90 degrees; 90 degrees.
If I have a shape that has
90-degree angles and all the sides are the same length, it's a square.
Yeah,
but wait, you know as well as I do if I go to the Earth; go on the Earth and say
"Oh, I'm going to start on the equator, go up to the North Pole,
then move over 90 degrees and take the line of longitude down
that goes right through the Gulf of Mexico and Yucatan, and hit there,
oh, I've got a 90-degree angle here,
a 90-degree angle here, and a 90-degree angle here."
On a sphere, I can have triangles that have more than 180 degrees in them.
I can cut a piece of paper on here that would be a three-sided square.
I could sort of shape this down here, and
here,
90 degrees down here.
Not sure I drew it correctly, but let's try this one.
Each of the angles here
are 90 degrees. Here's a 90 degree angle, here's a 90 degree angle,
And here's
almost a 90 degree angle.
Here's a three-sided square.
All three sides are the same length
and
all the angles are 90 degrees.
A sphere, not to be confused with a ball, but a sphere
has constant Gaussian curvature, which says that if I pick a point, that point there,
and draw two lines through it and choose the maximum and the minimum curvatures
of those things, multiply them together,
positive number times a positive number: I'll get positive. A sphere everywhere has constant positive curvature.
Okay, neat.
Can I get something that has constant negative curvature?
Is it possible to make something that is the opposite of a sphere?
Something that instead of always bulging outward,
is something that's always curving inward?
Can I make something-
and, and, and, the answer is yes!
I can make a pseudo-sphere, a fake sphere.
It's funnel-shaped. It's horn shaped.
And it has the delicious property that if I pick any point at all on it, say right here,
in one direction
it'll bulge inward and the other direction it bulges outward.
I have to make a little, little point here that I have to pick the minimum inward-going and the,
and the maximum outward-going, or the maximum of either one of them. I can't just rotate this 90 degrees
I have to, or a few degrees this way so it's like an X, I have to pick these so that in one direction
it's bulging out and the other one it's bulging in. It turns out that
a pseudo-sphere
has everywhere-constant negative curvature.
That means that if I multiply
to get the Gaussian curvature here where
the
positive
going here is not very big, but the negative going is huge.
And over here, the positive, the
negative-going curvature is kind of small, but the positive curvature along here is quite big.
A pseudo-sphere everywhere has constant negative curvature.
Let me go back for a minute; breathe.
On an ordinary Euclidean piece of paper
every triangle,
if I sum the angles, I'll get 180 degrees.
On a sphere, if I draw a triangle, any triangle at all I draw on it, the sum of the three angles
will be always
greater than 180 degrees.
It's true on the globe; true on a sphere.
On a pseudo-sphere, no matter how I draw a triangle,
its angles will always sum to less than 180 degrees.
They'll be pointy in an odd way. They'll come to little pointy shapes. We can take advantage of that to draw
a shape that has
90-degree corners and five sides. Origamist Bob Lang showed me how to do this in a paper that he wrote;
and uses all sorts of hyperbolic secants and cosecants and and hyperbolic functions abound on this surface!
But the cool thing is
you can take a piece of paper,
get it wet, wrap it around here. Everywhere has negative Gaussian curvature.
By choosing my points just right I can find that eac-
here's a five sided figure; one, two, three, four, five sides.
Each of the corners is 90 degrees.
90 there, 90 there 90 there!
Okay check. Keep me honest here. Let's take a piece of paper that's ninety degrees, right? There's a,
here's a 90 degree angle right there right there.
Yep, ninety degree angle there!
Over here? Sure enough, 90 degrees.
90 degrees.
Ninety.
Here's a five-sided figure. A five- catch this, a five sided figure!
All corners are ninety degrees!
Let's act- and cut out of paper like this, here see.
Here's a piece of
paper
folded to fit,
shaped to fit on top the pseudo-sphere.
Each of these corners are ninety degrees.
(Brady: That looks like a pentagon to me, though)
It's a Pentagon. It's a Pentagon whose corners are all ninety degrees.
(Brady: So is it a pentagon or a square?)
If I say: "Hey, a square is a shape that has
equal sides, equal length edges and all corners are ninety degrees,"
then I could claim that this is a five sided square.
(Brady: So its edges are equal length as well?)
Each of those edges
is the same length.
What happens if I bash it flat?
Well, a nifty thing about topology says it shouldn't be possible.
If I try bashing this flat
It's like taking a section of a sphere, taking a section of the globe, and bashing it flat. There's not a good
mapping that will preserve
areas, that will preserve directions.
We get into the problem of
conformal mapping and non-conformal mapping and things like this. We can do this,
but as I push down here that bends out. Push down here - oh, I'll just bend over here.
Oh! Maybe if I push down here, oh! Then it bumps up here, bumps up there. Ah, no!
How about if I push this one and this one down? This one, this one, this- I've got five fingers; one, two, three-
one, two, three, four, five;
and up. But now it's bumping up here. I can sort of put it in a press,
but,
it's trying to pop out the bottom of the table.
You can't perfectly map a spherical or a pseudo-spherical surface
onto a Euclidean plane.
It's a problem that map makers have had for a long time, and it's a problem that shows up when you
start mapping our universe, because our universe,
apparently, seems to have something of negative curvature.
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