Okay so let's examine vision in
hyperbolic space, so we're just be going
to be in the in two dimensional
hyperbolic space here. Right so now
suppose that you're standing here and
you're in this ball of radius R, now let's
first think about Euclidean space and
you have a beach ball here. So that has
size 1. How far away can you see objects?
Why can I see those mountains out there
which are 10 or 20 miles away and I
can't see a baseball that's 10, 10 miles
away? It's because it's the percentage of
your field of vision that it takes up. So
if you look at- so your field of vision
at radius R, in Euclidean space this
field of vision here has length pi times
R, roughly 3 times R. So then this beach
ball takes up, at radius R, takes up 1
over pi times R of your field of vision.
So at 30 feet that's roughly 1%
of your field of vision, at 300 feet it's
roughly 0.1%, at 3,000 feet it's 0.01%,
you can't possibly see it. So
what happens in hyperbolic space? We make
the same estimate; so here's your thing
of radius R, there's your eye sitting there
at the middle, well how big is your field
of vision? Your field of vision is now
half the length of the circle, so it's pi
sinh R which is roughly pi over 2e to
the R. So even if it's 30 feet away it
takes up like 0.0000- they're not sure
how many zeros but- times 1%
of your field of vision. So it turns out
that your- that the beach ball doesn't
take up as much as 1 1000th of your
field of vision until you're within less
than seven feet. So you can't see that
beach ball until it gets six or seven
feet from you. So a beach ball 300 feet
from you is going to be a speck of sand,
you know in a very large beach. Suppose
Randy Johnson is on the mound; Randy
Johnson is a retired baseball player who
used to throw threw the baseball 100
miles per hour, so if you think about it
he's throwing that baseball 100 miles
per hour you don't see it till it's 6
feet from you.
Which sounds rather dangerous. On the
other hand if you think of it he's, you
know, 70 or 80 feet away so he can't see
you at all; so- and and to throw it
accurately at you he'd have to show- throw
it with about 0.001%
to get the ball to you so you're
most likely to walk. And wh- you're
never gonna see- he's not gonna see you,
you're not gonna see him, they're going
to throw the ball you're going to walk
the bases and go around. And if you think
about what this what this means for the
outfielders, the outfielders if you
actually were somehow- they got the ball
across the plate and you, despite not
seeing until with 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. Now people sometimes in your
audience may wonder this, why are the
units of hyperbolic space feet? Well it
turns out that this is American
hyperbolic space and in European
hyperbolic space the units are meters. A
lot of people ask about their favourite
sport so sometimes we talk about what we
call soccer and you call football, or
we'll talk about you know hockey; hockey
is big where I come from in Michigan.
(Brady: Tell me what you tell them about soccer)
(because we do have a lot of a lot of)
(European and English viewers. What do you tell)
(them when they ask?) - Well it turns out soccer
isn't going to work so well either
because you imagine your soccer field,
let's just say it's a hundred feet by
three hundred feet - I don't know what the exact
dimensions are but it's probably
something like that. So in in Euclidean
space that's going to be roughly 30,000
square feet. And it turns out that what
happens in hyperbolic space, if you make
these both a hundred and this 300, that
these very quickly get very close to one another.
So what happens in fact, the area is
about two pi; or you know around
6.3. You could imagine that you
could put a player from one team here and
a player from another team there and
that's all you'd have room for, right? And
now you imagine you're trying to kick it
towards the goal that's over here,
suppose you're on this team trying to
kick it towards the goal; it's almost
always going to go out of bounds because
this line has become very thin. Your only
chance is
to sort of roll it down exactly down the
line towards the goal. As we've learned
in hyperbolic space it's extremely hard to
keep things on a straight line. - (Brady: So that's)
(what- in soccer we would call that a very)
(congested midfield.) - Yes. You know- and it
turns out that hyperbolic space may be
rather sedentary or you're only going to be
use like fixed exercise equipment. If
you think about it, even if you decide to
walk around the block the odds of
actually ever getting home again are
quite low - unless you like leave a trail
of breadcrumbs.
[Preview] ...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 2pi
times 300, and then you take 1
degree worth of that that's 1/360th. That turns out
to be - if I remember right - something
like about 5.23.