So today we're going to talk about
hyperbolic geometry, it's a different
kind of geometry than what you're used
to, we all grew up with Euclidean
geometry. And we're going to talk a
little bit about the space then we're
going to talk about what it is to live
in the space so we get a sort of
visceral understanding of it.
Particularly we're gonna talk about what
it's like to play baseball, golf or
other sports in hyperbolic space. There's
lots of ways to sort of make a model for
it; so one is it's based on- it's just the
inside of the unit disc. It's the set of
points which are within distance 1 of
the origin, a Euclidean distance 1, but
now I'm going to define a very different
way of measuring distances between two
points, and this is going to have the
property for instance that if you want
to travel along the shortest
path in hyperbolic space, you travel
along these semi circles which are
perpendicular to the boundary. That's one
property it has. And another property it
has is as you start at the origin and
you walk straight out to the boundary
that's going to take you an infinite
amount of time, as long as you keep
walking at the same speed; it's an
infinite distance away, yeah. So this this
boundary isn't really there, which is why
I tried to draw it sort of in a dotted
way. So these lines are straight from the
hyperbolic viewpoint, which means that if
I wa- if I'm standing here at say
position a and I want to get over here
into position b this is the fastest way
to do so in hyperbolic space, that takes
the least time. - (Brady: What happens if I were to)
(go here and cut my way from here to here?)
(That's not-) - That would be longer. That would
take longer, it's sort of like in here if
I were to start here at position a and I
wanted to get to position b in Euclidean
space, and I were to do that; that- I'd
get there but it would take me take me
somewhat longer. Let me show you just
a couple facts that are true in
hyperbolic space and we'll use them and
see how they play out when we start
trying to do sports in hyperbolic space.
So one of them if I draw- so here
I'm in hyperbolic space and I have a
ball of radius R, ball of radius R in
hyperbolic space, what is its area? So
this is 2 pi
cosh R minus 2 pi - most people aren't
familiar with that, but that's roughly pi
times e to the R.
Now e is a number which is roughly 2.7
to the R. So that means every time you increase
you increase your radius by 1 you
multiply your area by about 2.7. So we
all know that if you start doubling and
you double over and over you grow faster
and faster, this is how compound interest
works. And if you look at the
circumference, or what's the length of
the circle, so mathematicians would
call a circumference, that turns out to
be equal to 2 pi hyperbolic sin times R;
and again that's roughly pi(e)to the R.
So the length of the boundary grows just
as fast as the area does. Now let's let's
compare that to Euclidean space. So the-
remember the area of a Euclidean ball of
radius R was pi(R) squared, and the
circumference in Euclidean space, or the
length of the circle in Euclidean space, is
2pi(R). So this grows quadratically, this
grows linear; and here both area and
circumference grow exponentially at the
same rate. Okay, let's think about what
happens in Eucli- in baseball.
So in baseball - we'll idealize this a
little bit. So we're going to have a
field, it's going to- made up of a
quadrant. And the infield has radius 100
and the outfield has radius 300; and so
if you look at this guy the area of this
outfield is just over 6- is just over
60,000, if I remember right it's about
63,000 okay?
And so in American baseball we have
three outfielders out here and they are
able to cover most of the outfield. And
so we can imagine that an outfielder can
cover about 20,000 square feet. Well what
would happen in hyperbolic baseball if
you took this same picture? Well now the
area of the outfield would be- it's 1/4
the area of the ball of
radius 300 minus 1/4 of the area of the
ball of radius 100; and that's much
bigger than 10 to the 99.
It's ginormous. So how many outfielders do we
need? Well if each of them can cover 20
thousand square feet, which is sort of a
dubious assumption, you need more than 10
to the 99 over 20,000 outfielders; which
is bigger than 10 to the 94. So one thing
you can see here is you're not going to
make any Disney movies about having a
family big enough to have an entire
baseball team, that's not going to work. I
move to the next next sport, we'll talk
about golf. So imagine you just- you're a
golfer here- - (Brady: [laughs] Stick- stick to)
(mathematics, professor!) - Anyway. So suppose
you know you're standing here and you-
your flag is over here 300 feet away,
have a nice little wedge in to the hole,
and you hit a very good shot - it's 1
degree off but it's exactly 300 feet. How
would you estimate that in Euclidean
space? You'd say, well, that's going to be
roughly 1 degrees worth of a circle of
radius 300, the circle of radius 300 has
length 2 pi times 300, and then you take
1 degree worth of that,
that's one 360th. That turns out to be, if
I remember right, something at like
about 5.23. So you're
about 5.23 feet from the
hole, that's pretty good. - (Brady: You should be)
(able to putt that in.) - Well you should but I
don't always do so. Now what would happen
if you did the same thing in hyperbolic
space? Now you're 300 feet, here's your
pin, you're 1 degree off; you hit it
exactly 300 feet you're feeling pretty good.
So we- suppose we make that same estimate.
Well what's the size of the circumference
of a circle of radius 300? Well that was
2 pi sinh of 300, which, remember, is like pi
e the 300. And now we're going to divide by
360, well that's bigger than 10 to the
100. So this estimate suggests you're
you're some number of feet which has
over a hundred digits away from the hole.
Okay so this can't be accurate, because
you know that you started 300 feet from
the hole and you hit it 300 feet so
you're at most 600 feet from the hole. So we
learned that this is not a very good
thing to do and not providing a very
good estimate so and- while as in
Euclidean space walking around the
circle isn't so very inefficient; in the
hyperbolic space that's going to be a
very inefficient way to get around.
(Brady: So how far are we from the pin then?)
So you can do a little bit of hyperbolic
trigonometry and it turns out you're
more than 590 feet from the pin, which is
almost as bad as turning around and hitting
it backwards. - (Brady: So- hang on, I was- what
(happened? Did it- it curved away or? What did the ball do?)
One way of thinking about this is, what does
this look like at- you can you're- suppose
you start at the origin, you travel 300
hyperbolic feet towards the boundary, and
then you're 1 degree off and you
travel 300 hyperbolic feet towards the
boundary, it turns out that the shortest
way to get there goes all the way, way
back in and comes back out.
(Brady: Okay so- so my second shot,)
(my next shot, is going to have to follow)
(a curve that's so sharp it's another big)
(whack?) - Right so you know you're likely- you
could- now you're 590 feet, now you're
likely to be 1180 feet away after your
next shot even if it's very good et
cetera. So you're probably never going to
get there. - (Brady: To my Euclidean eyes I)
(thought, oh wow I'm really close to the)
(pin now, this is easy. But in fact that)
(path isn't open to me - or is it open to me?)
Well it is but this turns out to be distance 10 to
the 100. And if you want a path that is
distance 590 it does something more like
that; it goes back in towards the center
and comes back out.
[Preview] The outfielders if you actually were-
somehow they got the ball across the
plate and you, despite not seeing until
it was six feet from you managed to hit it,
then the outfielders wouldn't see it
till it was six or seven feet from them
so their only strategy would be to stand
there with their mitts open.