on the east coast of Sicily.
This was during the, well,
we'll talk a little bit
about what's going on,
what else is going on in
the world at the time,
during Archimedes' life.
But one of the things that happens
is the Second Punic War.
And so you can sort of see,
this is a war that involved
all the Mediterranean powers
at the time.
And some historians say sort
of in terms of its impact
or its magnitude, this was
sort of the largest war
in human history, even
bigger than World War One
or World War Two.
So they're sort of tumultuous times.
And Sicily and Syracuse are
sort of strategically important
for the navies in their location.
It's strategically
important for its location
in the Mediterranean.
So this is just a little context
and just what's happening
in the world right now
in the life of Archimedes.
The Roman Empire is about to sort of begin
its hundreds of years of
influence.
So here's some of the art from the period.
This is the
Nike of Samothrace, the
Winged Victory of Samothrace.
It's one of the only statues
that survives from the time.
But I think it's just incredible.
There're some other
statues that are similar,
that reflect the art
of Hellenistic Greece.
But most of what survives is Roman copies
of the original Greek statues.
But this statue still survives,
and it's been prominently
displayed at the Louvre
for some years now.
And just a close up, it's amazing.
It looks like real feathers almost,
but it's made out of stone.
So, sorry, I was giving
just a little context
and background, that what's
happening politically
in the world now.
It's sort of the peak of the Golden Era,
Hellenistic Greek mathematics and art.
And the Roman Empire is about to start
to begin waxing in its influence.
So the first story about
Archimedes I wanna tell
is the story of the golden crown.
So
we hear a story maybe 100 years
after Archimedes dies from Vitruvius.
And Vitruvius tells us
that the King of Syracuse,
Hiero the Second, had asked Archimedes
for his help with something.
The king had given some
gold to a goldsmith
and asked him to make him a crown.
And then he made him a crown.
And the crown weighs exact same amount
as the gold that he gave him.
But the king gives
Archimedes this problem.
He says, how do I know
that he hasn't cheated me?
How do I know that he hasn't
cut off some of the gold
and replaced it with an
equal mass of silver?
Kind of cut the gold with silver.
So by the way, this
crown that we have here
on this slide is also
roughly the same era,
same time and place.
So this might be what
the crown looked like.
So it's a very irregular shape.
You can't just compute its volume
with a simple formula like
it's a cylinder or something.
So people have, so then Vitruvius goes on
to tell the story of how
maybe Archimedes did this.
He says what Archimedes did is
he took an amount of gold
that was equivalent to,
that weighs the same
amount that the crown does
and then first submerged
the crown in water
and just made sure the water's just up
to the top of the bin.
And then he'd take the crown out
and now put the mass of gold in
and see if it comes up
right up to the same level.
If you replaced it by
something less dense,
silver is about half the density of gold,
then you would need more volume
to make it weigh the same amount.
So then the little tub should overflow.
So this is the story sometimes people tell
of how he figured out if
the crown was gold or not.
And here's one artist's interpretation
of the moment that he
comes up with this idea.
Eureka.
The bathtub's overflowing, so
Archimedes has discovered
the principle of displacement
of liquids.
But there's a number of problems
with this explanation that Vitruvius gave.
And
so one, and maybe the biggest problem,
is that it wouldn't work.
(audience laughing)
We could just cut straight
to the chase with that.
(audience laughing)
The reason it wouldn't work is
you can kind of do the
math and figure out,
I jotted down some notes 'cause I knew
I wouldn't remember the numbers.
So you made some reasonable assumptions
and you assumed that,
here's how big the crown is,
and let's say he cuts it with some silver,
but less than, say, 30% or more.
And then if he cuts it with
even more silver than that,
then the problem becomes even greater.
So you try to figure out
what's the difference
in displacement between the actual crown
and equivalent amount of gold.
And you assume you're
in a tub that's like,
let's say 20 centimeters
across or something
that you could just barely
submerge the crown in
and you could just barely
submerge the gold in.
Well, what's gonna be the
difference in the displacement
of the water, it ends up being less
than half a millimeter.
So the problem is there's all these errors
in the proposed experiment,
like surface tension of the water.
There could be air bubbles in the sort of
leaves of the crown.
And also, when you put the first thing in,
the crown itself, you take it out,
some water sticks to the crown.
So all those things end
up having a bigger effect
than
the difference in displacement
of the mass of gold.
So another problem with this, by the way,
is you have to come up
with a bunch more gold.
And maybe the king just
had an equal supply of gold
on hand, but
(audience laughing)
it seems a little suspicious.
So a lot of people have
thought about this story
over the course of the
last couple thousand years.
And one of them is Galileo.
And Galileo says no.
(audience laughing)
No, Vitruvius, that's not how he did it.
And I love,
yeah, I love Galileo
talking about Archimedes.
So he studied Archimedes' writings.
And he says, to him the biggest problem
is not just that it wouldn't work
and that maybe the king doesn't have
an exactly equal amount of gold on hand
to try this silly experiment.
But he says that if you read Archimedes
and how creative and how intelligent he is
that there's no way he would've done it
in such a sloppy way.
So,
yeah, so what Galileo
says Archimedes did is,
oh, sorry for this slide,
he says take the crown
and then take a lever
and take a little piece of gold,
just a little nugget of gold
and just put it far
enough out on the lever
that they balance, okay?
So we could do that.
We could take a crown and just
a small amount of pure gold
and just get them to balance on the lever.
As you know, Archimedes
is quite fond of levers.
And then you put them in water.
Then you submerge the crown
and the gold nugget in water.
So Archimedes wrote treatises on levers
that Galileo read.
And he also wrote
treatises on the principle
of buoyancy, that the amount of force
that's pushing, so it
seems like you're lighter
when you're in the water, right?
But it's just that gravity
is still acting on you
the same way, but now
buoyancy is pushing you up.
And the buoyancy is
proportional to the volume.
So when you submerge these both in water,
if they weigh the same, or at least
with the lever they weigh the same,
so they balance.
But now it's,
the principle of buoyancy should say
if they're the same density, then
the buoyancy of the
water should push them up
an equivalent amount.
But if they're not the same density,
then all of a sudden your lever tips.
And you could just do this,
you could actually just
have a small amount of water
to put each object in.
You don't even have to put
them all in one big tub.
So you can do the
back-of-the-envelope calculation
on this, too, in how much of a difference
in force would it make.
And it's something like, again,
making a bunch of reasonable
assumptions along the way,
something like 13 grams of force.
And so, easily a balance can detect that.
The balance'll just go ooh, you know?
So that makes a much bigger difference
than any of these other sources of error
that we were talking about before
with surface tension of the water
or anything like that.
So Galileo says that this
is how Archimedes did it.
I have to go back to that
slide, 'cause I love this
Galileo quote.
He says, "how inferior all other minds are
"to Archimedes's" mind,
"and what small hope is left
"to anyone of ever
discovering things similar."
So most of the slides that we have today
are various artists' paintings,
renditions of what they think Archimedes
might've looked like or of
his inventions and so on.
But in Galileo's case, we actually have,
this was a portrait in
his time by Sustermans.
Sustermans painted a couple
of portraits of Galileo.
So this is actually what he looked like.
And he looks a little frustrated, to me
(audience laughing)
in this picture.
And when you juxtapose it with this quote,
seems like maybe he's frustrated with,
he'll never equal Archimedes
in his accomplishments.
But probably more likely
he was just frustrated
that he was under house arrest
for the whole rest of his life.
(audience laughing)
So he thinks that this is
the way Archimedes did it
because besides being a more
accurate kind of experiment,
it's exactly based on the kinds of things
Archimedes was interested in,
the principle of the level,
sorry, principle of the lever,
principle of buoyancy, and so on.
So the next story about
Archimedes is "The Sand Reckoner."
This is sort of a, you could
say mathematical treatise
that he wrote and one of
the first mathematical
things that we'll talk about.
So he was interested in
very, very large numbers.
So he wrote a letter to King Gelon,
who, this is the son of Hiero the Second,
who we were talking about before.
And I think he was only in power
for maybe a year or two between King Geron
and the grandson, Hieronymus.
So Archimedes writes him a letter,
or his treatise is sort
of in the form of a letter
to the king.
It said, "some people think the
number of sand is infinite."
But some people think it's just,
he goes on to say that some
people think it's finite
but very large, and you
couldn't ever name a number
that large.
He says, but I'll show you
that, and it's easy to follow,
I'll show you a simple argument
that we can name numbers
that are very big,
so big, some of the numbers named by me,
"not only the number of the mass of sand
"equal in magnitude to the Earth,
"but also the mass equal in
magnitude to the universe."
So let's again put this into context.
Where are we historically?
They didn't have
any base-10 notation like
we take for granted now
for writing numbers.
And they also didn't have
the kind of algebraic notions
we have, or they had the notions,
but they didn't have the notation
for exponentiation and things like this.
So first of all, Archimedes
shows off a little bit
by just trying to construct
very large numbers.
And it's very wordy, because it has to be.
He doesn't have the notation in place.
So in this letter he describes
some very, very large numbers.
And it's sort of, well, if you multiply
a bunch of times, it's
essentially exponentiation,
but then you can do that operation.
So he essentially kind of ends up assuming
this nested exponentiation.
And the biggest number that he names here
ends up being equivalent in our notation
to 10 to the eighth to
the 10th to the eighth
to the 10th to the eighth,
but in that order.
So you can think about this,
10 to the eighth to the
10th to the eight to the
10th to the eighth power
is actually a bit larger number.
But, so the order in which
you exponentiate matters.
But here you can simplify this expression.
A to the B, if you raise
that to the C power,
that's just A to the B times C.
So by the way, in this treatise,
Archimedes derives some of the basic rules
of exponentiation.
And he says, essentially, in there
that 10 to the A times 10 to the B
is 10 to the A plus B.
So first he's just kind of playing around
and saying, look, I can
name some big numbers.
But then it gets interesting.
And he starts trying to contemplate, well,
how many grains of sand would you need
to fill the universe?
That's a fun question, right?
(audience laughing)
So he draws on, for inspiration,
some work of his
contemporary, Aristarchus.
I don't know if Archimedes and
Aristarchus knew each other,
but they were alive during the same time.
Aristarchus was maybe 20
years older than Archimedes.
So one of the things that Aristarchus did
that's really remarkable, and I think
it's one of the triumphs of Greek science,
he estimated the distance to the moon
and the size of the moon.
So this is an account
of Aristarchus's work
from maybe the 10th century or so.
And the smallest circle
is supposed to be the moon
and the bigger one the earth
and the bigger one the sun.
But it's not supposed to be to scale.
And even to Aristarchus, that picture
would not be to scale.
It's just drawn this way
so that we could make sense
of what are the relevant
angles and lengths
that we need to work with.
So let's back up a little bit.
First of all, Eratosthenes
had already computed
the circumference of the earth.
So have people heard this story before?
So Eratosthenes was a
I think sort of librarian at
the Library of Alexandria.
And he read an account
that on a certain day of the year,
the summer solstice, that the sun shined
straight down this well, which is
I think in Alexandria or someplace
that's very close to the equator.
And so then he decides to
actually check this himself.
But he also looks at
another place that's about
50 kilometers away on the same day.
And there the sun doesn't
shine straight down
at noon on the summer solstice.
There's a little bit of an angle.
And you can figure out
what the shadow cast is.
So then from that, Eratosthenes
deduces the circumference of the earth.
And so he already knew
the earth was round.
And then he says, here's how big it is.
And he was, this was just
naked eye observations,
but he was within, let's say five or 10%
of the right answer.
The Greeks knew how big the earth was.
I think a lot of people
have heard this story
of Eratosthenes, but far fewer people
have heard of Aristarchus's work.
So Aristarchus goes even further.
And he says, well, how big is the moon?
How far away to the moon?
I love asking people about this,
'cause if you ask one of your math
or physics savvy friends, they might say,
yeah, I think I've heard that.
And you say, well, then how did he do it?
And they'd say, well,
trigonometry or something?
It's like, yeah, or something, you know?
(audience laughing)
Trigonometry, except you don't know
the sides of any of the triangles.
Where are we even gonna start
with trying to figure out
how far it is to the moon?
So what Aristarchus does is
he waits for a lunar eclipse.
And by the way, the
evidence is that not only
could the Greeks predict lunar eclipses
but also solar eclipses,
which are much harder
to predict, it turns out.
You need much more accurate
kind of astronomical observations
and more complicated math
to predict solar eclipse.
But Aristarchus waits for a lunar eclipse
and observes it very carefully.
He sees the shadow of the earth
going across the moon.
And he can see how big the
shadow of the earth is.
But he already knows how big the earth is
by Eratosthenes' calculation.
And so he knows how big
the earth actually is.
And then he sees how big it appears to be
when it's projected against this screen
very far away, namely the moon.
And so then from that, he can deduce
how far away the moon is
and how big the moon is.
It's amazing.
And he was, so it's not quite
clear how accurate he was.
But some people say
that he was also within
about 10% error or something,
that he, just with naked eye observations,
figured out how far away the moon was.
And then he went on to try to figure out
how far away the sun was.
This is the most difficult
part of this calculation.
And the basic idea is you wanna measure
the angle that the
earth, sun, and moon make
when the moon is at half moon.
And
you need more delicate
information than what he had.
To really do this, his experiment
was sound philosophically.
His ideas were all correct.
But he just didn't have
sensitive enough instruments.
You're trying to figure out exactly
when the half moon is, and
you need to be accurate
within 15 minutes or something,
not just which night did it happen on.
You need to be accurate
more like 15 minutes
or something to get a good
measure of this angle.
But so they think that he thought
the distance to the sun was maybe
on the order of like nine million miles.
So that sounds maybe,
on one hand, not great,
because we know now it's more
like 90 million miles, right?
But it's really impressive, right,
that he's, again, just
naked eye observations
and deep thought, he's thinking about
how far away is the sun.
And what he realizes is the
sun is much farther away
than the moon.
And then it must be much larger,
'cause the sun and moon are each
about the same size in the sky.
That's part of what makes
eclipses so fascinating, right?
So, solar eclipses especially.
So Aristarchus concludes that the sun
is very far away and is much larger
than the earth or the moon.
And so then he proposes a
heliocentric solar system.
So I don't know why everybody
still says Copernicus, right?
'Cause in fact, Copernicus said
it was Aristarchus's
idea, the heliocentric,
or he's following Aristarchus's, right?
It's sort of one of these things
that humans discovered, that the sun
is at the center of the solar system,
and then sort of Aristotle and Ptolemy
and others thought, no,
we like the geocentric system better.
And then they kind of won
that intellectual debate.
And then we just continued to try to work
with the geocentric system
for hundreds of years.
And it ends up being a big mess
by the time you get to Kepler
and people who really wanna try
to kind of predict the
motions of the planets
more accurately.
So Archimedes is aware
of Aristarchus's work.
And he really likes it.
And he is totally on board
with the heliocentric system
for our solar system.
So Archimedes, going back
to the problem at hand,
he's interested in
trying to figure out
how many grains of sand
can fill the universe.
So the other thing that
Aristarchus had thought about
and that Archimedes comments on
in "The Sand Reckoner" is that the stars
are very far away.
All the other stars besides the sun,
they're even farther away.
And they think that's true because
they don't observe any parallax.
They try to measure the angle
of that star, of where that
star is in the sky here
and then where the earth is
kind of as far away as it's gonna be
like six months later, on
the other side of the sun,
measure where that star is again,
and it just seems to be
a fixed point in the sky.
So again, their methods were sound.
Philosophically, this is very keen,
their idea to try to
observe star parallax.
It's exactly what astronomers do now.
They just didn't have telescopes,
sensitive enough instruments
to really measure it.
Okay, so on to Archimedes' calculation,
he just says, well, we don't really know
how far away those stars are,
what kinda sphere they live in.
But let's just for fun say
that the diameter of the whole universe
to the diameter, essentially
of the solar system
or at least the earth's
part of the solar system
is the same as the
diameter of the earth orbit
to the diameter of the earth.
And remember that they know the diameter
of earth's orbit around the sun,
or at least they tried to compute it.
Aristarchus has, and thinks it's maybe
nine million miles or so.
And they have a pretty good guess
to the diameter of the
earth by Eratosthenes' work.
So then the missing thing in this equation
is the diameter of the universe.
And he makes some assumptions.
Maybe there's some nice
proportions in the universe.
So then what does he come up with?
He decides that the number of grains
needed to fill the universe is
something like 10 to the 63.
He just comes up with this big number.
One thing that's kind
of intriguing is that
there's a sort of coincidence.
So if you broke those
grains of sand down further
into nucleons, I mean, so most of the mass
of the universe is in
protons and neutrons.
So those are called nucleons.
Electrons are much, much lighter, right?
So this is Eddington's number,
is the number of nucleons
in the observable universe.
And now, with modern observations,
that's what we think the answer is.
It's 10 to the 80 nucleons.
So Archimedes has a bunch of
kind of wrong assumptions.
The universe is much larger
than what he thought it was.
And also, he's saying, well,
in that smaller universe,
what if it's completely
filled up with sand,
just, there's nothing
else in there, just sand.
Then how much sand?
And he said 10 to the 63 grains of sand.
But it's just kind of
amazing that that answer
ends up being what we think
the mass of the universe is,
basically, more or less.
That's what we think today.
So just a coincidence, I guess.
Or maybe Archimedes was a
time traveling scientist.
(audience laughing)
We're not sure.
So especially until the Middle Ages,
Archimedes was best
known for his inventions.
It's really the last
500 years or something
that we've really come to
appreciate his mathematics.
And that continues to unfold today.
But let's talk about
some of his inventions.
A lot of these come into play
in the defense of Syracuse.
So this is around 212 to 210 BC now.
And
Hiero the Second had been
kind of a loyalist to Rome.
And then he passes away, and
his grandson, Hieronymus,
is in power.
And they're maybe wanting
to align more with Carthage,
other powers.
And so Rome isn't having it.
And Rome starts to attack
the island of Sicily.
And this is the king
asking Archimedes for help
in this painting.
Archimedes is sitting on the ground.
The color didn't come
up right on this slide,
but he's working on his math.
And then the king rides up on a horse
and asks him for help
defending the city, Syracuse.
So what did Archimedes do to help defend?
We have a lot of stories about this,
most of them written 100 or 200 years
after Archimedes died.
So one is that he invented
new kinds of catapults,
really powerful catapults,
catapults that could
shoot farther than catapults had before,
maybe trebuchet-like devices.
So that's one of the
things that we have records
of him doing.
Another thing that they talk about
after Archimedes died
is the Archimedean claw,
or literally the iron hand.
And they say, so you can
think of the catapults
as your sort of long-range
defenses, right?
Like when the ships are way out there,
you can just see them.
You wanna hit them before
they get too close.
But what about the
ships that do get close?
Then they say that Archimedes
had some kind of iron hand
that would come out of the cliff wall
and tip the ships over.
And we're not really sure what that was.
But the speculation is
essentially some kind of crane
with something hanging on
it and some kind of lever
that, maybe something that
could pick the ship up
out of the water and then drop it again,
and it would actually sink.
So people have actually,
I was googling this
over the last couple days.
There was some TV show called
something like Super
Weapons of the Ancient World
or something. (audience laughing)
And so that was one episode, was devoted
to the Archimedes' claw.
And these people made some kind of crane,
lever sort of system that could actually
pick a ship up and drop
it, and it sunk the ship.
That was their experiment.
But I love Parigi's, the artist,
his interpretation of
it, 'cause it's actually
literally an iron hand
just picking the ship up
out of the water.
It's just, it's too good.
I love this painting. (audience laughing)
So here's another painting by Parigi.
And this is maybe the most famous,
the most notorious, you could
say the most controversial
of Archimedes' inventions
to defend Syracuse.
Did he make a heat ray?
(audience chuckling)
So a lot of people have
wondered about this.
They say that he did.
The people, again, it's like 100
or a couple hundred years
after the Siege of Syracuse,
there was accounts of him having
some kind of mirror and
catching ships on fire,
actually catching them on fire.
So keep that in mind.
Some historians now, they say, well,
we don't know if he actually
caught the ships on fire.
Maybe he just kinda tried to use the light
to blind the sailors or something.
So this is a controversial point.
And
Obama actually asked MythBusters
to pick up this question.
(audience laughing)
MythBusters asked Obama,
is there something
you want us to do for our show,
for an episode of our show?
This is what he wanted to know.
Did Archimedes really
make a heat ray or not?
And he may have had
national security reasons
for asking that that we don't know about.
(audience laughing)
Obama wanted to know.
So the MythBusters kinda
tried to do it with mirrors,
and they couldn't get it
to catch the ship on fire.
So they were saying, well, maybe
it's just kind of blinding light
or something like that.
But then some students from
MIT took up the challenge.
And they took a bunch of mirrors.
And this is part of what's clever.
So let's say you have a bunch of mirrors.
And some versions of the
story are that it was
a bunch of soldiers' shields standing up
on the walls of the city.
But one problem with that
is how do you get them all
to focus on the same spot?
We could all be chasing each other around
and just have a bunch
of little spots, right?
So you could say, no,
everybody go to my spot.
But all the spots look the same.
So these guys had a really nice idea.
Look at, in the center there is a big X.
And it's only the reflective
part that is the X.
The rest of it is, so they
first kind of aligned that
to make the target on
the side of the ship.
And then they went around and aligned
all the other mirrors very quickly
to get them all on the same point,
running around, they've
got the X in place.
And now they're running around
aligning all the mirrors.
And yeah, they caught that ship on fire.
(audience laughing)
So then I'm not even sure what
the controversy is anyway,
you know, anymore.
Could Archimedes have thought of that?
We're talking about Archimedes.
I think he could've thought of that.
So this is the defense of Syracuse,
various inventions along the way.
So now we come to the
Antikythera mechanism.
And this is something that was discovered
in a shipwreck in the
Mediterranean and near Sicily
in the early 20th century.
And then since then, they've
continued to study it.
And scientists can shine x-rays through it
and various kinds of light and figure out
what's going on inside of this machine.
So here's the reconstruction of it.
It seems like it had about 30 or 40 gears.
And so they reconstructed
what the gears were,
and they say it's very advanced machinery.
Nobody had any idea that there
were machines this complex
in that time.
And they can, what's
really amazing is then
they're not just shining light on it
and saying, what's in there.
Then they kinda try to figure out,
what was it used for?
So they start figuring out what the ratios
of the gears were.
So what they think this machine could do
is predict the motions
of the five known planets
at the time.
And then it could also predict eclipses.
And so you can actually find on YouTube
a video of one of these that
somebody made with Legos.
And so they used I think
twice as many gears
as the actual Antikythera mechanism.
But all the gear ratios are the same.
And then there's these
two dials in the front,
and it looks like when the dials point
in the same direction that
it's predicting an eclipse
on this other dial, or something.
So it's a really remarkable machine.
By the way, this whole thing, with 30
or 40 gears and these dials, it's in a box
about this size.
So, small and pretty advanced computer.
Well, what does this have
to do with Archimedes?
It's not thought that he made this one,
and the best dating on
it is maybe a little bit
after Archimedes' time,
maybe 100 years later.
But there is stories of
after the sack of Syracuse,
after Rome eventually
successfully took over Syracuse
and the whole island of Sicily,
that General Marcus Marcellus came back
with a couple souvenirs and
that he had two machines
like this, made by Archimedes
that could predict the motions
of the planets and so on.
And he kept one for
personal, just trinket,
(laughs) the spoils of war.
And then the other one went in a temple
or sort of on display for
people to see in public.
So there's reason to think that Archimedes
was making machines like this.
And the time and place is
very close to Archimedes.
So even if he didn't make this one,
the Antikythera mechanism,
it might've been
one of his students or someone
in the Archimedes school
who sort of actually made it.
It's definitely the most
sophisticated computer
we know of from ancient times.
Okay, so let's get a little bit more
into the mathematics of Archimedes.
His approximation of pi is something
that he's famous for.
So the idea is that you
can approximate a circle
by inscribing and circumscribing polygons,
regular polygons.
So in the center, we have the hexagon.
So you could think that,
so if we wanna know
what pi is, that's the
ratio of circumference
to diameter for the circle.
So that inscribed hexagon, whoops, sorry.
The inscribed hexagon
must have circumference
a little bit less than the circle, right?
If you go around that inside hexagon,
you're shortcutting, and you didn't go
all the way around the circle.
Well, so it's easy to compare the diameter
of the circle to the
circumference of that hexagon.
The ratio is three.
That's the ratio of the hexagon,
the diameter of the hexagon
to the circumference
of the hexagon is three.
So that tells you already that pi
is at least three, from
the inside hexagon.
And then the outside hexagon,
the circumference must be bigger
than the circumference
of the circle, right?
And if you work it out a little,
you need to know what the
triangles in there are
and maybe do a little trigonometry,
you get that the
circumference of that hexagon
is the square root of 12, or
two times square root of three.
So then that tells you pi is at most 3.46.
So just from that picture, we know
that pi is between three and 3.5.
But what Archimedes did is
he used a regular 96-gon.
So then it's not just the idea,
but you actually have to do it, right?
You have to do the
trigonometry to figure out,
well, what is the circumference
of that regular 96-gon,
inscribed and circumscribed?
And so then he does that in nice ways
and sort of says, well, I can
estimate that angle by this.
And he ends up with nice
fractions in the end.
So he makes estimates that are simple
but accurate.
So what does he conclude?
He concludes that pi
is at most 22 sevenths.
That's actually the fraction.
That's often what you'll hear people say
is a decent approximation for pi.
And then he showed that
it's at least 223 over 71.
So
the Greeks were very into
integers and rational numbers.
And you hear stories about the discovery
of irrational numbers.
I think Archimedes was
perfectly aware of this.
And I think he probably
also realized that pi
is an irrational number
and can't be expressed
as a ratio of two integers.
I bet he suspected that was true,
that the best you could do is just sort of
put it between two fractions.
That's a way of kind of pinning it down.
So if you're keeping
track, if you work out
what the fractions are, he showed that pi
is about 3.14, and the next
digit probably one or two.
So it's a pretty nice
approximation of pi that he did.
Okay, so another thing that he did,
and this is unfortunately
in one of his lost works,
is he classified what we
now call Archimedean solids.
So first of all, there's
the platonic solids.
And these are the sort of
most symmetric polyhedrons
that you could have.
And there's the cube, the tetrahedron,
the octahedron, dodecahedron,
and icosahedron.
So what characterizes
those is that every face
is a regular polygon,
and it looks the same
at every vertex.
Like, say the dodecahedron, you have three
regular pentagons all meeting at a corner.
And every corner looks like that.
So here you don't have to
have every face be the same.
But they all have to be regular polygons.
And then it has to look
the same at every vertex,
at every corner.
But these get quite complicated.
So for example, here we have something
that's called the truncated
icosadodecahedron.
(audience laughing)
And it has 120 vertices.
And you don't just have
triangles, squares,
pentagons, and hexagons for your sides.
This is a dodecagon.
So there's an account
that, I think from Pappus,
again, some time after
Archimedes passed away,
that he classified these objects
and that he came up with a list of 13.
And now that's how we know that,
now we know there are 13.
So we think that that's what
Pappus is talking about.
So Kepler rediscovered this
in his "Music of the Spheres."
He was interested in this,
and he rediscovered this,
these 13 shapes.
But I would really love to know one day,
how did Archimedes do this?
How did he think of it,
and how did he prove it?
It's a tricky theorem,
and one of the reasons
it's a tricky theorem to
prove there's exactly 13
is it's not true.
There's infinitely many polyhedrons
that satisfy what I just said.
Every face is a regular polygon,
and it looks the same at every corner.
'Cause like any prism, so you could take
a regular 100-gon and
then make it into a prism.
And then every side around
it, there's 100 square sides
and then 100-gon on top
and 100-gon on bottom.
And then similarly, there's an anti-prism,
where you just kind of rotate the top,
and then you make all the side faces
equilateral triangles.
So this isn't just a curiosity
that these come along.
It's that if you do the theorem,
if you prove the theorem correctly,
which Kepler did, these show
up in your classifications.
That has to be part of your proof.
And you can say we don't wanna count those
as Archimedean solids, but you somehow
have to account for them if you wanna say
that the only polyhedra
that have these properties
are the following 13.
So really the following 13
plus two infinite families.
But this sounds like a very
modern mathematical theorem
where you're asked to classify
certain mathematical objects.
And there's sort of infinite families,
and then there's just
finitely many counterexamples.
So the classification
of finite simple groups,
for example, is considered
one of the triumphs
of 20th century mathematics,
that they said all of
the finite simple groups
fall in three infinite families,
and then there's 28 exceptions.
And it's a very difficult
theorem that took
hundreds of pages, tens of thousands of,
or sorry, hundreds of papers,
tens of thousands of pages to prove
the classification of
finite simple groups.
So this theorem isn't
difficult in the same way,
but it's similar.
It's that you wanna classify
some kind of symmetries,
and you get some infinite families.
And then those 13 are
the kind of special ones,
the exceptions.
So unfortunately, how
did Archimedes classify
the Archimedean solids is lost.
Maybe it'll be rediscovered.
By the way, this is one of
the things that comes up.
And it's another thing
that's not quite right.
What I said, that every
face is a regular polygon
and every vertex looks the same,
that's kind of the way they
would've talked about it,
say, in Euclid's "Elements."
But what we would say mathematically today
is something like vertex transitive,
that you have symmetries,
but take any vertex
to any other vertex.
So this is the cuboctahedron,
sorry, rhombicuboctahedron on the right.
So just imagine, by the way,
if somebody in Scrabble
played cuboctahedron
and you dropped rhombi out in front of it
(audience laughing) to have
the rhombicuboctahedron.
(audience laughing)
I think that would be a legal
Scrabble move, I'm not sure.
So this is very nicely symmetric.
And just by rotating this thing around,
you can move any corner
to any other corner
and have the shape back
in the same position.
But this one on the left is
the pseudo-rhombicuboctahedron.
That's the other, your
Scrabble opponent puts
pseudo on the front. (audience laughing)
Pseudo-rhombicuboctahedron.
It still looks the same at every vertex.
Every vertex you have three squares
and an equilateral triangle.
But you don't have the
global symmetries anymore.
So it just looks, so we
would say it's the difference
between a local isometry
and a global isometry.
What's changed is the bottom.
You've taken the bottom
and turned it a little turn
like a Rubik's cube or something.
And now you've kind of
messed up the symmetry.
So does that belong in
your classification or not?
It depends on what you
make your definition.
So this is something that the literature
is full of mistakes with for this reason.
You'll see lots of textbooks and
popular math articles that say, well,
you just wanna have regular polygons
and the same, it looks
the same at every corner.
But
if you do that, then there's
two infinite families,
prisms and anti-prisms.
And then there's also this
pseudo-rhombicuboctahedron.
So, okay. (audience members laughing)
So quickly I'll say something
about the palimpsest.
So one of the things
that was found in here,
so the palimpsest was a text,
a recording of some of Archimedes' work.
And then they reused paper.
They reused books then.
So somebody had bleached out all the pages
and then printed a bible
or something else on it.
And so, but then with modern technology,
they can sort of shine
ultraviolet light on it
and recover this lost Archimedes works.
And so that's been just in
the last 10 or 20 years,
people have been studying the palimpsest.
But one of the things that
comes up is the stomachion.
And it's this dissection
puzzle with 14 pieces.
And Archimedes asks
how many ways are there
to rearrange these 14 pieces.
And it's remarkable to us now
because that seems like a
question in combinatorics.
It's a very modern kind of question,
like how many ways are there to arrange
these pieces to make this shape?
And
people thought that combinatorics really,
maybe it's only really been enacted
for maybe 50 years, 60 years.
But some people say,
well, it began with Euler,
two or three hundred years ago.
Those were some of the first
real studies in combinatorics.
But it seems like
Archimedes might've done it.
Other people wrote about the stomachion
at the same time, and it seems
like it was probably a popular puzzle.
And you could just see what animals,
what shapes you could
make with those pieces.
But then Archimedes looked at it
and asked interesting
mathematical questions about it.
How many ways are there to
shuffle these pieces around?
By the way, this is sort
of like the closest thing
to my own research as
anything I'm gonna say
in the talk.
It's that kind of question
I'm very interested in.
I'm interested in sort of
randomly rearranging shapes.
What kind of random shapes can you make?
And then to do that, you need to be able
to count how many ways there are,
how many different rearrangements.
At least, you need to be able
to count that approximately.
Okay, so this is, I wanna just tell you
one last story, which I think is the most
remarkable story, the most remarkable part
of Archimedes' mathematical work,
which is the computation of
the volume of the sphere.
So first of all, let's review.
How do you know the area of
the sphere is pi r squared?
And this is kinda the schematic proof.
You divide the circle up
into sectors like this.
And then you can kind of
rearrange these sectors.
And as the number of the sectors
is going to infinity, they
get thinner and thinner.
And this shape here ends
up being closer and closer
to a rectangle.
And it's r by pi r, this
rectangle, it turns out.
And so we get, you can kind of guess
that the area of the
circle is pi r squared
from method of exhaustion
kind of argument, this.
But another way to say this,
it's very clear that
Archimedes was doing calculus,
although people like
to say it was invented
by Newton or Leibniz.
He could do everything that
we consider calculus today.
He could differentiate,
and he could integrate.
And another way to say this is
the circumference of
the circle is two pi r.
That's just the derivative
of the area, pi r squared.
So if you know calculus, the derivative
of pi r squared is two pi r.
So if you defined the
circumference of your circle
to be two pi r and you
know a little calculus
or you can make an argument like this,
then that gives you immediately
pi r squared for the area.
The problem is in three dimensions,
you can make a similar argument.
But what it's gonna tell
you is the derivative
of the volume is the surface area.
That's more or less what you're gonna get.
The problem is you don't know either one.
If you knew one, you could find the other.
But you don't know either one.
So Archimedes did something amazing.
This is, we know of
one or two other proofs
he made of this, by the way.
But this was the most remarkable.
And people speculate this
is what he did first,
how he actually computed
the volume of the ball.
So he compares it to
the volume of the cone
and the volume of the cylinder.
And here's a picture with
all of them inscribed.
What Archimedes' theorem tells you
is that if you take
the ratios of the cone,
the sphere, and the cylinder,
the ratio is one, two, three.
That's beautiful, right?
So what's much easier is that the ratio
of the cone to the
cylinder is one to three.
But how did he figure out that the volume
of the ball is two?
Well, he actually used a lever.
In the palimpsest, he describes
the method of mechanical,
the mechanical method, or something.
And so I don't have a
picture of them on a balance,
but what he does is he sliced,
he does integration by slices.
And so he looks at a
slice of the hemisphere
and the cone and the cylinder.
So here the cylinder and
the cone are shown together.
Take a slice at the
top, a horizontal slice.
Those circles both have the same area.
And over here on the hemisphere,
we don't have any area.
It's just a point.
And now you know where that slice
through the shapes, and then
say you're at the bottom,
then you've got the circle at the bottom
of the hemisphere and a circle
at the bottom of, but the
cone has gone to a point.
So what Archimedes
noticed, and all you need
is the area of a circle is pi r squared,
at every level, not just
at the top and bottom,
the area of the sphere
plus the area of the cone
equals the area of the cylinder.
They're always circles.
He's just saying the area of this circle
plus the area of this circle equals
the area of this circle.
So then if you know that the ratio
of the volumes of the
cone and the cylinder
is one, technically then the volume
of the sphere must be two.
So he computes that the
volume of the sphere
is four thirds pi r cubed.
The surface area of the
sphere is four pi r squared.
And he thought this was so important,
he did it several different ways.
But I think this is
the most beautiful way.
And again, him using the lever
is just amazing.
So this is sometimes called
Cavalieri's principle.
And it was rediscovered much later,
the idea that if you take little slices
and you get the same area in your shape
at every slice, then these
must have the same volume.
Here it's illustrated with coins.
Even though they're very different shapes,
these have to have the same mass, right?
They have the same volume.
So, oh.
Well, I think I'm out of time.
So I won't tell you about another proof
that Archimedes did.
He didn't consider this, the
mechanical method rigorous.
So he tried to give
more convincing proofs.
Nowadays, we would consider
it perfectly rigorous,
'cause we know how to make
integration by slices,
that's what we teach
our calculus students.
We know how to make it precise.
But he was worried that
it wasn't quite precise,
so he also gave method of
exhaustion kind of proof,
where he kinda takes a little polygon
and spins it around
halfway around the sphere.
That gives a really
complicated polyhedron.
And then he gets in and
computes the volumes
of all these little fulstrums.
So it's a really difficult proof.
But by the method of exhaustion,
he again computes the volume of the ball
and gets the same answer.
So, in the Siege of Syracuse,
Marcus Claudius Marcellus,
the attacking general,
had orders to take Archimedes alive.
And there're various apocryphal stories.
And some of the stories
are that some soldier
came across Archimedes and
asked him to come with him.
And Archimedes said,
don't disturb my circles,
like I'm doing math, leave me alone.
(audience laughing)
And he got killed for his insolence.
And Marcus, the general was upset
that Archimedes was killed
in the Siege of Syracuse.
He wanted to talk to him.
He wanted him alive.
I like this artwork.
And you can see various things in here.
Some of these dodecahedrons
in the background,
notably the sphere and the
cylinder on his chalkboard,
and also the Archimedean screw,
which is also attributed to him.
Archimedes considered the computation
of the volume of the sphere
his greatest accomplishment.
For me, I might've thought
maybe the heat ray or something.
(audience laughing)
He knew how important it was,
how deep this mathematics was.
It's just a fundamental thing in geometry.
How can you do three minutes of geometry
if you don't even know what
the volume of a ball is?
He figured it out.
And so according to legend,
there was, on his tomb,
there was a sphere,
a cone, and a cylinder, a sculpture.
And Cicero tells us 100 years later,
he found Archimedes' tomb.
And this is a painting
of Cicero and his friends
discovering Archimedes'
tomb, brushing away,
pulling away the brush and finding it.
So you can see the
sphere way up at the top
with the cylinder.
So it's a good trivia question
to ask mathematicians.
It's a good year to ask,
who's on the Fields Medal?
Fields Medal is often
considered the highest award
given to mathematicians.
And there's all these great mathematicians
it could be, Gauss, maybe Newton.
So it's Archimedes,
Archimedes who's on the Fields Medal.
And I really like that on the back
of this year's, of the Fields Medal,
there it is again.
It's the sphere and the cylinder,
this amazing theorem of Archimedes.
So I think I'm out of time.
Thank you for your time and attention.
(applause)
(man talking quietly)
- [Audience Member] Do
you share these stories
with your students?
- I do share these
stories with my students.
I have to tell you, where
this first came from,
my interest in Archimedes was,
I was teaching a Canada-USA math camp
for high school students.
And I saw a talk by my friend Sam Payne,
who's now a math professor at Yale.
At that time he was a graduate student
at University of Michigan.
And he gave a talk about Archimedes
and the mechanical method to
these high school students.
And you could just see
little minds exploding
around the room.
I mean, our minds of the other teachers
who watched the same talk did, too.
But one thing I loved was
these high school kids,
then they went out, they had
a little cult of Archimedes,
I think, after that. (audience laughing)
I don't really know what it was.
I wasn't initiated into the club.
(audience laughing)
But especially, I think
there's lots of opportunities
as a math teacher to talk about this,
especially when we do calculus,
when we teach calculus, which is sort of
our bread and butter
in the math department,
is teaching calculus to
thousands of students
every year, all the engineers
and all the scientists,
is that we should show
them how he computed
the volume of the ball.
What a beautiful proof
of integration by slices
that you have.
And I think often it's not
in a calculus textbook.
It's a missed opportunity.
Yeah.
(man talking quietly)
I don't know much about it, dude.
I think a lot of the
mathematical discoveries
were probably when he was younger.
And then I think the inventions came
out of necessity, like when it seems
like Syracuse is being sieged,
and then he comes up
with the Archimedes claw
and the heat ray and all that.
But I like to think that as a young man,
that he had plenty of time to just think
and go for long walks.
I think he was a healthy person.
And he lived to be 70 or 80 years old.
And that must've been a very long time,
I think, in ancient Greece.
So I'd like to think that he was,
I don't have any evidence of this,
I just like to think that
he took lots of walks
(audience laughing) and
thought about math a lot
and that he was free
to just kind of explore
what was interesting to him
before sort of duty called
toward the end of his life.
(person talking quietly)
It's the, so can you,
so you were saying you didn't think
that surface tension
would really be an error
in Vitruvius's proposed method of-
(person talking quietly)
That's right.
(person talking quietly)
Right, so I think that people
tried to replicate Archimedes' experiment
by doing what Vitruvius said
and what other people said,
which was take equal amounts of gold,
put the crown in and pull it out
and make sure the water's
right up to the top,
then take it out and put
an equal mass of gold in.
And if the water comes
just up to the top again,
then you'd say it was pure gold.
And if it overflows, then it would be,
maybe it's cut with silver.
But the thought is it doesn't work because
the level of the water only increases
by a half millimeter.
And that might be such a
small amount of water that,
and also some of the water
came out with the crown,
sticks to the crown, that then even
when it's a little past full, the thing
might not overflow because surface tension
is still holding it together.
So, but I think, so it's controversial.
Could he have done it that way?
Maybe, but I like to
think Galileo is right,
that Archimedes would've
thought of a more subtle way
and a more accurate way to do it.
And it seems like right up his alley,
doing it on levers and also using
the principle of buoyancy.
And it's true, which is so then buoyancy
is gonna be proportional to the volume,
and so then we're gonna get
a more accurate measurement.
Yes?
(man talking quietly)
That's right, and I think he was worried
about, so the question
is, was Archimedes worried
about paradoxes that come up in calculus
about, can you cut something
up into infinitely many
thin slices, infinitely
thin and get the volume
of a shape?
So he was worried about that.
But he did, he thought
of it different ways
at different times.
So the method of
exhaustion, he wants to, say
find the area under a parabola,
and he kind of just says, well,
I could, maybe there's a little box here
and I could add up this
box and add up this box.
So then there's an infinite series
that just adds up to something.
And I think he was comfortable with that,
with the method of exhaustion,
that that could be made rigorous.
But then he liked mechanical method.
And people speculate
now that that might be
how he actually first
computed the right answer
for the volume of the ball,
four thirds pi r cubed.
But he was worried that
it wasn't rigorous,
integrating by slices.
It's a tricky notion to say,
just the areas of these circles
is all we're gonna compare.
That's two two-dimensional things.
How can you deduce something about
these three-dimensional shapes
by only two-dimensional notions?
Or another way to say it is
every one of those circles
has volume zero.
And then you're adding
up infinitely many things
of volume zero to get volume
four thirds pi r cubed.
That seemed, mm, so Archimedes was worried
about this kind of paradox.
And so he wrote down the method,
but then he tried to give
more rigorous arguments,
as well.
But they were more complicated,
more involved computationally.
But it's really neat
that 2,000 years later,
after a lot of people have thought
about infinitesimals and so on
that it turns out the basic method worked.
It can be made precise.
(man talking quietly)
(applause)
("Carmen Ohio")
