- 3 tháng trước
Danh mục
🤖
Công nghệPhụ đề
00:00A single sentence in one of the oldest math books held the key to understanding our universe.
00:07Euclid's Elements has been published in more editions than any other book except the Bible.
00:12It was the go-to math text for over 2,000 years.
00:16But for all that time, mathematicians were skeptical of a single line, which seemed like a mistake.
00:24Ultimately, some of the greatest math minds realized that Euclid wasn't wrong after all.
00:30But there was more to the story.
00:33Slight tweaks to this line opened up strange new universes out of nothing.
00:39Surprisingly, 80 years later, we found out those strange new universes are core to understanding our own universe.
00:47Around 300 BC, the Greek mathematician Euclid takes on a massive project to summarize all mathematics known at the time.
01:00To essentially create the one book that contains everything that everyone knows about mathematics.
01:07But that's no easy task.
01:09See, before Euclid, there was a bit of a problem with math.
01:13People would prove things, but they would just be going around in circles.
01:16Why does a triangle have 180 degrees?
01:18Because if you take two parallel lines, yeah, but why do parallel lines exist?
01:21Oh, that's because you can make a square.
01:22Why does a square exist?
01:23You have this infinite recursion of what the fundamental reason why something is true is.
01:29It's kind of like in the dictionary.
01:30Every word is defined in terms of other words.
01:34So how do you get to ground truth?
01:37Euclid used a solution that was pioneered by the Greeks.
01:40Let's just accept a few of the most simple, basic things as being true.
01:45These are our postulates.
01:47Then, based upon these postulates, we can prove theorems, one at a time, building up our math using logic.
01:54So that as long as those first statements are true, then everything else that follows from them must definitely be true.
02:02He had perfected the gold standard for rigorous mathematical proof that all modern math relies on.
02:08Euclid used this method when he published his 13-book series called The Elements, in which he proved 465 theorems covering almost all of mathematics known at the time, including geometry and number theory.
02:22And all these theorems depended on were some definitions, a few common notions, and five postulates.
02:30We go right to book one, and book one starts with definitions.
02:36You've got to start somewhere.
02:37And the definition is a point is that which has no part.
02:40A line is a breadthless length.
02:42And the ends of lines are points.
02:45By line, he really means a curve.
02:47And then it has some ends.
02:49A straight line is a line that lies evenly within the points on itself.
02:53And then so on and so forth.
02:55He's got 23 definitions.
02:57Then he's got the five postulates.
03:00The first four are simple.
03:02One, if you have two points, you can draw a straight line between them.
03:06Two, if you have a straight line, you can extend it indefinitely.
03:11Three, given a center and a radius, you can draw a circle.
03:15And the fourth is that all right angles are equal to each other.
03:19But postulate five gets the big guns.
03:21Which is that if a straight line falling on two straight lines makes the interior angles
03:27on the same side less than two right angles, the two straight lines, if produced indefinitely,
03:33meet on that side on which are the angles less than the two right angles.
03:38What the hell is he talking about?
03:40That's a postulate?
03:40Like all of these are, you know, half of a sentence.
03:43And they're all blatantly obvious.
03:44And then comes five out of left field.
03:46And it's like an entire paragraph.
03:49What is he doing?
03:49This made mathematicians suspicious.
03:53It seemed like Euclid made a mistake.
03:56Greek philosopher Proclus thought postulate five ought even to be struck out of the postulates
04:01altogether, for it is a theorem.
04:04But if it's a theorem, we should be able to prove it from the first four postulates.
04:08So that's what many people tried.
04:11Some, including Ptolemy and Proclus, believed they had succeeded.
04:14But they hadn't.
04:17In fact, all they'd managed to do was just restate postulate five in different words.
04:23Here's one formulation.
04:25If you have a line and a point that is not on that line, then there is a single unique
04:30line which will be parallel to the first line.
04:34For this reason, the fifth postulate is often called the parallel postulate.
04:38When the method of direct proof failed, other mathematicians, including Al-Haytham and Omar
04:44Kayyem, tried a different approach, proof by contradiction.
04:49The idea is simple.
04:51You keep the first four postulates the same, but assume that the fifth postulate is false.
04:56Then you use those new postulates to prove theorems.
04:58And if that leads to a contradiction, for example, true equals false, well, then it means your
05:03new fifth postulate must be wrong.
05:05Therefore, the only remaining option would be that Euclid's version of the fifth postulate
05:09is correct, and you would have proven the fifth postulate.
05:13So what would it look like if the fifth postulate were false?
05:16Well, according to Euclid, through a point not on a line, there could only be one line
05:21that is parallel to the first.
05:23One alternative is that there are no parallel lines that you could draw through that point.
05:28Well, people tried that, and they realized that then lines had to be finite in length.
05:33You're like, well, that can't be.
05:35So this option was ruled out.
05:37It contradicted the second postulate, which states that lines can be extended indefinitely.
05:41The other alternative is that you can draw more than one parallel line through a point
05:46not on the first line.
05:48So that's what they would do.
05:49They would assume the postulate five fails.
05:51They're like, this has got to be wrong.
05:52Where's the contradiction?
05:53They couldn't find the contradiction.
05:55So proof by contradiction also failed.
05:58In total, mathematicians spent more than 2,000 years trying to prove the fifth postulate.
06:03But everyone who tried failed.
06:05Then around 1820, Janos Bollii, a 17-year-old student, started spending his days and nights
06:15working on the mystery.
06:17His father became worried, and he wrote to his son,
06:20But the young Bollii didn't listen to his father.
06:42He could not leave the science of parallels alone.
06:46After years of work, he realized that maybe the fifth postulate can't be proven from the
06:51other four.
06:52It could be completely independent.
06:56See, according to Euclid, you could have only one parallel line through a point.
07:00But Bollii imagined a world where there could be more than one parallel line through that
07:05point.
07:06But how?
07:07Well, who said you needed to have a flat surface?
07:10On a surface that is curved like this, you can draw more than one line that is parallel
07:15to the original line.
07:17But wait a second.
07:17Those lines don't look straight.
07:20Well, what makes straight lines special is that they're the shortest paths between two
07:24points.
07:25On this surface, those shortest paths just look bent because the surface is curved.
07:31Here's a more familiar example.
07:33Airplanes always try to fly the shortest path between two cities.
07:36They're basically flying in a straight line.
07:38But that line doesn't look straight on a map because the surface is curved.
07:43These shortest paths on curved surfaces are called geodesics.
07:48So, all these lines are straight.
07:51They just don't look it because the world Bollii had imagined turned out to be curved.
07:55We now know this as hyperbolic geometry.
08:00You know, when I used to think of the hyperbolic plane, I would just imagine it as one giant
08:05saddle.
08:06But that's not really what it is.
08:08The hyperbolic plane is much more like this piece of crocheting.
08:12So it starts out pretty flat and even in the middle.
08:15But as you move outwards, more and more fabric is created.
08:20And that would push parallel lines apart.
08:24And the further and further out you go, the amount of fabric grows exponentially.
08:29And that ends up causing this crumpling effect.
08:32So if you really want to think about the hyperbolic plane, I think you've got to think about saddles
08:37on saddles on saddles.
08:38Like it's an infinite crumpling mess.
08:41But that little piece of crochet isn't the full hyperbolic plane.
08:45To show that, we need to make a map.
08:48One that fits the entire plane into a disc.
08:52To show how this works, we're going to fill the entire plane with these triangles.
08:55Starting in the middle, just as with the crochet, things look pretty normal.
09:00But as you go farther out from the center, you get all this extra space.
09:04So you can fit more and more triangles.
09:06So they appear smaller, but they are actually the same size.
09:11Now, since the hyperbolic plane is infinite, you can keep adding triangles forever.
09:16And they all need to fit on the disc.
09:18So as you get closer to the edge, the triangles will appear smaller and smaller and smaller.
09:24Infinitely smaller, never ending.
09:26And you can never quite reach the edge.
09:29This is known as the Poincaré disc model.
09:32Here, straight lines are arcs of circles that intersect the disc at 90 degrees.
09:37And just like on our original shape, a straight line down the middle appears straight.
09:42While straight lines next to it appear to curve away.
09:45What's remarkable is that Bollii didn't have a model of hyperbolic geometry yet.
09:50He was just drawing Euclidean triangles with the assumption that Euclid's fifth postulate didn't hold.
09:57And while Bollii found that the behavior in hyperbolic geometry is very different than Euclid's,
10:02mathematically, it seemed just as consistent.
10:07In 1823, the 20-year-old Janos wrote to his dad,
10:11I have discovered such wonderful things that I was amazed.
10:16Out of nothing, I have created a strange new universe.
10:23But Bollii had been doing more than just tackling ancient math mysteries.
10:28In his 20s, he joined the army, where he continued developing two of his other passions,
10:33playing the violin and dueling.
10:36He had mastered both, but with a sword in particular, he was unmatched.
10:40Perhaps because of his many talents, Bollii grew arrogant and found it difficult to accept authority from his superiors.
10:47That made him hard to get along with.
10:50This reached a peak when, during one of his deployments,
10:5313 cavalry officers from his garrison challenged him to a duel.
10:58Bollii accepted their challenge on the condition that,
11:01after every two duels, he could play for a little while on his violin.
11:04Bollii fought each of them in succession, winning all 13 duels and leaving behind all his adversaries on the square.
11:15While Bollii loved dueling, his first love was still mathematics.
11:19In 1832, nine years after he discovered his strange new universe,
11:24he published his findings as a 24-page appendix to his father's textbook.
11:30Extremely proud and excited about his son's work,
11:32Farkas Bollii sent it to perhaps the greatest mathematician of all time,
11:36Carl Friedrich Gauss.
11:38After careful examination, Gauss replied a few months later,
11:42Years earlier, Gauss had wandered a similar path.
12:00In 1824, he wrote a private letter to one of his friends
12:03in which he describes discovering a curious geometry,
12:07one with paradoxical and, to the uninitiated, absurd theorems.
12:12For example, Gauss writes,
12:14The three angles of a triangle become as small as one wishes,
12:18if only the sides are taken large enough.
12:21Yet the area of the triangle can never exceed a definite limit.
12:26In other words, you can have a triangle that's infinitely long,
12:30but the area is finite.
12:33You can see why by using the Poincaré disk model.
12:36A small triangle looks pretty ordinary,
12:38but as you make it bigger,
12:40the angles start to become smaller and smaller.
12:44Eventually, all those angles go to zero,
12:47because all these lines intersect the disk at 90 degrees.
12:51Now, these lines are infinitely long,
12:53but because of the geometry,
12:55the area is finite.
12:58In the same private letter,
13:00Gauss wrote,
13:01All my efforts to discover a contradiction
13:03and inconsistency in this non-Euclidean geometry
13:07have been without success.
13:09Just like Bollii,
13:11Gauss had found that this geometry seemed
13:13thoroughly consistent.
13:15He named it non-Euclidean geometry,
13:17a name that stuck.
13:19It describes geometries where Euclid's first four postulates hold,
13:22but the fifth doesn't.
13:24But Gauss decided not to publish his findings
13:26for fear of ridicule.
13:29This aversion to a different kind of geometry
13:31should at least be a little surprising,
13:34because there is one other geometry
13:35that we should be very familiar with,
13:38spherical geometry,
13:39since we all live on a sphere.
13:42On a sphere,
13:43straight lines are parts of great circles.
13:46These are the circles
13:46with the largest possible circumference.
13:49On Earth, the equator and circles of longitude
13:52are examples of great circles.
13:55And we can use this
13:56to see how straight lines behave.
13:58These lines seem to go in the same direction,
14:01but as you keep extending them,
14:02you find that they intersect,
14:04once and then again
14:06on the other side of the Earth.
14:08And this will always happen
14:10for any two great circles,
14:12because they must each
14:13have the largest possible circumference.
14:15So on a sphere,
14:17there are no parallel lines.
14:21Gauss had long been fascinated
14:22by spherical geometry.
14:24He was also a geodesist
14:25and frequently took measurements of the Earth.
14:28In the 1820s,
14:29he was tasked with surveying
14:30the Kingdom of Hanover
14:31to help make a map.
14:33As part of this survey,
14:34he climbed the mountains near Göttingen.
14:37With the help of people
14:37situated on other landmarks,
14:39they were able to carefully measure
14:40the angles of several triangles,
14:43which would then be used
14:44to determine the position
14:45of one place relative to another.
14:47As a reference for the survey
14:48and to help determine
14:50the roundness of the Earth,
14:51they also precisely measured
14:53the angles of a large triangle
14:54formed by three mountains.
14:58But for all of Gauss' romantic notions
15:00of taking measurements
15:01on top of mountains,
15:03he was not the kindest correspondent.
15:06When Bollii received
15:07his hero's response,
15:08he was devastated,
15:10believing that Gauss was trying
15:11to undermine him
15:12and steal his ideas.
15:14He was so embittered
15:15by Gauss' response
15:16that he never published again.
15:20In 1848,
15:21Bollii had to endure
15:22another hardship
15:23when he found out
15:24that Russian mathematician
15:25Nikolai Lobachevsky
15:26had independently discovered
15:28non-Euclidean geometry
15:29several years before
15:31Bollii had published
15:32his 24-page-long appendix.
15:35When Bollii died in 1860,
15:38he left behind
15:3920,000 pages
15:40of unpublished
15:41mathematical manuscripts.
15:43He would not know
15:44that Gauss did independently
15:46discover non-Euclidean geometry,
15:48nor would he know
15:49that upon receiving
15:50the appendix,
15:50Gauss had written
15:51to a friend,
15:53I regard this young
15:54geometer, Bollii,
15:55as a genius
15:56of the first order.
16:00While Bollii was embittered,
16:03non-Euclidean geometries
16:04continued to develop.
16:06Until 1854,
16:08spherical geometry
16:09wasn't actually considered
16:10a non-Euclidean geometry.
16:13That's because on a sphere,
16:14lines can't be extended indefinitely.
16:17This is what earlier
16:18mathematicians had stumbled upon
16:20and therefore
16:20dismissed this geometry,
16:22since Euclid's second postulate
16:23wouldn't hold.
16:25But in 1854,
16:27Riemann changed
16:28the second postulate
16:29from an infinite extension
16:30to something that is,
16:32quote,
16:32unbounded
16:33so that the second postulate
16:35still holds on a sphere.
16:38With this change,
16:39spherical geometry
16:40became another valid
16:41non-Euclidean geometry.
16:43By using the generalized
16:45four postulates
16:45and taking the fifth
16:46as there being
16:47no parallel lines,
16:48you could now derive
16:50spherical or elliptic geometry.
16:53Would you consider
16:54the fifth postulate
16:55a mistake?
16:57Would it have been better
16:58if he just never
16:59wrote that down?
17:00If he never wrote that down,
17:01he would have stunted
17:02his geometry
17:03because he wouldn't
17:03be able to prove
17:04a lot of the things
17:05that he claimed to have.
17:06It's beautiful
17:07that he wrote this.
17:09It's beautiful
17:10that people spent
17:102,000 years
17:11trying to refute him
17:12only to discover
17:13that in fact he was right
17:15in writing this
17:16in the first place.
17:18So while Euclid was right
17:20to write down
17:20the fifth postulate,
17:21he did make
17:22a different mistake.
17:24So here's the problem
17:25with what Euclid was doing.
17:26Definition one,
17:27a point is that
17:28which has no part.
17:30What does it mean
17:31to have a part?
17:31What is a part?
17:33What does it mean
17:33not to have a part?
17:34A line is a breathless length?
17:37What does it mean
17:37to have breadth?
17:38Lying evenly within itself?
17:40Within the points in itself?
17:41What the hell
17:42is he talking about?
17:43We read this
17:43two minutes ago
17:44and we were nodding along
17:45like yeah,
17:46completely makes sense
17:46what he's saying.
17:48It's all nonsense.
17:49Don't give me
17:50a definition
17:51that's going to have
17:51an infinite recursion.
17:53If you give me
17:53a definition
17:54in terms of other things
17:55then you have to tell me
17:55what those things are.
17:56If you tell me
17:57what that is
17:57you have to tell me
17:58what the thing before it is.
17:59Are the definitions
18:00a bad idea?
18:03You shouldn't have definitions.
18:05You should have
18:05undefined terms.
18:07I'm not going to tell you
18:07what a point is.
18:08I'm not going to tell you
18:09what a line is.
18:09I'm not going to tell you
18:10what a plane is.
18:11All I'm going to tell you
18:12is what the postulates are
18:14that they're assumed
18:15to satisfy.
18:16It's the relationships
18:17between the objects
18:18that's important
18:19not the definitions
18:20of the object themselves.
18:22And once you free your mind
18:23to that possibility
18:24all of a sudden
18:25you realize
18:26that there's a perfectly
18:27good geometric world
18:28in which by line
18:30you mean great circle
18:31and by plane
18:32you mean sphere
18:33and by point
18:34you mean a point
18:34on a sphere
18:35and then four of those
18:37axioms are satisfied
18:38just not the fifth.
18:40And similarly
18:41there's another model
18:42something called
18:43the disk model
18:44for hyperbolic space
18:45which the disk
18:46is the plane.
18:48What I mean by
18:49straight lines
18:49is arcs of circles
18:51that are orthogonal
18:51to the disk
18:52and then points
18:53are points
18:54inside the disk
18:55and the disk
18:56is the plane.
18:58See you can think
18:59of geometry
18:59as a game.
19:00The first four postulates
19:01are like the minimum
19:03rules required
19:04to play that game
19:04and then the fifth
19:06postulate selects
19:07the world
19:08that you'll play in.
19:09If you pick
19:10that there are
19:10no parallel lines
19:11you're playing
19:12in spherical geometry.
19:13If you choose
19:14one parallel line
19:15you're playing
19:16in flat geometry
19:17and if you go
19:18for more than
19:18one parallel line
19:19then you're playing
19:20in hyperbolic geometry.
19:23But Riemann
19:24decided to take it
19:25one step further.
19:26Instead of selecting
19:27just one world
19:28to play in
19:29why not combine
19:30them all into one?
19:33During his inaugural
19:34speech in 1854
19:35he laid out
19:36the groundwork
19:37for a geometry
19:37where the curvature
19:38could differ
19:39from place to place.
19:41One part might be flat
19:42another part
19:43might be slightly curved
19:44and yet another part
19:45might have a very
19:46strong curvature.
19:48And this geometry
19:48wouldn't be limited
19:49to two-dimensional
19:50planes either.
19:51It could be extended
19:52to three or more
19:54dimensions.
19:57Another breakthrough
19:58came in 1868
19:59when Eugenio Beltrami
20:01unequivocally proved
20:02that hyperbolic
20:03and spherical geometry
20:04were just as consistent
20:06as Euclid's
20:06flat geometry.
20:08That is
20:08if there were
20:09any inconsistencies
20:10in hyperbolic
20:11or spherical geometry
20:12then they must also
20:13be present
20:14in Euclid's
20:15flat geometry.
20:17The prospects
20:17for these new geometries
20:19were looking great
20:20and it turns out
20:21this was just
20:22the beginning.
20:24In 1905
20:25Einstein proposed
20:26the special theory
20:27of relativity
20:28which is based
20:29on just two postulates.
20:30One,
20:31the laws of physics
20:32are the same
20:32in all inertial frames
20:33of reference
20:34and two,
20:35the speed of light
20:36in a vacuum
20:37is the same
20:38for all inertial observers.
20:40So as a result
20:41space and time
20:43must be relative.
20:44But that created
20:47a problem
20:47for Newtonian gravity
20:48because according
20:50to Newton
20:50the force of gravity
20:51is inversely proportional
20:53to the distance
20:54between the two objects
20:55squared.
20:56But in Einstein's
20:56special relativity
20:57that distance
20:58is no longer
20:59well defined.
21:01In whose reference frame
21:02are we measuring?
21:04And so
21:04Einstein had to find
21:05a way to reconcile
21:06relativity
21:07and gravity.
21:09Two years later
21:10in 1907
21:11Einstein had the
21:12happiest thought
21:12of his life.
21:13He imagined a man
21:15falling off the roof
21:16of a house
21:16and what made
21:17Einstein so joyful
21:18is that he realized
21:20that while the man
21:21is falling
21:21he would feel
21:22absolutely weightless
21:23and if he let go
21:24of an object
21:25it would just
21:25remain in uniform
21:27motion relative
21:27to him.
21:29It would be
21:29just like being
21:30in space
21:31not near any masses
21:32floating around
21:33in a spaceship
21:34at constant velocity
21:35and that
21:37is an inertial observer.
21:39Now here's
21:39the big breakthrough.
21:41Einstein realized
21:42that they're not
21:42just similar
21:43they are
21:44identical
21:44because there is
21:45no experiment
21:46you could do
21:47to determine
21:47whether you're
21:48in free fall
21:49in a uniform
21:49gravitational field
21:50or whether you're
21:51in deep space
21:52not near
21:53any massive objects.
21:56And so
21:56the free falling man
21:57too must be
21:59an inertial observer
22:00meaning
22:00he is not
22:01accelerating
22:02and he's not
22:03experiencing
22:03any force
22:04of gravity.
22:06But if gravity
22:06is not a force
22:07then how do you
22:08explain things
22:09like the space station
22:10orbiting Earth?
22:11shouldn't it just
22:12fly off
22:13in a straight line?
22:14Well astronauts
22:15in the space station
22:16also feel weightless
22:17and that's the key.
22:19It feels
22:20just as if
22:20they're traveling
22:21at constant velocity
22:22in a straight line.
22:23It feels like that
22:24because that's
22:25precisely what
22:26they're doing.
22:27They're traveling
22:28in a straight line.
22:30How then could
22:30that straight line
22:31appear curved
22:32to a distant observer?
22:33The answer is
22:35because the
22:35space time
22:36that straight line
22:36is on
22:37is curved.
22:40See
22:41massive objects
22:41curve space time
22:43and objects
22:44moving through
22:44curved space time
22:45will follow
22:45the shortest path
22:46through that
22:47curved geometry
22:48the geodesic.
22:50So while
22:50astronauts in the
22:51space station
22:51are following
22:52a straight line
22:53it appears curved
22:54to a distant observer
22:55because the Earth
22:57curves the space
22:58time around it.
22:58So the behavior
23:01of straight lines
23:02in curved geometries
23:03is core
23:04to understanding
23:05the universe
23:06we live in
23:06and in the more
23:07than 100 years
23:08since it was published
23:09the general theory
23:10of relativity
23:11has been
23:12remarkably successful.
23:14In 2014
23:15astronomers
23:16briefly observed
23:18a supernova
23:18a violent
23:19and extremely bright
23:20death of a star.
23:22In fact
23:22they saw the exact
23:23same supernova
23:24in four different places.
23:26How?
23:27Well
23:27in between the supernova
23:28and Earth
23:29there was a massive galaxy
23:31which curved space time.
23:33So light
23:33from the supernova
23:34which was spreading out
23:35in all directions
23:36had several different paths
23:38to reach the Earth
23:39and four of those
23:40reached Earth
23:40at approximately
23:41the same time.
23:42The galaxy
23:43had functioned
23:44as a massive
23:45gravitational lens.
23:47The astronomers
23:48realized that
23:48other galaxies
23:49in the cluster
23:50might also lens
23:51the light
23:51from that supernova
23:52but with different
23:54path lengths
23:54and gravitational potentials
23:56so the light
23:57would reach
23:57Earth
23:58at different times.
24:00After careful
24:01modeling
24:01they predicted
24:02that they should
24:03see a replay
24:04of that supernova
24:04just a year later.
24:07And on the 11th
24:08of December 2015
24:09just as predicted
24:11they saw the same
24:12supernova
24:13once more.
24:15In addition
24:16to being able
24:16to observe
24:17the effects
24:18of curved space time
24:19we can now
24:19even measure
24:20the ripples
24:21of space time itself.
24:22Gravitational waves
24:23formed by cosmic events
24:25far far away.
24:26like the merger
24:27of black holes.
24:30And according
24:30to a recent survey
24:31by Nanograv
24:32the fabric
24:33of space time
24:34seems to be buzzing
24:35with the remnants
24:36of grand cosmic events.
24:39In the hundred years
24:39since General Relativity
24:40was published
24:41countless findings
24:42have supported
24:43its predictions
24:43and at its very core
24:45are the curved geometries
24:47of Bollii and Riemann.
24:49But so far
24:50all the effects
24:51we've looked at
24:52are local
24:53distortions
24:53of space time.
24:55What
24:56is the shape
24:56of the entire universe?
25:00Using the differences
25:01between the geometries
25:02we can find that out too.
25:05In flat geometry
25:05we expect
25:06all the angles
25:07of a triangle
25:08to add up to
25:08180 degrees
25:09without fail.
25:11But in spherical geometry
25:12the angles
25:13don't add up
25:14to 180 degrees
25:15but to more.
25:16Similarly
25:17in hyperbolic geometry
25:18the angles
25:18add up to
25:19less than
25:20180 degrees.
25:21So to determine
25:22the shape
25:23of the universe
25:23you just need
25:24to measure
25:25the angles
25:26of a triangle.
25:27And measuring
25:28a triangle
25:28is precisely
25:29what Gauss
25:29was doing
25:30200 years ago.
25:31In fact
25:32this led some
25:33to speculate
25:33that he was
25:34actually trying
25:34to measure
25:35the curvature
25:35of space itself.
25:38The angle
25:38he found
25:39180 degrees
25:40within observational error.
25:43But that
25:44shouldn't be
25:45very surprising.
25:46Take this balloon
25:47for example
25:47which approximates
25:48a sphere.
25:49If I draw
25:50a small triangle
25:51on it
25:52well the surface
25:53I'm drawing on
25:53is basically
25:54flat.
25:55So the angles
25:56inside the triangle
25:56will add up
25:57to essentially
25:58180 degrees.
25:59Only if I make
26:00the triangle
26:01large enough
26:02will the effects
26:03of curvature
26:04come into play
26:06and then the
26:06angles in the triangle
26:07will add up
26:08to more than
26:08180 degrees.
26:10And this was
26:10the problem
26:11with Gauss's
26:11experiment.
26:12Even if he was
26:13trying to measure
26:14the curvature
26:14of space itself
26:15for which
26:16there is no
26:17solid evidence
26:17the triangle
26:18he measured
26:19would have been
26:19far too small
26:21relative to
26:22the size
26:22of the universe.
26:25So in order
26:26to overcome
26:27the scale issue
26:28that Gauss
26:28encountered
26:29we need to
26:30scale the triangles
26:31formed between
26:31mountains up to
26:32the largest
26:33triangles we can.
26:35And since
26:35looking further
26:36and further away
26:37is the same
26:37as looking
26:38further back
26:38in time
26:39we need to
26:40look back
26:40as far
26:41as possible
26:42to the very
26:42first light
26:43we can see
26:44the cosmic
26:45microwave background
26:47or CMB
26:47a picture
26:48from when the
26:49universe was
26:50just 380,000
26:52years old.
26:53While the CMB
26:54is almost
26:55completely uniform
26:56there are some
26:57spots that are
26:58slightly hotter
26:58or colder.
27:00Now we know
27:00how far away
27:01the CMB is
27:02so if we can
27:04figure out
27:04how large
27:05such a spot
27:05is then we
27:06can draw
27:06a cosmic
27:07triangle.
27:09It's thought
27:10that the first
27:10density and
27:11temperature variations
27:12originated from
27:13quantum fluctuations
27:14in the very
27:15early universe
27:16which were then
27:17blown up
27:17as the universe
27:18expanded.
27:19But due to
27:20this rapid
27:20expansion
27:21not all regions
27:22were in causal
27:23contact with
27:24each other.
27:25So using
27:26the information
27:26we have
27:27of how the
27:28early universe
27:28evolved
27:29astronomers
27:30can predict
27:30how often
27:31spots of
27:32different sizes
27:32should appear
27:34in the CMB.
27:35This is what
27:36this power
27:37spectrum shows
27:38essentially a
27:39histogram of
27:40how often
27:40each spot
27:41size should
27:42occur if
27:43the universe
27:43is flat.
27:45So now we
27:45have something
27:46to compare
27:47our measurement
27:47to.
27:48If the universe
27:49is flat
27:49the angle
27:50we measure
27:50on the sky
27:51should be
27:51the same
27:51as we'd
27:52expect.
27:53But if the
27:54universe is
27:54curved like
27:55a sphere
27:55the angles
27:56of the triangle
27:57should add up
27:57to more
27:58than 180
27:59degrees
27:59so the angle
28:00we'd measure
28:01would be
28:01larger than
28:02predicted
28:02and this peak
28:04would shift
28:04to the left.
28:05Similarly
28:06if the universe
28:07has hyperbolic
28:08geometry
28:08then spots
28:09should appear
28:10smaller than
28:11predicted
28:11and this peak
28:12would shift
28:12to the right.
28:14So what
28:15do we measure?
28:16This is the data
28:17from the Planck
28:17mission which is
28:18almost exactly
28:19what you'd expect
28:20if the universe
28:21were flat.
28:22This mission
28:23also gives us
28:23the current
28:24best estimate
28:25for the curvature
28:25of the universe
28:26which is
28:27.0007
28:29plus minus
28:30.0019.
28:31So that's
28:32basically
28:33zero within
28:34the margin
28:34of error.
28:35So we're
28:36fairly certain
28:36that the universe
28:37we live in
28:38is flat.
28:41But living
28:42in a flat
28:43universe
28:43seems to be
28:44remarkably
28:44serendipitous.
28:46Right now
28:46the average
28:47mass energy
28:47density comes
28:48down to
28:48the equivalent
28:49of about
28:49six hydrogen
28:50atoms
28:50per cubic meter.
28:52If on average
28:53that was just
28:54one more
28:54hydrogen atom
28:55the universe
28:56would have been
28:56more spherically
28:57curved.
28:58If there were
28:59just one less
29:00the curvature
29:00would be
29:01hyperbolic
29:02geometry.
29:03And so far
29:04we're not
29:04entirely sure
29:05why the universe
29:05has the
29:06mass energy
29:06density it has.
29:08What we do
29:09know is that
29:10general relativity
29:11is one of our
29:11best physical
29:12theories of reality
29:13and at the
29:14very heart of it
29:15are those
29:16paradoxical
29:17and seemingly
29:17absurd geometries.
29:20Ones we found
29:20because
29:21mathematicians
29:22spent over
29:222,000 years
29:23thinking about
29:24a single sentence
29:25from the world's
29:26most famous
29:27math text.
29:28When it comes
29:33to human
29:34brilliance
29:35it's not just
29:35what you know
29:36it's also
29:37how you think.
29:38And a great way
29:39to improve
29:40how you think
29:40is by building
29:41your knowledge
29:42and problem
29:43solving skills.
29:44So if you're
29:44looking for a
29:45free and easy
29:46way to do
29:47just that
29:47then check out
29:48this video's
29:49sponsor
29:49brilliant.org
29:51With Brilliant
29:52you can master
29:52key concepts
29:53in everything
29:54from math
29:54and data science
29:55to programming
29:56and technology.
29:57All you have
29:58to do
29:58is set
29:59your goal
30:00and Brilliant
30:00will design
30:01the perfect
30:01learning path
30:02for you
30:02giving you
30:03all the tools
30:04you need
30:04to reach it.
30:05Want to
30:06follow in the
30:06footsteps of
30:07Euclid
30:08and Bollii?
30:08Then Brilliant's
30:09latest course
30:10Measurement
30:11is the perfect
30:12addition to
30:13your problem
30:13solving toolkit.
30:14The course
30:15takes you
30:15on a tour
30:16through the
30:16essentials
30:17of geometry
30:18getting you
30:19hands-on
30:20with concepts
30:20to help
30:21sharpen your
30:22spatial reasoning
30:23skills.
30:24A solid
30:24foundation
30:25in geometry
30:25can be a
30:26launchpad
30:27to hundreds
30:27of applications
30:28everything from
30:29computer graphics
30:30and data clustering
30:31in AI algorithms
30:32to understanding
30:33Einstein's theory
30:34of general relativity.
30:36Beyond measurement
30:36Brilliant has a
30:37huge library
30:38of things to learn
30:39but what I love
30:40most about Brilliant
30:42is that they
30:42connect what you
30:43learn to
30:44real world examples
30:45and because each
30:46lesson is hands-on
30:47you'll build
30:48intuition so you
30:49can put what
30:50you've learned
30:50to good use.
30:52To try everything
30:52Brilliant has to
30:53offer for free
30:54for a full 30 days
30:55visit brilliant.org
30:57slash veritasium
30:58or click that link
30:59down in the
30:59description.
31:00For the first 200
31:01of you you will
31:02get 20% off
31:03Brilliant's annual
31:04premium subscription.
31:05So I want to
31:06thank Brilliant for
31:07sponsoring this video
31:07and I want to
31:08thank you for
31:09watching.
Hãy là người đầu tiên nhận xét