Mathematician Jordan Ellenberg answers the internet's burning questions about geometry. How are new shapes still being discovered? Where are we using Pythagorean theorem in real life? How many holes are in a...straw? Ellenberg answers all these questions and much, much more!Jordan Ellenberg's book Shape is available on Amazon or Penguin Random Househttps://www.amazon.com/gp/product/1984879073?tag=randohouseinc7986-20https://www.penguinrandomhouse.com/books/612131/shape-by-jordan-ellenberg/Director: Lisandro Perez-ReyDirector of Photography: Constantine EconomidesEditor: Richard TrammellExpert: Jordan EllenbergLine Producer: Joseph BuscemiAssociate Producer: Brandon WhiteProduction Manager: D. Eric MartinezProduction Coordinator: Fernando DavilaCasting Producer: Nick SawyerCamera Operator: Christopher EustacheGaffer: Rebecca Van Der MeulenSound Mixer: Michael GugginoProduction Assistant: Sonia ButtPost Production Supervisor: Alexa DeutschPost Production Coordinator: Ian BryantSupervising Editor: Doug LarsenAdditional Editor: Paul TaelAssistant Editor: Billy Ward
Category
🤖
TechTranscript
00:00 I'm Jordan Ellenberg, mathematician.
00:01 Let's answer some questions from the internet.
00:03 This is Geometry Support.
00:05 (upbeat music)
00:08 @S39GSY asks, "Who the F created geometry?"
00:14 Nobody created geometry.
00:17 Geometry was always there.
00:18 It's just part of the way we interact
00:20 with the physical world.
00:21 The person who first codified it and formalized it
00:24 was somebody named Euclid,
00:25 who lived in North Africa around 2000 years ago.
00:27 And we also know that a lot of what he wrote down
00:29 was the work of a lot of other people
00:31 that he was collecting and putting in written form.
00:33 But this idea of geometry as the set of formal rules
00:37 we use to carefully put together demonstrations of facts
00:40 about angles, triangles, circles, et cetera,
00:43 that's when it sort of stops being purely intuitive
00:45 and starts being something we can put in a book.
00:48 @AlienSearcher asks, "New shapes are just discovered?"
00:52 Yes, absolutely.
00:53 New shapes are just discovered all the time.
00:55 One of the big misconceptions about math that people have
00:58 is that math is finished.
01:00 People who are geometers are typically thinking about
01:04 crazy stuff that's going on in high dimensions
01:06 with all kinds of crazy curvature.
01:08 But four-dimensional shapes are in some sense
01:10 just as real as three-dimensional shapes.
01:13 We just have to kind of train our minds
01:15 to be able to perceive what shapes in those dimensions
01:17 like a hypercube or a tesseract would look like.
01:20 @Inkbot_Kowalski asked, "Wait, wait,
01:22 "a tesseract is a real thing?"
01:25 Definitely, yes.
01:26 A tesseract is another name for what's usually called
01:28 in math a hypercube.
01:30 MCU did not create the idea of the tesseract
01:33 being in popular science fiction.
01:34 That really comes in in Madeline L'Engle's book,
01:37 "A Wrinkle in Time."
01:38 All right, here you have a square, a two-dimensional figure,
01:41 and here you have its three-dimensional counterpart, a cube.
01:45 A cube you can think of as two squares,
01:47 the top square and the bottom square,
01:49 and then you sort of connect them together.
01:51 If the cube is the three-dimensional figure
01:53 and the square is the two-dimensional figure,
01:55 what would be the four-dimensional figure?
01:57 I guess the hypercube would have to be something
02:00 that was two cubes joined together
02:02 and it would have to have twice as many corners
02:04 as the cube does or 16.
02:06 And now I've got to connect each corner of the little cube
02:09 to the corresponding corner of the big cube.
02:12 This is our picture of the hypercube.
02:14 And now you may say like,
02:15 "Are there really four dimensions
02:18 "or is that just an invention?"
02:19 Well, you know what?
02:20 When we do regular geometry,
02:21 we're working in a perfectly flat plane.
02:23 Does that exist in the real world?
02:25 Like probably not as a physical object.
02:27 The two-dimensional plane or three-dimensional space
02:29 are just as much of an abstraction
02:31 as four-dimensional space.
02:32 Okay, Claudio Jacobo asks,
02:34 "If algebra is the study of structure, what is geometry?"
02:38 Algebra is the logical and symbolic, right?
02:41 It's that side of your brain.
02:42 Geometry is different.
02:43 Geometry is physical.
02:44 Geometry is primal.
02:45 And like doing mathematics makes use of this tension
02:47 between the algebraic side of our mind
02:49 and the geometric side.
02:50 3Omega2 asks, "How can I use the Pythagorean theorem
02:54 "to solve my problems in life?"
02:56 Look, I'm gonna be honest with you.
02:57 I can't quite imagine what problem you might have
03:00 in your life that would be solved
03:01 by the Pythagorean theorem.
03:03 The problem that the Pythagorean theorem solves
03:05 is the following one.
03:06 If for some reason I have some distance I wanna traverse,
03:10 and if I know how far west you have to go to get there,
03:14 and then how far north,
03:16 and I happen to know these two distances,
03:19 then the Pythagorean theorem allows you
03:21 to compute this diagonal distance, which we call C,
03:24 but we can also write it as the square root
03:26 of A squared plus B squared.
03:29 Is this the problem you're facing in your everyday life?
03:32 If it is, you're in luck.
03:34 The Pythagorean theorem is here for you,
03:36 but in most cases it is not.
03:39 TMSion asks, "What is special
03:42 "about a Pringle's hyperbolic paraboloid geometry?"
03:45 The Pringle is a wonderful geometric form.
03:47 What's special about it is this point right here
03:50 at the center of the Pringle.
03:51 If I move from left to right,
03:54 I can't help but go up,
03:55 so it seems like I'm at the bottom of the Pringle.
03:57 But if I move from front to back,
04:00 I can't help but go down from the center,
04:01 so it's somehow simultaneously at the top.
04:03 It's a peak and a valley at the same time,
04:06 and this special kind of point,
04:08 which is called a saddle point in math,
04:10 is what gives the Pringle
04:12 its particularly charming geometry.
04:14 Dr. Funky Spoon asks, "Sucker MCs maintain cool
04:19 "under pressure, but who (beep) with geometry
04:22 "like MC Escher?"
04:24 What a good question.
04:25 MC Escher, beloved artist of all mathy people.
04:29 Escher was famous for studying and using in his art
04:32 what are called tessellations,
04:34 ways of taking a flat plane and covering it with copies.
04:38 It was something he learned actually in part
04:39 from hanging out at the Alhambra,
04:41 this incredible palace from Islamic Spain.
04:44 When you go to the Alhambra,
04:45 you see these incredibly intricate,
04:47 but also very repetitive figures,
04:50 which by repetition across the entire wall,
04:52 it becomes very complicated and rich.
04:54 That's the feature of a tessellation.
04:55 Who (beep) with geometry like MC Escher?
04:58 The answer is the unnamed architects
05:00 of the Alhambra in Granada, Spain.
05:02 Raspberry Pie asks, "How many holes are there in a straw?"
05:07 Fortunately, I always bring a straw with me wherever I go.
05:09 How many holes are there in it?
05:10 There are the one-holers who feel that,
05:13 well, look, there's like one hole.
05:14 It goes all the way through.
05:15 Like what more is there to say?
05:16 And there are the two-holers whose view is,
05:19 there's a hole at the top of the straw
05:20 and there's a hole at the bottom of the straw.
05:22 For the people who think there's two holes,
05:24 I would say, imagine this straw, if you can,
05:26 getting shorter and shorter.
05:27 Like imagine I sort of cut it and it was half as long.
05:29 I cut it again until it's so short
05:31 that it's actually like shorter than the distance around.
05:33 A little bit like this.
05:34 Does this have one hole in it or two?
05:38 How many holes does a bagel have in it?
05:40 That's basically the same shape as this.
05:41 If you say a bagel has two holes,
05:43 I think we all agree that would be like a very weird thing
05:45 to say about a bagel.
05:46 So now I'm talking to you, triumphant one-holers.
05:49 If you think this straw is one hole,
05:51 let's say I take it and I pinch the bottom like this.
05:54 How many holes are there in it now?
05:55 There's just like the one hole at the top.
05:57 I mean, you could fill this with water, right?
05:58 It's basically a bottle.
06:00 How many holes are there in the water bottle?
06:02 Just the one at the top that you drink out of, right?
06:04 But if it has one hole now
06:06 and I poked a hole in the bottom
06:07 and I opened up the bottom, how many holes would it have?
06:09 It's gotta have two, right?
06:10 I think the way to think about the straw
06:12 is that yeah, there's two holes,
06:14 but one of them is the negative of the other.
06:16 Top hole plus bottom hole equals zero.
06:19 That sounds like an insane thing to say.
06:21 Both the one-holers and two-holers are right in a way,
06:25 as long as they're willing to learn
06:27 about the arithmetic of holes.
06:28 Liberated Soul asks, "The golden ratio in art photography,
06:33 "is that something to do with perfect composition?"
06:35 Yes, the golden ratio is very popular.
06:38 It's a number, a kind of unassuming number.
06:40 It's about 1.618.
06:42 And there have always been people
06:44 who felt that this particular number
06:46 had some kind of mystical properties.
06:49 Why that number?
06:51 Well, one way of describing it is that
06:53 if I have a rectangle whose length and width
06:56 are in that proportion, a so-called golden rectangle,
06:58 it has a special property,
07:00 which is that if I cut the rectangle
07:02 to make one part of it a square,
07:04 what's left is again, a golden rectangle.
07:07 No other kind of rectangle has that property.
07:09 Some people would say, like, you can find it in nature.
07:11 Like for instance, I have here the shell
07:13 of some kind of in birdbrain, like a whelk.
07:15 In here, we could find the golden ratio.
07:18 They say you can find it in a pine cone,
07:20 or I mean, I think it's mystical significance
07:22 has been much overrated.
07:23 So I don't wanna sound too salty about this,
07:25 but I think you shouldn't look to it
07:27 to improve your stock portfolio, help you lose weight,
07:31 or help you find the prettiest rectangle.
07:34 Zohai Rafik83 asks, "Why are honeycombs hexagons?"
07:39 One thing I can tell you is that
07:41 when the bees build the honeycombs, they're not hexagons.
07:45 They actually build them round
07:46 and then something forces them into that hexagonal shape.
07:49 So there's a lot of controversy about this.
07:51 For instance, why hexagons
07:53 and not a grid of squares or triangles?
07:55 And there are people who will say,
07:56 well, there's an efficiency argument.
07:57 Maybe this is the way to give
07:59 the honeycomb structural integrity
08:00 using the least amount of material.
08:01 I'm not sure that's completely convincing,
08:03 but that's at least one theory that people have.
08:05 Bibit E asks, "How are there so many
08:08 different types of triangles?"
08:09 This actually speaks to kind of a deep division in math.
08:12 The so-called three body problems,
08:13 one of the hardest problems in mathematics.
08:15 With two points, you're making a line segment
08:17 that looks like this.
08:18 And there's not a lot of variety among line segments.
08:21 They're all basically the same.
08:22 Three points, totally different story.
08:24 Triangles come in an infinite variety of variations.
08:27 I mean, you could have one that's like very narrow,
08:30 like this.
08:31 You could have one that's nice and symmetric.
08:33 Our friend, the equilateral triangle, like that.
08:36 You could have a right triangle with a nice right angle.
08:39 I could just keep on drawing triangles in this little board
08:41 and each one would look different from all the others.
08:44 And that is the difference between two and three.
08:46 Problems involving two points, simple.
08:48 Problems involving three points,
08:50 already a completely infinite variety.
08:52 Tzack16 asks, "What is the random walk theory
08:56 and what does it mean for investors?"
08:57 Imagine a person with no sense of purpose.
09:00 Every day they wake up and they walk a mile
09:03 in one direction or another.
09:04 You could track that person's motion
09:07 over a long period of time.
09:09 That purposeless, mindless, unpredictable process,
09:13 a lot of people think the stock market
09:14 basically works pretty much like that.
09:17 This is something that was worked out
09:18 actually a really long time ago,
09:19 around 1900 by Louis Bachelier.
09:21 He was studying bond prices,
09:24 trying to understand what are the forces
09:26 that govern these prices.
09:28 And he had this sort of incredible insight,
09:29 which is to say, what if those prices,
09:31 just every day, they might happen to go up
09:33 or they might happen to go down, purely by random chance.
09:36 And what he found is that if you model prices that way,
09:40 it looks exactly like the prices in real life.
09:42 Vikram Punt asks, "Can you believe
09:44 you can take the circumference of any circle
09:46 and divide it by its diameter
09:48 and you will always get exactly pi?"
09:50 Yeah, I totally believe that.
09:51 In fact, I would say I relish it
09:53 because it's one of the things that makes circles circles.
09:55 There's really only one kind of circle.
09:58 It could be small or it could be big,
10:00 but this one is just a scaled up version of this one.
10:04 Whatever the diameter of this circle is,
10:06 and this guy has a diameter too.
10:08 If this diameter is seven times as big as this one,
10:11 then also this circumference,
10:13 that's the total distance around the circle,
10:15 is seven times the size of this one.
10:17 So in particular, the ratio between the circumference
10:20 and the diameter is the same in both cases.
10:22 And that constant ratio, pi, it's about 3.1415.
10:27 I don't care so much what pi is to 10 decimal places
10:30 or 20 decimal places.
10:31 Mathematically, what's important
10:33 is that there is such a thing as pi,
10:35 that there is a constant that governs all circles,
10:38 no matter how big or how small.
10:40 Taskine Hansa asks,
10:42 "What is the worst section in math
10:44 and why is it Euclidean geometry?"
10:47 Okay, that stings a little bit.
10:48 Geometry is the cilantro of math.
10:50 Everybody either loves it or hates it.
10:52 It's the only part of math
10:54 where you're asked to prove that something is true
10:56 rather than just getting the answer to a question.
10:59 Euclidean geometry is geometry of the plane.
11:02 There's lots of other geometries,
11:04 non-Euclidean geometries.
11:05 You guys probably know the fact
11:07 that the sum of the angles of a triangle
11:10 is supposed to be 180 degrees.
11:11 And in Euclid world, that's true.
11:13 But on a curved surface like a sphere,
11:15 that's totally wrong.
11:16 All right, my lines are not as straight as they might be,
11:19 but if you look at this kind of bulgy triangle
11:21 and it's three angles,
11:22 their sum is gonna be around 270,
11:24 like way bigger than 180.
11:26 And that's a fundamentally non-Euclidean phenomenon
11:29 that can only happen in a curved space.
11:31 We now know, thanks to Einstein,
11:33 that space actually is curved.
11:35 When he revolutionized physics
11:36 in the beginning of the 20th century,
11:37 the miracle is that non-Euclidean geometry
11:39 was already there for him to use.
11:41 The mathematicians had already understood
11:43 how curved space could work well in time
11:46 for Einstein to realize
11:47 that the world we actually live in is like that.
11:50 - WissonCMK asks, "Inception,
11:53 is it really a thesis on manifold and geometry
11:55 and four-dimensional space?"
11:57 - Inception is a little bit more like
11:59 what we call in geometry a fractal,
12:00 which has the property that it's self-similar,
12:02 that if you zoom in on it,
12:04 you see a smaller replica of the whole thing.
12:06 The more you zoom in, the more detail you see.
12:09 And that seems to me the sort of spirit
12:11 of the movie "Inception."
12:12 So I think I'm gonna call that a fractal movie.
12:14 ThinkBigKids asks, "Is there any better way
12:17 to teach transformational geometry
12:19 than original Nintendo Tetris?"
12:21 I spent way too much time playing Tetris in college,
12:24 so I've thought about it a lot,
12:25 tried to make excuses for why
12:26 that was actually a productive use of my time.
12:28 If you take a modern geometry class,
12:30 it's not just about angles and circles and shapes.
12:32 They also talk about transformations.
12:34 They say, "What happens if you take this shape
12:37 and reflect it, or take this shape and rotate it?"
12:39 Tetris teaches you that skill.
12:42 Imagine this little dude like marching down the screen.
12:45 You have to very quickly, mentally figure out
12:48 what it's gonna look like rotated
12:50 and which version of it is gonna fit
12:52 into a space where you need it.
12:54 And so I think you can think of Tetris
12:56 as like a very, very efficient
12:58 and somewhat stressful training device
13:00 for exactly that mental rotation skill
13:02 that we're now trying to teach kids in geometry.
13:04 - Maris Crabtree has a joke for me.
13:06 A mobius strip walks into a bar sobbing.
13:09 The bartender asks, "What's wrong, buddy?"
13:11 The mobius strip replies, "Where do I even begin?"
13:14 You'd think in my profession,
13:16 you'd think I would've heard all the math jokes there are,
13:18 but every once in a while, I hear a new one.
13:20 So a mobius strip is a geometric figure
13:23 with a rather unusual quality
13:25 that's not visible to the naked eye,
13:27 which is that it only has one side.
13:28 I'm gonna mark a little spot with an X.
13:31 And now I'm gonna take my finger, put it on the X,
13:33 and I'm gonna start moving my way around the band.
13:37 Watch me very closely.
13:38 I'm not switching sides.
13:40 I'm moving, I'm moving.
13:41 My finger is staying on the band.
13:45 And look where I am.
13:47 I'm sort of in the same spot, but I'm on the other side.
13:49 Somewhat miraculously, what appear to be
13:51 two different sides of the band are actually connected.
13:54 Rebecca57219 asked, "Anyone currently in a position
13:59 "where you use Pascal's triangle?"
14:01 I definitely use Pascal's triangle
14:03 and the numbers in it all the time.
14:05 Here, I have one with me.
14:06 There's these numbers written in the form of a triangle.
14:09 And the rule, if you wanted to make one of these yourself,
14:11 is just that each number is the sum
14:14 of the two numbers above it.
14:16 So right, see how this six is the sum of three and three?
14:18 And then if I didn't know what went in here,
14:20 I could look above it and see a four and a six.
14:22 Oh, those add up to 10, so I have to put a 10 there.
14:24 But the cool thing is that these numbers
14:25 actually mean something.
14:26 Actually, they mean a lot of different things,
14:28 but one of my favorite things that they mean
14:29 is they record the likelihood of various outcomes
14:33 in a random scenario, like flipping coins.
14:36 So how do you turn these numbers into probabilities?
14:38 Well, if you were to add up all six of these numbers,
14:41 you would get 32.
14:42 So you should really think of these numbers as fractions,
14:44 like one out of 32, five out of 32, 10 out of 32.
14:48 Those fractions are probabilities.
14:50 If I flip a coin five times,
14:53 there's six things that can happen.
14:54 I can get zero heads, one head, two heads,
14:58 three heads, four heads.
14:59 Okay, well, I ran out of fingers, but, or five heads.
15:02 That's a sixth possibility.
15:03 And those correspond exactly to these six numbers
15:07 in the fifth row of Pascal's triangle.
15:09 If you did an experiment and you flipped five coins
15:12 thousands and thousands of times,
15:14 the proportion of those times
15:15 that you would get two heads out of five
15:17 would converge to 10 out of 32.
15:20 HarperWest71Burner asks,
15:22 why does the shape of a district matter?
15:24 And I'm gonna assume that the question here
15:26 is about congressional districts.
15:27 The reason is that if you see one
15:30 with a very strange shape,
15:31 that is an indication that someone has designed that district
15:36 for a political purpose.
15:37 I'm sorry to say that like rather advanced
15:39 mathematical techniques are used
15:41 in order to effectively explore that geometric space
15:44 to find the most partisan advantage
15:45 that you can squeeze out of a map.
15:47 Legislators choosing their voters
15:49 instead of the voters choosing their legislators.
15:51 So that's why we care.
15:52 PW1111, okay, I don't know how many ones there are.
15:55 There's a lot of ones.
15:56 Why do GPS systems need to use geometry
15:59 based on a sphere in order to work?
16:01 What GPS essentially does is there's a bunch of satellites
16:04 which are in positions that we know.
16:05 They can tell you what is your distance
16:08 when you're somewhere on the earth
16:09 from each one of those satellites.
16:10 And knowing those numbers is actually enough
16:13 to specify your exact location.
16:15 Let's say I know I'm exactly 5,342 kilometers
16:19 from a given satellite.
16:20 The set of all points that are at exactly that distance
16:24 from the satellite is a sphere
16:26 whose center is that satellite.
16:27 That's what the definition of a sphere is.
16:29 It's the set of all points
16:30 at a fixed distance from a given center.
16:32 If I have two satellites,
16:33 I'm at the intersection of two spheres.
16:35 Once you have four or more of those spheres,
16:37 they're never gonna have more than one point in common.
16:39 That's exactly the geometry that underlies GPS.
16:42 Quantumstat asked,
16:44 what can the geometry of deep learning networks
16:46 tell us about their inner workings?
16:48 I'm gonna tell you the strategy that he uses.
16:50 It's basically a very intensive form of trial and error.
16:54 We make sort of some modest change to our behaviors
16:57 and sort of see if it gives us better results.
16:58 And if it does, we keep doing that thing.
17:00 I think of that as a kind of exploration of a space.
17:03 Geometry in the modern sense is any context
17:07 in which we can talk about things being near and far.
17:10 We know what it means for two people
17:12 to be near each other geographically.
17:14 Similarly, the space of all strategies
17:17 for recognizing a face,
17:19 those have geometries too.
17:20 There are some strategies that are near each other
17:22 and some that are far away.
17:24 Any context in which we can talk about near and far,
17:27 whether that's the surface of the earth
17:29 or a social network or your family,
17:31 where you can talk about close relatives or far relatives.
17:33 I know I'm kind of sounding like I'm just saying
17:36 geometry is everything, but I'm gonna be honest,
17:37 that is kind of what I think.
17:38 Okay, so those are all the questions we have time for today.
17:41 I hope my answers made some sense
17:43 or messed with your mind a little,
17:44 or best of all, maybe did some combination
17:46 of those two things.
17:47 Thanks for watching Geometry Support.
17:49 (dramatic music)