Mathematician Answers Geometry Questions From Twitter | Tech Support | WIRED

I'm Jordan Ellenberg mathematician let's answer some questions from the internet this is geometry [Music] support at s39 gsy asks who the f created geometry nobody created geometry geometry was always there it's just part of the way we interact with the physical world the person who first codified it and formalized it was somebody named uclid who lived in North Africa around 2,000 years ago and we also know that a lot of what he wrote down was the work of a lot of other people that he was collecting and putting in written form but this idea of geometry is this set of formal rules we Ed to carefully put together demonstrations of facts about angles triangles circles Etc that's when it sort of stops being purely intuitive and starts being something we can put in a book alien Searcher asks new shapes are just discovered yes absolutely new shapes are just discovered all the time one of the big misconceptions about math that people have is that math is f finished people who are geometers are typically thinking about crazy stuff that's going on in high Dimensions with all kinds of crazy curvature but four-dimensional shapes are in some sense just as real as three-dimensional shapes we just have to kind of train our minds to be able to perceive what shapes in those Dimensions like a hypercube or a tesseract would look like inkbot Kowalski asked wait wait a tesseract is a real thing definitely yes a Tess Act is another name for what's usually called in math a Cube MCU did not create the idea of the Tesseract being in Popular Science Fiction that really comes in in meline langel's book A Wrinkle in Time all right here you have a square a two-dimensional figure and here you have its three-dimensional counterpart a cube a cube you can think of as two squares the top square and the bottom square and then you sort of connect them together if the cube is the three-dimensional figure and the square is the two-dimensional figure what would be the four-dimensional figure I guess the hyper Cube would have to be something that was two cubes joined together and it would have to have twice as many corners as the cube does or 16 and now I've got to connect Each corner of the little Cube to the corresponding corner of The Big Cube this is our picture of the hyper Cube and now you may say like are there really four dimensions or is that just an invention well you know what when we do regular geometry we're working in a perfectly flat plane does that exist in the real world like probably not as a physical object the two-dimensional plane or threedimensional space are just as much of an abstraction as four-dimensional space okay claudo jacobo asks if algebra is the study of structure what is geometry algebra is the logical and symbolic right it's that side of your brain geometry is different geometry is physical geometry is Primal and like doing mathematics makes use of this tension between the algebraic side of our mind and the geometric side three Omega 2 asks how can I use the Pythagorean theorem to solve my problems in life look I'm going to be honest with you I can't quite imagine what problem you might have in your life that would be solved by the Pythagorean theorem the problem that the Pythagorean theorem solves is the following one if for some reason I have some distance I want to Traverse and if I know how far west you have to go to get there and then how far north and I happen to know these two distances then the Pythagorean theorem allows you to compute this diagonal distance which we call C but we can also write it as the square root of a 2 + b 2 is this the problem you're facing in your everyday life if it is you're in luck the Pythagorean theorem is here for you but in most cases it is not TM San asks what is special about a pringle's hyperbolic paraboloid geometry the Pringle is a wonderful geometric form what's special about it is this point right here at the center of the Pringle if I move from left to right I can't help but go up so it seems like I'm at the bottom of the Pringle but if I move from front to back back I can't help but go down from the center so it's somehow simultaneously at the top it's a peak and a valley at the same time and this special kind of point which is called a saddle point in math is what gives the Pringle its particularly Charming geometry Dr Funky spoon asks sucker MC's maintain cool Under Pressure but who with geometry like MC eer what a good question MC eer beloved artist of all matthy people was famous for studying and using in his art what are called tessellations ways of taking a flat plane and covering it with copies it was something he learned actually in part from hanging out with the alra this incredible Palace from Islamic Spain when you go to the alra you see these incredibly intricate but also very repetitive figures which by repetition across the entire wall it becomes very complicated and Rich that's the feature of a Tessellation who with geometry like mcer the answer is the UN named architects of the alra in Granada Spain Raspberry Pi asks how many holes are there in a straw fortunately I always bring a straw with me wherever I go how many holes are there in it there are the one holers who feel that well look there's like one hole it goes all the way through like what more is there to say and there are the two holers whose view is there's a hole at the top of the straw and there's a hole at the bottom of the straw for the people who think there's two holes I would say imagine this straw if you can getting shorter and shorter like imagine I sort of cut it and it was half a long and I cut it again until it's so short that it's actually like shorter than the distance around a little bit like this does this have one hole in it or two how many holes does a bagel have in it that's basically the same shape as this if you say a bagel has two holes I think we all agree that would be like a very weird thing to say about a bagel so now I'm talking to you triumphant one H holders if you think this straw is one hole let's say I take it and I pinch the bottom like this how many holes are there in it now there's just like the one hole at the top I mean you could fill this with water right it's basically a bottle how many holes are there in the water bottle just the one at the top that you drink out of right but if it has one hole now and I poked a hole in the bottom and I opened up the bottom how many holes would it have it's got to have two right I think the way to think about the straw is that yeah there's two holes but one of them is the negative of the other top hole plus bottom hole equals zero that sounds like an insane thing to say both the one holers and two holers are right in a way as long as they're willing to learn about the arithmetic of holes liberated Soul asks the golden ratio in art photography is that something to do with perfect composition yes the golden ratio is very popular it's a number a kind of unassuming number it's about 1.618 and there have always been people who felt that this particular number had some kind of mystical properties why that number well one way of describing it is that if I have a rectangle whose length and width are in that proportion a so-called golden rectangle it has has a special property which is that if I cut the rectangle to make one part of it a square what's left is again a golden rectangle no other kind of rectangle has that property some people would say like you can find it in nature like for instance I have here the shell of some kind of EMB bird braid like a wel in here we could find the golden ratio they say you can find it in a pine cone or I mean I think its mystical significance has been much overrated so I don't want to sound too salty about this but I think you shouldn't look to it to improve your Stockport P folio help you lose weight or help you find the prettiest rectangle zohi rafik 83 asks why are Honeycombs hexagons one thing I can tell you is that when the bees build the Honeycombs they're not hexagons they actually build them round and then something forces them into that hexagonal shape so there's a lot of controversy about this for instance why hexagons and not a grid of squares or triangles and there are people who will say well there's an efficiency argument maybe this is the way to give the honeycomb structural Integrity using the least amount of material I'm not sure that's completely convincing but that's at least one theory that people have bibbit e asks how are there so many different types of triangles this actually speaks to kind of a deep division in math the so-called three body problems one of the hardest problems in mathematics with two points you're making a line segment that looks like this and there's not a lot of variety among line segments they're all basically the same three points totally different story triangles come in an infinite variety of variations I mean you could have one that's like very narrow like this you could have one that's nice and symmetric our friend the equilateral triangle like that you could have a right triangle with a nice right angle I could just keep on drawing triangles in this little board and each one would look different from all the others and that is the difference between two and three problems involving two points simple problems involving three points already a completely infinite variety Tac 16 asks what is the random walk Theory and what does it mean for investors imagine a person with no sense of purpose every day they wake up and they walk a mile in One Direction or another you could track that person's motion over a long period of time that purposeless mindless unpredictable process a lot of people think the stock market basically works pretty much like that this is something that was worked out actually a really long time ago around 1900 by Lou bashel he was studying bond prices trying to understand what are the forces that govern these prices and yeah this would have been incredible Insight which is to say what if those prices just every day they might happen to go up or they might happen to go down purely by random chance and what he found is that if you model prices that way it looks exactly like the prices in real life vicam punt asks can you believe you can take the circumference of any Circle and divide it by its diameter and you will always get exactly Pi yeah I totally believe that and in fact I would say I relish it because it's one of the things that makes Circle circles there's really only one kind of circle it could be small or it could be big but this one is just a scaled up version of this one Whatever the diameter of this circle is and this guy has a diameter too if this diameter is seven times as big as this one then also this circumference that's the total distance around the circle is 7 times the size of this one so in particular the ratio between the circumference and the diameter is the same in both cases and that constant ratio Pi it's about 3.1415 I don't care so much what pi is is to 10 decimal places or 20 decimal places mathematically what's important is that there is such a thing as Pi that there is a constant that governs all circles no matter how big or how small tasking hanza asks what is the worst section in maths and why is it ukian geometry okay that stings a little bit geometry is the cilantro of math everybody either loves it or hates it it's the only part of math where you're asked to prove that something is true rather than just getting the answer to a question ukian geometry is geometry of the plane there's lots of other geometries non- ukian geometries you guys probably know the fact that the sum of the angles of a triangle is supposed to be 180° and a uid world that's true but on a curved surface like a sphere that's totally wrong all right my lines are not as straight as they might be but if you look at this kind of bulgy triangle and it's three angles their sum is going to be around 270 like way bigger than 180 and that's a fundamentally nonukan phenomenon that can only happen in a curved space we now know thanks to Einstein that space actually is curved when he revolutionized physics in the beginning of the 20th century the miracle is that non- ukian geometry was already there for him to use the mathematicians had already understood how curved space could work well in time for Einstein to realize that the world we actually live in is like that wion cmk asks Inception is it really a thesis on manifold and geometry and four-dimensional space Inception is a little bit more like what we call in geometry of fractal which has the property that it's self-similar that if you zoom in on it you see a smaller replica of the whole thing the more you zoom in the more detail you see and that seems to me the sort of spirit of the movie Inception so I I think I'm going to call that a fractal movie think big kids asks is there any better way to teach transformational geometry than original Nintendo Tetris I spend way too much time playing Tetris in college so I've thought about it a lot tried to make excuses for why that was actually a productive use of my time if you a modern geometry class it's not just about angles and circles and shapes they also talk about Transformations they say what happens if you take this shape and reflect it or take this shape and rotate it Tetris teaches you that skill imagine this little dude like marching down the screen you have to very quickly mentally figure out what it's going to look like rotated and which version of it is going to fit into a space where you need it and so I think you can think of Tetris as like a very very efficient in somewhat stressful training device for exactly that mental rotation skill that we're now trying to teach kids in Geometry Maris Crabtree has a joke for me a mobia strip walks into a bar sobbing the bartender asks what's wrong buddy the mobia strip replies where do I even begin You' think in my profession You' think I would have heard all the math jokes there are but every once in a while I hear a new one so amobia strip is a gric figure with a rather unusual quality that's not visible to the naked eye which is that it only has one side I'm going to Mark a little spot with an X and now I'm going to take my finger put it on the X and I'm going to start moving my way around the band watch me very closely I'm not switching sides I'm moving I'm moving my finger is staying on the band and look where I am I'm sort of in the same spot but I'm on the other side somewhat miraculously what appeared to be two different sides of the band are actually connected Rebecca 57219 asks anyone currently in a position where you use Pascal's triangle I definitely use Pascal's triangle and the numbers in it all the time here I have one with me there's these numbers written in the form of a triangle and the rule if you wanted to make one of these yourself is just that each number is the sum of the two numbers above it so right see how this six is the sum of three and three and then if I didn't know what went in here I could look above it and see a four and a six oh those add up to 10 so I have to put a 10 there but the cool thing is that these numbers actually mean something actually they mean a lot of different things but one of my favorite things that they mean is they record the likelihood of various outcomes in a random scenario like flipping coins so how do you turn these numbers into probabilities well if you were to add up all six of these numbers you would get 32 so you should really think of these numbers as fractions like one out of 32 5 out of 32 10 out of 32 those fractions are probabilities if I flip a coin five times there's six things that can happen I can get zero heads one head two heads three heads heads four heads okay well I I ran out of fingers but or five heads that's a sixth possibility and those correspond exactly to these six numbers in the fifth row of Pascal's triangle if you did an experiment and you flipped five coins thousands and thousands of times the proportion of those times that you would get two heads out of five would converge to 10 out of 32 harpa 71 burner asks why does the shape of a district matter and I'm going to assume that the question here is about congressional districts the reason is that if you see one with a very strange shape that is an indication that someone has designed that district for a political purpose I'm sorry to say that like rather Advanced mathematical techniques are used in order to effectively explore that geometric space to find the most partisan advantage that you can squeeze out of a map legislators choosing their voters instead of the voters choosing their legislators so that's why we care pw11 one1 okay I don't know how many ones there are there's a lot of ones why do GPS systems need to use geometry based on a sphere in order to work what GPS essentially does is there's a bunch of satellites which are in positions that we know they can tell you what is your distance when you're somewhere on the Earth from each one of those satellites and knowing those numbers is actually enough to specify your exact location let's say I know I'm exactly 5,342 km from a given satellite the set of all points that are at exactly that distance from the satellite is a sphere whose Center is that satellite that's what the definition of a sphere is it's the set of all points at a fixed distance from a given Center if I have two satellites I'm at the intersection of two spheres once you have four or more of those spheres they're never going to have more than one point in common that's exactly the geometry that underlies GPS Quantum stat asked what can the geometry of deep learning networks tell us about their inner workings I'm going to tell you the strategy that it uses it's basically a very intensive form of trial and error we make sort of some modest change to our behaviors and sort of see if it gives us better results and if it does we keep doing that thing I think of that as a kind of exploration of a space geometry in the modern sense is any context in which we can talk about things being near and far we know what it means for two people to be near each other geographically similarly the space of all strategies for recognizing a face those have geometries too there are some strategies that are near each other and some that are far away any context in which we can talk about near and far whether that's the surface of the Earth or a social network or your family where you can talk about close relatives or far relatives I know I'm kind of sounding like I'm just saying geometry as everything but I'm going to be honest that is kind of what I think okay so those are all the questions we have time for today I hope my answers made some sense or messed with your mind a little or best of all maybe did some combination to those two things thanks for watching geometry support

Channel: WIRED

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Keywords: geometry, geometry 101, geometry expert, geometry explained, geometry math, geometry mathematician, geometry question, geometry questions, geometry shapes, geometry tech support, geometry theories, geometry wired, innovation, jordan ellenberg, jordan ellenberg shapes, math, math expert, math explained, math questions, mathematician, mathematicians, ott tech support, science & technology, tech support, triangles, wired, wired geometry, wired tech support

Id: n_cS6S6bWX4

Length: 17min 53sec (1073 seconds)

Published: Tue Dec 05 2023