The Curvature of Space Jack Lee Professor of Mathematics UW Seattle

Euclid (around 300 BCE)

The Elements by Euclid

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

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

Euclid’s Postulates for Geometry Postulate 1:A straight line segment can be drawn from any point to 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. 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. Postulate 2:A straight line segment can be extended in either direction to form a longer straight line segment. Postulate 3:A circle can be drawn with any point as its center and any other point on its circumference.

Euclid’s Postulates for Geometry Postulate 1:A straight line segment can be drawn from any point to any other point. Postulate 2:A straight line segment can be extended in either direction to form a longer straight line segment. Postulate 3:A circle can be drawn with any point as its center and any other point on its circumference.

Euclid’s Postulates for Geometry Postulate 1:A straight line segment can be drawn from any point to any other point. Postulate 2:A straight line segment can be extended in either direction to form a longer straight line segment. Postulate 3:A circle can be drawn with any point as its center and any other point on its circumference. Postulate 4:All right angles are equal.

Euclid’s Postulates for Geometry Postulate 1:A straight line segment can be drawn from any point to any other point. Postulate 2:A straight line segment can be extended in either direction to form a longer straight line segment. Postulate 3:A circle can be drawn with any point as its center and any other point on its circumference. Postulate 4:All right angles are equal.

Euclid’s Postulates for Geometry Postulate 1:A straight line segment can be drawn from any point to any other point. Postulate 2:A straight line segment can be extended in either direction to form a longer straight line segment. Postulate 3:A circle can be drawn with any point as its center and any other point on its circumference. Postulate 4:All right angles are equal.

Euclid’s Postulates for Geometry Postulate 1:A straight line segment can be drawn from any point to any other point. Postulate 2:A straight line segment can be extended in either direction to form a longer straight line segment. Postulate 3:A circle can be drawn with any point as its center and any other point on its circumference. 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

Euclid’s Postulates for Geometry Postulate 5:If a straight line crossing two other straight lines makes two interior angles on one side

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 .

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, then the two straight lines, if extended far enough,  1 +  2 < 180 .

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, 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

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? Wrong! Ptolemy’s proof had a mistake. Ptolemy

Trying to Prove the Fifth Postulate Around 400 CE (700 years after Euclid): The Greek mathematician Proclus criticized Ptolemy’s proof. Proclus

Trying to Prove the Fifth Postulate Around 400 CE (700 years after Euclid): 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

Trying to Prove the Fifth Postulate So Proclus offered his own proof of the fifth postulate … Proclus

Trying to Prove the Fifth Postulate So Proclus offered his own proof of the fifth postulate … …which was also wrong! Proclus

Trying to Prove the Fifth Postulate Around 1100 CE (1400 years after Euclid): 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

Trying to Prove the Fifth Postulate Around 1100 CE (1400 years after Euclid): 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

Trying to Prove the Fifth Postulate 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.”

Trying to Prove the Fifth Postulate 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!

Trying to Prove the Fifth Postulate 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)

Trying to Prove the Fifth Postulate 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)

Trying to Prove the Fifth Postulate 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 Postulate

But 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 Postulate But 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 Postulate But 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.

Trying to Prove the Fifth Postulate But 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.

Trying to Prove the Fifth Postulate Equidistance Postulate Euclid’s Fifth Postulate

Trying to Prove the Fifth Postulate Equidistance Postulate Euclid’s Fifth Postulate The Parallel Postulate (uniqueness)

Trying to Prove the Fifth Postulate Equidistance Postulate Euclid’s Fifth Postulate The Parallel Postulate (uniqueness)

A Bold New Idea In 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 Idea Maybe 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 Idea Maybe 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 Idea HOW 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.”

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.” A line or a curve is 1-dimensional, because it only takes one number to describe where a point is. 

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.” 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 Dimensions A 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”.

Curvature in 2 Dimensions A 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…

Curvature in 2 Dimensions A 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… And yet he can tell if he lives in a curved world!

Curvature in 2 Dimensions ZERO CURVATURE CASE: Parallel lines are equidistant. How can Bart tell if his world is curved or not?

Curvature in 2 Dimensions 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 . How can Bart tell if his world is curved or not?

Curvature in 2 Dimensions POSITIVE CURVATURE CASE: From the “outside,” it looks spherical. How can Bart tell if his world is curved or not?

Curvature in 2 Dimensions POSITIVE CURVATURE CASE: From the “outside,” it looks spherical. “Lines” are great circles—the path he would follow if he went as straight as possible. How can Bart tell if his world is curved or not?

Curvature in 2 Dimensions POSITIVE CURVATURE CASE: Lines get closer together as you follow them in either direction. How can Bart tell if his world is curved or not?

Curvature in 2 Dimensions POSITIVE CURVATURE CASE: 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 . How can Bart tell if his world is curved or not?

NEGATIVE CURVATURE CASE: From the outside, it looks “saddle-shaped.” Curvature in 2 Dimensions How can Bart tell if his world is curved or not?

NEGATIVE CURVATURE CASE: Triangles have angle sums less than 180 . Curvature in 2 Dimensions How can Bart tell if his world is curved or not?

NEGATIVE CURVATURE CASE: Triangles have angle sums less than 180 . Parallel lines diverge from each other. Curvature in 2 Dimensions How can Bart tell if his world is curved or not?

Curvature in 3 Dimensions Now here comes the surprising part … It’s possible for our 3-dimensional universe to be curved!

Curvature in 3 Dimensions Now 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.

Curvature in 3 Dimensions Now 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. 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 Dimensions Gauss’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?

Curvature in 3 Dimensions 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.

Curvature in 3 Dimensions Einstein’s Theory: 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.

Curvature in 3 Dimensions Einstein’s Theory: This has actually been observed.

Curvature in 3 Dimensions Einstein’s Theory: 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.

Curvature in 3 Dimensions Einstein’s Theory: 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.”

Curvature in 3 Dimensions Einstein’s Theory: The Big Question: When the bumps are smoothed out, does the universe have positive curvature (“spherical”), zero curvature (“flat”), or negative curvature (“saddle-shaped”)?

Curvature in 3 Dimensions Einstein’s Theory: 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!)

Curvature in 3 Dimensions 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.

Curvature in 3 Dimensions 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

Curvature in 3 Dimensions 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. 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!!

Curvature in 3 Dimensions Einstein’s Theory: 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.

Curvature in 3 Dimensions 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?

The Shape of the Universe 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) …

The Shape of the Universe 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.

The Shape of the Universe For example, the surface of a hot dog is topologically equivalent to a sphere.

The Shape of the Universe For example, the surface of a hot dog is topologically equivalent to a sphere.

The Shape of the Universe The surface of a one-handled coffee cup is topologically equivalent to a doughnut surface (a torus).

The Shape of the Universe The surface of a one-handled coffee cup is topologically equivalent to a doughnut surface (a torus).

The Shape of the Universe 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… The Shape of the Universe

Thought Experiment: Imagine you’re a 2-dimensional being living on a closed surface.

The Shape of the Universe Thought Experiment: Imagine you’re a 2-dimensional being living on a closed surface. 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.

The Shape of the Universe Thought Experiment: Imagine you’re a 2-dimensional being living on a closed surface. 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.

The Shape of the Universe 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.

The Shape of the Universe 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

The Shape of the Universe 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

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. 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

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. 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

The Poincaré Conjecture But it turned out not to be so easy.

The Poincaré Conjecture 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

The Poincaré Conjecture 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

The Poincaré Conjecture 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

Hamilton got stuck for 20 years … Richard Hamilton ??

The Poincaré Conjecture 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).

The Poincaré Conjecture 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 Poincaré Conjecture The Clay Institute announced last March that he had won the million-dollar prize. Grigori Perelman

The Poincaré Conjecture 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 Besides those problems, there are thousands of other unanswered questions in mathematics…