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, um 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 at a 90 degree angle here on a sphere.

I can have triangles that have more than 180 degrees in him.

I can cut a piece of paper on here.

That would be, Ah, three sided square.

I could sort of shape this down here and here, 90 degrees down here.

It's not sure I do it correctly, but let's try this one.

Each of the angles here aren't 90 degrees.

Here's a 90 degree angle.

Here is a 90 degree angle, and here's almost a 90 degree angle.

Here's Ah, three sided square.

All three sides of the same length end all the angles, her 90 degrees plus here not to be confused with a ball, but it's fair 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 curvature Sze of those things multiply them together.

Positive for times positive, I'll get positive.

A sphere everywhere has constant positive current er, OK, 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, its horn shaped, and it has the delicious property that if I pick any point, it all on it, say right here in one direction, it'll bulge inward in the off 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 in the and the maximum outward going, or the maximum of either one of them.

I can't just rotate this 90 years.

I have to or a few degrees this way.

So it's like an ex.

I have to pick these so that in one direction it's bulging out, and the other one, it's bulging in.

Turns out that a pseudo sphere has everywhere constant negative curvature.

That means that if I multiply to get the golfing 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 courage.

Let me go back for a minute break on an ordinary Euclidean piece of paper.

Every triangle.

If I sum the angles, I'll get 100 80 degrees on a sphere.

If I draw a triangle, any triangle it all, I drawn it.

The sum of the three angles will be always greater than 180 degrees.

It's around the globe, true on the sphere on a pseudo sphere.

No matter how you draw a triangle, it's angles will always some 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 in five sides.

Origami ist Bob Lang showed me how to do this paper that he wrote and uses all sorts of hyperbolic see Can'ts and CO.

C can't stand and hyperbolic functions abound on the surface.

But the cool thing is, you can take a piece of paper, get wet, wrap it around here everywhere, has negative gassing curvature.

By choosing my point's just just right, I can find that eat.

Here's a five sided figure.

12345 sides.

Each of the corners is 90 degrees.

90 they're 90.

They're 19.

Check.

Keep me honest here.

Let's take a piece of paper.

That's 90 degrees, right?

There's a Here's a 90 degree angle right there, right there.

Yep, 90 degree angle There, over here.

Sure enough, 90 degrees.

90 degrees.

90 90.

Here's a five sided figure.

Five kitchens, the five sided figure.

All corners are 90 degrees.

Let's ax and cut out of paper like this here and say, Here's a piece off paper folded to fit, shaped to fit on top.

The suit of fear.

Each of these corners are 90 degrees.

It's a Pentagon.

It's a Pentagon.

Who's corners are all 90.

Riis.

There's a Pentagon or square.

If I say Hey, a square is a shape that has equal sides equal length edges in all corners, air 90 degrees than I could claim that this is a five sided square.

So its edges of equal length as well each of those edges is the same length.

What happens if I bash it flat?

Well, a nifty thing about apologies 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 conform will mapping and nonconformity, mapping and things like this We can do this.

But as I pushed down here, that bends out pushed down here.

Oh, I'll just bend over here.

Oh, maybe if I push down here Oh, that it bumps up here.

Bumps up there?

Uh, no.

How about if I push this one and this one down.

This one, this one this.

I've got five fingers.

123 123 45 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 Spirit kal or a pseudo spherical surface onto 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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