So I've got something to say about these balls today, and also
something called
anti-podal points or
antipodal points. I don't know which way people like to say it,
but I'm just going to call them
antipodal points. The best way you can think about it is on the planet Earth,
you have to believe the Earth is round, then it has a North and a South Pole.
Now those points are
antipodal (or anti-podal) which means if you draw a line between the North and South Pole
it'll pass straight through the center of the Earth.
So they're exact opposites on a sphere. But I want to just show you first a map of the world.
So here what I've got is a flattened out map of the world. So this isn't the world,
this is a flat version of it. Don't get confused between the two,
okay? This is a flat representation of a spherical world.
So North and South is up here, the North and South Pole,
but there's many, many more points we can have. There's the equator right? And so if we picked a point on the equator,
say boom, right here - Indian Ocean.
We could find out what the antipodal point is. And that would be
180 degrees around.
So for us that would be, we find that we end up here.
But also if we got a point here, there's an antipodal point here, or here, or here. Basically, there's an infinite number,
right? But it doesn't have to just be on the equator. There's also antipodal points in different places.
Most interestingly you can actually pick different towns or different locations. So we could pick - say here in Auckland,
there's a famous spot for rugby fans, Eden Park,
which is right here. And
the antipodal point for that is, again, we go back
180 degrees, which we end up somewhere over here.
But then we have to flip and we go up and then we end up down here in
southern Spain. So these two points
are as far away apart as you can be from one another. If I look at my tennis ball,
there's something interesting. We've got a seam, right?
Now the seam
actually cuts the
surface
into two equal areas.
That little patch there equals that patch there. So this seam is nice and smooth. It's a loop,
cuts the surface into two equal areas.
But if I just do a little bit of a research here,
look what I find;
antipodal points.
So on the baseball, again
we've got the same situation; two pieces of leather that are exactly the same shape, exactly the same area. And they have this seam,
it's nice and smooth. So it cuts the surface into two equal areas and
yes, I've got my finger on the- on the seam and my thumb. And look, I could rotate it like it's North and South Pole.
And then of course a cricket ball, which is sort of the simplest, because it's just got an equator running straight through.
So you could pick - you can really run your fingers all around the cricket ball.
And you can find heaps and heaps. But yep, there it goes. This is
special. If
we've got a situation where we have a surface of a sphere, if we have the surface cut into two equal areas
with a simple loop,
then that loop will contain antipodal points.
And now that, to a mathematician, is
a very exciting proposition. There has to be a reason why that happens. Because I can tell you
any loop which cuts the surface of a sphere into two equal areas
will always contain antipodal points. This has actually got a really special history for me, because when I was studying
maths at Melbourne Uni, a friend of mine who was just another student, his name is Jan Hi, he was
brilliant at mathematics. And he actually came up with this.
So he was just looking at a tennis ball, and sort of looked at the seam and found antipodal points, and he said
exactly that. And he proved it! And he explained to me the proof. And for me this was like one of the most beautiful experiences,
just someone coming up with maths. So I don't even know if this is a theorem, if it's out there,
I don't really care because it happened to me. It's a personal experience.
So I want to prove this to you right now Brady, okay? Right.
We need to have an understanding about antipodal
points to be able to make this work. And I think the best way to do it is looking at something we're familiar with,
which is the world we live on. I've already shown you that the equator has antipodal points by going
180 degrees in the opposite direction. And then also we've got these two towns.
So in actual fact, the way you find an antipodal point from wherever you are,
so where you are right now, is
that you rotate by 180 degrees, and then you flip your location
from - if you're in the northern hemisphere to the southern hemisphere, or vice versa.
So it's the same latitude.
So you basically go 180 degrees and then you use the equator as a mirror. So this here is rotating by
180 degrees, and
then if we take
the same distance here to here, which is taking the mirror image, and then we'll end up in the south of Spain.
So that's how we get an antipodal point.
But what if
we took the entire map of the world and
mapped it to its antipodal points? I've got another map of the world,
like so, and so first of all
I need to move it over 180 degrees. And I can do that really simply by
cutting straight through the Greenwich Meridian, which is 0.
I'm gonna cut straight through here. Ok, so this is now going to go
180 degrees, which means I move this over here. And
I move this behind.
But then we need to do a mirror image, which is really simple. If you've got transparencies,
you just turn it upside down. So there we go. So now we have the two maps
flipped. And we can see here, there we had
Auckland, so Eden Park, and Eden Park is now over there. And it's - I've got it pretty spot on. So it's pretty close.
What's interesting as well
is you can see a lot of the land mass of the world actually kind of maps to oceans.
So you've got maybe Argentina here and Chile, goes to parts of China.
You have Borneo going to parts of the northern parts of South America,
but yeah
a lot of seems - Australia, look it's in the middle of the Atlantic! And you've got Africa
sort of taking up this area here in the Pacific.
So in most parts of the world, if you go straight through the centre,
so wherever you are and go through the centre; you'll end up wet.
Yeah - well you'll burn to death first. But anyway, you'll end up wet on the other side.
And that's a good thing 'cause you'll probably be burning, so there. If you look at, say, our beautiful
home country, Brady. So the continent of Australia.
If we flip it, you can see what Australia looks like in the antipo- antipodal - antipodal map.
But what do you notice about the area of Australia, with the real Australia and then the antipodal Australia?
What do you notice about those two areas?
(Brady: Well they'll be the same?) - Yes.
Yes, so what that means is, an antipodal map
preserves area.
So if you have a region, and it has an area inside it, if you map it to another region
that area is the same. And that's really important for what we're doing here. I'm gonna take this tennis ball and
I'm going to transpose it to the map, the map that I was using.
So there's lots of different ways that you can map a a sphere to,
to a map; but the way we were using it,
well - that's a separate video really. You kind of come down like so. I'm not used to being this imprecise,
but that's alright. I'm just going to think it's a poorly manufactured tennis ball. Here's one we prepared earlier, and it's perfect.
So now, what I'm gonna do is I'm going to
create a transparency so I can do a
anti-podal map, or an antipodal map, which doesn't involve
calculating. Moving around.
Now I should have mentioned, if this was nice and perfect, which it is, then this whole area in here
is exactly half the total area. And so is this, and this is half total.
So this is half total, and this is half total. And so now if I take this little doddle-whacky here-
I could probably just make this, make my life easy - there like that, and go like that.
And I'll give it a cut down the middle. So I'm going to craft my antipodal map. So let's do that now,
so now we've got this.
180 degrees,
voila.
And
then flip.
And so
that's rough! That's rough, I've missed that altogether. Here's one we prepared earlier.
You should have six pairs of antipodal points. So you'll have twelve intersections, which correspond to six antipodal pairs,
yeah? Six North/South. - (Brady: How many did you get, hand- hand drawn?)
Hand drawn - wow. It's terrible.
So hand drawn I got,
one...got
hmm. I don't even know what's going on here.
Okay, we have everything we need to now prove that if you have a simple closed loop, that
cuts the surface of the sphere into two equal areas, then that loop must always contain antipodal points (or anti-podal points).
Now the way we're going to do that, is that we're going to create a loop
that doesn't
have antipodal points, but
we're going to just assume that it actually cuts the surface area of the sphere into two equal areas,
all right? And we can do that! Because this is the way mathematicians think - what if?
So I'm gonna create a curve here, and I'm gonna come and bring it down here, and now I'm going to go all the way up now here.
And maybe - where should I pop out?
Well, I've got to pop out somewhere there,
so to make it actually work,
I'm gonna need my ruler.
Now, this loop
doesn't have antipodal points,
but, let's just say;
and please don't check - because it's not true. It doesn't matter. This is half,
area total. We can do that. Now, let's create
an antipodal map.
So I'm an old hand at this now.
And not only that, I'm gonna copy these little lines here.
So first of all, 180 degrees.
Doesn't look good - but then we flip,
and it worked! So as you can see, the, the loop here
doesn't have antipodal points. Because I did the antipodal map and it didn't intersect. It did when we did the tennis ball, it
did - it would when you have the baseball or the cricket ball.
They would intersect. These don't intersect. But I'm claiming that this is half the area,
but that means
this is half the area.
So 1/2 plus 1/2 is already the whole, and what's all this bit in the middle?
That should be equal to 0, but it's not. And
so, as we do in maths,
it's a contradiction. And
so that's the reason why,
on a tennis ball, on a baseball, on a cricket ball;
you have these areas which are equal, cutting the surface into two equal areas. The seam is nice and smooth,
it's a loop. They have to contain antipodal points. It comes down to the beautiful
continuous symmetry of the sphere.
You know,
if you're the sort of person who holds a baseball and finds yourself thinking; you know what? I wonder how many
antipodal points there are along the seam?
Then you're the sort of person that might want to check out today's show sponsor, Brilliant. Brilliant is a website full of puzzles,
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(Brady: What colour do you think a tennis ball is?) - What colour? - (Is a tennis ball green or yellow?) - Oh
I'd say it's like a lemon yellow, isn't it? - (Brady: Just asking) - Really?
I'm colour blind with with reds and browns, but I would actually say it's kind of yellowy green.