the curvature of space - university of washington

The Curvature of Space

Jack Lee

Professor of Mathematics

UW Seattle

Euclid (around 300 BCE)

The Elementsby Euclid

Euclid’s Postulates for Geometry • Postulate 1: A straight line segment can be drawn from any point to

any other point.


any other point.

• Postulate 2: A straight line segment can be extended in either

direction to form a longer straight line segment.

Euclid’s Postulates for Geometry• Postulate 1: A straight line segment can be drawn from any point to

any other point.




any other point.



• Postulate 3: A circle can be drawn with any point as its center and

any other point on its circumference.


any other point.





• Postulate 4: All right angles are equal.


any other point.





• Postulate 4: All right angles are equal.

• Postulate 5: If a straight line crossing two other straight lines makes two interior angles on one side that add up to less than two right angles, then the two straight lines, if extended far enough, meet each other on the same side as the two interior angles adding up to less than two right angles.

Euclid’s Postulates for Geometry • Postulate 5: If a straight line crossing two other straight lines


makes two interior angles on one side



that add up to less than two right angles,

1 + 2 < 180.




then the two straight lines, if extended far enough,

1 + 2 < 180.




then the two straight lines, if extended far enough,

meet each other on the same side as the two interior

angles adding up to less than two right angles.

1 + 2 < 180.

Euclid’s Postulates for Geometry Using only these five postulates, Euclid was able to prove all of the facts

about geometry that were known at the time.

For example,

Theorem: The interior angles of every triangle add up to exactly 180.

1 + 2 + 3 = 180.

Trying to Prove the Fifth Postulate… but that darned fifth postulate doesn’t really seem as “obvious” as the

other four, does it?

Many mathematicians thought it seemed more like something that

should be proved.

Trying to Prove the Fifth Postulate

Around 100 CE (400 years after Euclid):

The great mathematician and astronomer

Ptolemy, living in Egypt, wrote down a

proof of Euclid’s Fifth Postulate, based on

the other four postulates.

So the fifth postulate is not needed, right?

Ptolemy



The great mathematician and astronomer

Ptolemy, living in Egypt, wrote down a

proof of Euclid’s Fifth Postulate, based on

the other four postulates.

So the fifth postulate is not needed, right?

Wrong! Ptolemy’s proof had a mistake.

Ptolemy



The Greek mathematician Proclus criticized

Ptolemy’s proof.

Proclus



The Greek mathematician Proclus criticized

Ptolemy’s proof.

But he still believed that the Fifth Postulate

was a theorem that should be proved, not a

postulate that should be assumed:

“The fifth postulate ought to be struck out

of the postulates altogether; for it is a

theorem involving many difficulties.”

--Proclus

Proclus


So Proclus offered his own proof of the fifth

postulate …

Proclus


So Proclus offered his own proof of the fifth

postulate …

…which was also wrong!

Proclus



The great Persian mathematician and poet

Omar Khayyam published a commentary on

Euclid’s Elements, in which he offered his own

proof of the Fifth Postulate …

Omar Khayyam



The great Persian mathematician and poet

Omar Khayyam published a commentary on

Euclid’s Elements, in which he offered his own

proof of the Fifth Postulate …

… which was also wrong!

Omar Khayyam


Around 1700 (2000 years after Euclid):

The Italian mathematician Giovanni Saccheri set

out to “fix” Euclid once and for all. He wrote and

published an entire book devoted to proving

Euclid’s fifth postulate, and triumphantly titled it

“Euclid Freed of Every Flaw.”


Around 1700 (2000 years after Euclid):

The Italian mathematician Giovanni Saccheri set

out to “fix” Euclid once and for all. He wrote and

published an entire book devoted to proving

Euclid’s fifth postulate, and triumphantly titled it

“Euclid Freed of Every Flaw.”

Unfortunately, Saccheri forgot to free his own book

of every flaw. He, like every mathematician before

him, made a mistake. His proof didn’t work!


Other famous mathematicians who published “proofs” of the

Fifth Postulate:

•Aghanis (Byzantine empire, 400s)

•Simplicius (Byzantine empire, 500s)

•Al-Jawhari (Baghdad, 800s)

•Thabit ibn Qurra (Baghdad, 800s)

•Al-Nayrizi (Persia, 900s)

•Abu Ali Ibn Alhazen (Egypt, 1000s)

•Al-Salar (Persia, 1200s)

•Al-Tusi (Persia, 1200s)

•Al-Abhari (Persia, 1200s)

•Al-Maghribi (Persia, 1200s)

•Vitello (Poland, 1200s)

•Levi Ben-Gerson (France, 1300s)

•Alfonso (Spain, 1300s)

•Christopher Clavius (Germany, 1574)

•Pietro Cataldi (Italy, 1603)

•Giovanni Borelli (Italy, 1658)

•Vitale Giordano (Italy, 1680)

•Johann Lambert (Alsace, 1786)

•Louis Bertand (Switzerland, 1778)

•Adrien-Marie Legendre (France, 1794)


Why did so many great mathematicians make mistakes when

trying to prove Euclid’s fifth postulate?

It’s simple, really. Every one of these failed proofs had to use some

properties about parallel lines.

By definition, parallel lines are lines that are in the same plane but never

meet, no matter how far you extend them.


Trying to Prove the Fifth PostulateBut most of the failed proofs also accidentally used other properties of

parallel lines, such as:

• Equidistance: If two lines are parallel, they are everywhere the

same distance apart.

Trying to Prove the Fifth PostulateBut most of the failed proofs also accidentally used other properties of

parallel lines, such as:

• Equidistance: If two lines are parallel, they are everywhere the

same distance apart.

• Uniqueness: Given a line and a point not on that line, there is only

one line parallel to the given line through the given point.


Equidistance Postulate Euclid’s Fifth Postulate


Equidistance Postulate Euclid’s Fifth Postulate

The Parallel Postulate(uniqueness)

A Bold New IdeaIn the 1820s (more than 2100 years after Euclid), three

mathematicians in three different countries independently hit upon the

same amazing insight…

Janos Bolyai,

in Hungary

(18 years old)

Nikolai Lobachevsky,

in Russia

(38 years old)

Carl Friedrich Gauss,

in Germany

(58 years old)

A Bold New IdeaMaybe there is a simple explanation for why

nobody had succeeded in proving the Fifth

Postulate based only on the other four:

Maybe it’s logically impossible!

A Bold New IdeaMaybe there is a simple explanation for why

nobody had succeeded in proving the Fifth

Postulate based only on the other four:

Maybe it’s logically impossible!

Euclid thought his postulates were describing the

only conceivable geometry of the physical world

we live in.

But what if other geometries are not only

conceivable, but just as consistent and

mathematically sound as Euclid’s??

A Bold New IdeaHOW CAN THIS BE?

The key is curvature.

What are Dimensions?To see how to imagine the curvature of space, let’s start by pretending

we live in a 2-dimensional world.

Dimensions = “how many numbers it takes to describe the location

of a point.”




of a point.”

A line or a curve is 1-dimensional, because it only takes one number

to describe where a point is.




of a point.”

A plane is 2-dimensional, because it take 2 numbers, x and y, to

describe where a point is.

What are Dimensions?

The surface of a sphere is

also 2-dimensional, because it

takes two numbers—latitude

and longitude—to say where a

point is.

What are Dimensions?

Space is 3-dimensional, because it take 3 numbers, x, y, and z, to

describe where a point is.

Curvature in 2 Dimensions

Negative

curvature

Positive

curvature

Zero

curvature

Curvature in 2 DimensionsA 2-dimensional being, living in a 2-dimensional world, could not

possibly be aware of anything outside of his two dimensions, because

it would be “out of his world”.




Even if his 2-dimensional world is curved, he cannot look down on it

from “outside” to see the curvature…




Even if his 2-dimensional world is curved, he cannot look down on it

from “outside” to see the curvature…

And yet he can tell if

he lives in a curved

world!


ZERO CURVATURE CASE:

• Parallel lines are equidistant.

How can Bart tell if his world is curved or not?


ZERO CURVATURE CASE:

• Parallel lines are equidistant.

• Through a point not on a line, there’s

one and only one parallel.

• Triangles have angle sums of 180.



POSITIVE CURVATURE CASE:

From the “outside,” it looks spherical.




From the “outside,” it looks spherical.

“Lines” are great circles—the path he

would follow if he went as straight as

possible.




• Lines get closer together as you

follow them in either direction.




• Lines get closer together as you

follow them in either direction.

• Through a point not on a line, there

is no parallel, because all lines meet

eventually.

• Triangles have angle sums greater

than 180.


NEGATIVE CURVATURE CASE:

From the outside, it looks

“saddle-shaped.”




• Triangles have angle sums less

than 180.




• Triangles have angle sums less

than 180.

• Parallel lines diverge from each

other.



Curvature in 3 DimensionsNow here comes the surprising part …

It’s possible for our 3-dimensional universe to be curved!



Just like 2-dimensional Bart, we can’t step “outside the universe” to

see the curvature.

But we can detect it from inside our world by measuring angle-

sums of triangles.



Just like 2-dimensional Bart, we can’t step “outside the universe” to

see the curvature.

But we can detect it from inside our world by measuring angle-

sums of triangles.

Gauss tried to determine experimentally if our

world is flat or not—that is, if it follows the laws

of Euclidean geometry or not.

Curvature in 3 DimensionsGauss’s experiment:

Gauss set up surveying equipment on the tops of three mountains in

Germany, forming a triangle with light rays.

He then measured the angles of that triangle, and found …

The angles added up to

180, as close as his

measurements could

determine.

But maybe they just

weren’t accurate

enough?


Einstein’s Theory:

If there was any remaining doubt about whether it was possible for our

universe to be non-Euclidean, it was dispelled around 1900 by Albert

Einstein.



In physical space, “straight lines” are paths followed by light rays.

Einstein’s theory predicts that a large cluster of galaxies between us

and a distant galaxy will warp the space between us and the galaxy,

and cause its light to reach us along two different paths, so we see

two images of the same galaxy.



This has actually been observed.



Of course, this is a “small” region

of space in the larger scheme of

things. In a small region, space

is “lumpy,” with areas of positive

and negative curvature caused

by the gravitational fields of all

the stars and galaxies floating

around…

Just as the earth is “lumpy” if you

look at it up close.



But physicists expect that, if you

imagine looking at the universe

from far, far away, so that galaxies

look like dust scattered evenly

throughout the universe, then it will

have a nice smooth “shape.”



The Big Question:

When the bumps are smoothed out, does the universe have positive

curvature (“spherical”), zero curvature (“flat”), or negative curvature

(“saddle-shaped”)?



Einstein’s equations tell us how to determine the answer:

Just measure the average density of matter in the universe.

Einstein’s theory predicts that there’s a critical density – about 5 atoms

per cubic yard. (Remember, most of the universe is empty space!)


If the average density of the universe is …

• exactly equal to the critical density, then the universe has zero

curvature, is infinitely large, and will go on expanding forever.





• less than the critical density, then the universe has negative curvature,

is infinitely large, and will go on expanding forever.

Time





• less than the critical density, then the universe has negative curvature,

is infinitely large, and will go on expanding forever.

• greater than the critical density, then the universe has positive

curvature, is closed up on itself and only finitely large, and will eventually

stop expanding and collapse into a …

BIG CRUNCH!!



The Big Question:

When the bumps are smoothed out, does the universe have positive

curvature (“spherical”), zero curvature (“flat”), or negative curvature

(“saddle-shaped”)?

The current evidence suggests that it is positively curved, which means

that it is closed, like a sphere.


What does it mean for the universe to be closed?

On the surface of an ordinary (2-dimensional) sphere, if you start

traveling in any direction and go far enough, you’ll eventually come back

to the place where you started.

If our 3-dimensional universe is closed,

the same is true: if you start traveling in

any direction and go far enough (about

290,000,000,000,000,000,000,000 miles

probably), you’ll come back to the place

where you began!

The Shape of the Universe

If the universe is closed, what shape is it?


If the universe is closed, what shape is it?

If the universe were 2-dimensional, we would know what all the

possibilities were (after smoothing out the bumps) …


What do we mean by “smoothing out the bumps”?

Two surfaces are said to be topologically equivalent if one can be

continuously deformed into the other.


For example, the surface of a hot dog is topologically equivalent to a

sphere.


The surface of a one-handled coffee cup is topologically equivalent to a

doughnut surface (a torus).


But a sphere cannot be continuously deformed into a torus, because that

would require tearing a hole in the middle, a discontinuous operation.

The sphere is the simplest closed surface, in a very precise sense…



Thought Experiment:

Imagine you’re a 2-dimensional being living

on a closed surface.


Thought Experiment:



Start somewhere, and walk as straight as

you can, trailing a string behind you, until

you get back to the place where you

started.

Then try to pull the loop of string back to

where you are.


Thought Experiment:



Start somewhere, and walk as straight as

you can, trailing a string behind you, until

you get back to the place where you

started.

Then try to pull the loop of string back to

where you are.

If you’re on a sphere, there are no holes to

stop the string from shrinking back to you.

We say the sphere is simply connected.


Thought Experiment:

If you’re on another surface like a 1-holed

doughnut, this doesn’t work, because the

string can’t get across the hole.


Thought Experiment:

In 3 dimensions, there are many more possibilities.

But the simplest one is a 3-dimensional sphere-like object called a

hypersphere or a 3-sphere.

Fake picture of a hypersphere


Thought Experiment:

In 3 dimensions, there are many more possibilities.

But the simplest one is a 3-dimensional sphere-like object called a

hypersphere or a 3-sphere.

There are many other possible three-dimensional closed universes.

But are there any others that are simply connected?

Fake picture of a hypersphere

The Poincaré Conjecture

Around 1900, the French mathematician Henri

Poincaré tried to figure out if there are any

possible closed 3-dimensional spaces that are

simply connected, other than the 3-sphere.

Henri Poincaré






He couldn’t think of any others, and he conjectured

that the hypersphere is the only one.

The Poincaré Conjecture: Every simply

connected closed 3-dimensional space is

topologically equivalent to a 3-sphere.

Henri Poincaré






He couldn’t think of any others, and he conjectured

that the hypersphere is the only one.

The Poincaré Conjecture: Every simply

connected closed 3-dimensional space is

topologically equivalent to a 3-sphere.

He thought this would be a simple first step in

determining all closed 3-dimensional spaces.

Henri Poincaré


But it turned out not to be so easy.


Various bits of progress were made, until

In 1984, the American mathematician

Richard Hamilton thought of a

systematic way to “smooth the bumps”

on any simply connected surface, called

the Ricci flow.

Richard Hamilton


Various bits of progress were made, until

In 1984, the American mathematician

Richard Hamilton thought of a

systematic way to “smooth the bumps”

on any simply connected surface, called

the Ricci flow.

It’s like heating up a chocolate bunny

and watching the bumps smooth out.

Richard Hamilton

The Ricci Flow For Surfaces

The Ricci Flow in 3 Dimensions

The Ricci Flow in 3 Dimensions

Hamilton got stuck for 20 years …

Richard Hamilton

??


In 2000, the Clay Mathematics Institute announced its seven Millennium

Problems: unsolved math problems with a $1,000,000 prize to anyone who

solves one of them.

One of the problems was:

Either prove the Poincaré conjecture, or show that it’s false by finding a

counterexample (a simply-connected closed 3-dimensional space that isn’t a

hypersphere when the bumps are smoothed out).


Finally, in 2003, an eccentric Russian

mathematician named Grigori Perelman

figured out how to prove that when

pinching occurs, it always pinches along

a nice cylinder, and he completed the

proof of the Poincaré conjecture.

Grigori Perelman


The Clay Institute announced last March

that he had won the million-dollar prize.

Grigori Perelman


The Clay Institute announced last March

that he had won the million-dollar prize.

Reporter: You just won a million dollars.

Do you have any comment?

Perelman: I don’t need it! It’ll just make

me a target of the Russian mafia.

Grigori Perelman

Much more to do …

There are still many unanswered questions for aspiring

mathematicians to work on.

Check out the remaining six Millennium Prize problems at

www.claymath.org

Besides those problems, there are thousands of other unanswered

questions in mathematics…

http://www.claymath.org/

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