Mathematician Answers Geometry Questions From Twitter | Tech Support | WIRED

Mar 18, 2024

I'm Jordan Ellenberg,

mathematician

, let's answer some

questions

from the internet, this is

geometry

support

in s39gsy asking who the hell created

geometry

, no one created geometry, geometry was always there, it's just part of the way we interact with the physical world, the person who first coded. and he formalized it, he was someone called uclid who lived in North Africa about 2,000 years ago and we also know that a lot of what he wrote was the work of many other people that he was collecting and putting into writing, but this idea of geometry is This set of formal rules that we educate to carefully assemble demonstrations of facts about angles, triangles, circles, etc.

That's when it stops being purely intuitive and starts to be something we can put in a book. The alien seeker asks that new forms have just been discovered. Yes. Absolutely new ways are discovered all the time. One of the big misconceptions people have about mathematics is that mathematics is something finished. People who are geometers often think of crazy things happening in high dimensions with all kinds of crazy curvatures, but four... Dimensional shapes are, in a sense, as real as three-dimensional shapes. We just have to train our minds to be able to perceive what shapes would look like in those dimensions, like a hypercube or a tesseract.

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mathematician answers geometry questions from twitter tech support wired...

Inkbot Kowalski asked: Wait, wait, it's a tesseract. something real, definitely yes, Tess Act is another name for what is generally called in math a Cube MCU didn't create the idea of Tesseract being in popular science fiction that actually appears in meline langel's book A Wrinkle in Time, everything here it is. a square a two-dimensional figure and here you have its three-dimensional counterpart a cube a cube that you can consider as two squares the upper square and the lower square and then you connect them together if the cube is three-dimensional The figure and the square is the two-dimensional figure, what?

What would be the four-dimensional figure? I guess the hypercube would have to be something that was two cubes joined together and it would have to have twice as many corners as the cube or 16. and now I have to connect each corner of the small cube with the corresponding corner of the big cube. This is our image of the hypercube and now you can say: do four dimensions really exist or is it just an invention? Well, you know? What happens when we do regular geometry? We are working on a perfectly flat plane. Does that exist in the real world?

Probably not as a physical object. The two-dimensional plane or three-dimensional space is as abstract as four-dimensional space. claudo jacobo asks if algebra is the study of structure what is geometry algebra is the logical and symbolic right is that side of your brain geometry is different geometry is physics geometry is Primal and just like doing mathematics it 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 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 if for some reason I have a certain distance that I want to travel and if I know how far west to go to get there and then how far north and it happens to me If you know these two distances, then the Pythagorean theorem allows you to calculate this diagonal distance that we call C but we can also write it as the square root of a 2 + b 2.

Is this the problem you face in your daily life? You're lucky, the Pythagorean theorem is here for you, but in most cases it isn't. TM San asks what is special about Pringle's hyperbolic paraboloid geometry. Pringle is a wonderful geometric shape. What's special about it is this spot right here in the center. of the Pringle if I move from left to right I can't help but go up so it looks like I'm at the bottom of the Pringle but if I move from front to back I can't help but go down from the center so it's somehow simultaneously at the top , is a peak and a valley at the same time and this special type of point called a saddle point in mathematics is what gives the Pringle its particularly charming geometry.

Dr. Funky Spoon asks the goofy MCs to stay cool. Pressure but who with geometry like MC eer what a good question MC eer the beloved artist of all people matthy was famous for studying and using in his art what are called tessellations ways of taking a plane and covering it with copies was something he actually learned partly because it is with the alra, this incredible palace of Islamic Spain, when you go to the alra you see these incredibly intricate but also very repetitive figures that, when repeated throughout the wall, become very complicated and rich, that is the characteristic of a tessellation that with geometry like mcer the answer is the architects appointed by the UN of the alra in Granada Spain Raspberry Pi asks how many holes are in a straw fortunately I always carry a straw wherever I go how many holes are there in it there is one Holers that feel Well look, there is like a hole that goes completely through, like what else is there to say and there are the two Holers whose view is that there is a hole in the top of the straw and there is a hole in the bottom of the straw for the people who think there are two holes, I would say imagine this straw if you can get shorter and shorter, like imagine, I cut it and it was half long and I cut it again until it was so short that it's actually shorter than the distance around it a little like this.

Does it have one or two holes? How many holes does a muffin have? It basically has the same shape as this one. If you say a muffin has two holes, I think we can all agree that it would be like. That's a very strange thing to say about a bagel, so now I'm talking to you H-holder overachievers, if you think this straw has only one hole, let's say I take it and pinch the bottom like this, how many holes does it have? now? like the hole at the top, I mean you could fill this with water, it's basically a bottle, how many holes are there in the water bottle, just the one at the top that you drink from, but if it has a hole now and I made a hole in the bottom and opened the bottom, how many holes would it have?

It has to have two, right? I think the way to think about the straw is that yes, there are two holes, but one of them is the negative of the other. The top hole plus the bottom hole equal zero. It sounds crazy to say that both the one-hole maker and the two-hole maker are right in some ways, as long as they are willing to learn about the arithmetic of liberated holes. Soul asks what the golden ratio is in fine art photography. it is something to do with perfect composition yes the golden ratio is very popular it is a number, a kind of unassuming number, it is approximately 1.618 and there have always been people who felt that this particular number had some kind of mystical properties why that number is good one way to describe it is that if I have a rectangle whose length and width are in that proportion, the so-called golden rectangle has a special property which is that if I cut the rectangle to turn a part of it into a square , what is left is again a golden rectangle, no other type of rectangle has that property, some people would say it can be found in nature, like for example, I have here the shell of some kind of EMB bird braid like a well, here we can find the golden ratio, they say you can find it in a pineapple or I mean, I think its mystical meaning has been very overrated, so I don't want to sound too salty with this, but I think you shouldn't resort to it to improve your Stockport folio Q. lose weight or help you find the prettiest rectangle zohi rafik 83 asks why honeycombs are hexagons.

One thing I can tell you is that when bees build honeycombs, they are not hexagons, they actually build them round and then something forces them into that hexagonal shape. So there is a lot of controversy about this, for example why are there hexagons and not a grid of squares or triangles, and there are people who will say there is an efficiency argument, maybe this is the way to give structural integrity to the honeycomb using the Less amount of material. I'm not sure it's completely convincing, but that's at least one theory people have. Bibbit e asks how there are so many different types of triangles.

This actually speaks to a kind of deep divide in mathematics. The so-called three-body problems. One of the most difficult. problems in math with two points you are making a line segment that looks like this and there is not much variety between the line segments they are all basically the same three points totally different story triangles come in an infinite variety of variations, I mean you could have one that is very narrow like this you could have one that is nice and symmetrical our friend the equilateral triangle so you could have a right triangle with a nice right angle I could keep drawing triangles on this little board and each one looks different from all the others and that is the difference between two and three problems involving two points simple problems involving three points and 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 all days they wake up and walk a mile in one direction or another, you can follow that person's movement for a long period of time, that unpredictable process, without purpose, without meaning, many people think that the stock market basically works more or less like that.

It's something that was actually solved a long time ago, around 1900, by Lou Bashel. He was studying bond prices trying to understand what the forces are that govern these prices and yes, this would have been incredible insight, I mean, what if those prices just happened? every day they can go up or down by pure chance and what he discovered is that if you model prices that way, they look exactly like prices in real life. Vicam Punt asks: can you believe that you can take the circumference of any circle and divide it by its diameter and you will always get exactly Pi, yes I totally believe that and in fact I would say that I enjoy it because it is one of the things that makes the circles be circles;

There's actually only one type of circle that can be small or it can be big, but this is just an enlarged version of it. Whatever the diameter of this circle is and this guy also has a diameter, if this diameter is seven times bigger than this one, then also this circumference is the total distance around the circle is 7 times the size of this one, so In particular, the relationship between the circumference and the diameter is the same in both cases and that constant relationship Pi is approximately 3.1415. I don't care much what pi is. 10 decimals or 20 decimals mathematically the important thing is that there is something called Pi, that there is a constant that governs all circles no matter how big or small the task is.

Hanza asks what the worst section in math is and why it's British. geometry, okay that hurts a little, geometry is the coriander of mathematics, everyone loves it or hates it, it's the only part of mathematics where you're asked to prove something is true instead of just get the answer to a question, British geometry is plane geometry there are many other geometries, non-British geometries, you probably know the fact that the sum of the angles of a triangle is supposed to be 180° and a fluid world, that That's true, but on a curved surface like a sphere, that's totally wrong.

Alright, my lines are not as straight as they could be, but if you look at this kind of bulge triangle and it has three angles, their sum will be around 270, much larger than 180 and that is a fundamentally non-ukan phenomenon that can only happen in a curved space we now know thanks to Einstein that space is actually curved when he revolutionized physics at the beginning of the 20th century the miracle is that non-British geometry was already there for him to use

mathematician

s had already understood how the curved space Well, in time for Einstein to realize that the world we really live in is like this.

Wion CMK asks Inception. Is it really a thesis about the multiple, geometry and four-dimensional space? Inception is a little more like what we call in geometry fractal, which has the property that it is self-similar and if you get closer you see a smaller replica of everything, the closer you get, the more details you see and that seems to me to be the kind of spirit from the movie Inception, so I think I'm going to call that a fractal movie, I think the big kids are asking if there's any better way to teach transformational geometry than the original Nintendo Tetris.

I spend too much time playing Tetris in college, so I've thought about it a lot and tried to make excuses. Why was it really a productive use of my time? If you're a modern geometry class, it's not just about angles, circles, andshapes, they also talk about Transformations, they say what happens if you take this shape and reflect it or you take this shape and rotate it. Tetris teaches you that skill: imagine this little guy walking across the screen. You have to mentally figure out very quickly what it's going to look like rotated and which version will fit in the space where you need it, so I think I can think of Tetris as a very, very efficient and somewhat stressful training device, exactly for that mental rotation skill. which we are now trying to teach children in Geometry.

Maris Crabtree has a joke for me. A strip of mobia enters a bar sobbing and asks the waiter. What's up dude? The Mobia strip

answers

, where do I start? You think about my profession. You think I would have heard every math joke out there, but every once in a while I hear a new one, so the amobia strip is a gric figure with a rather unusual quality that is not visible to the naked eye, which is that it only has one side. I'm going to Mark a little dot with an X and now I'm going to take my finger, put it on the X and start moving in my direction. around the band look at me very closely I'm not changing sides I'm moving I'm moving my finger stays on the band and look where I am I'm like in the same place but I'm on the other side Somewhat miraculously, what They seemed to be two different sides of the band they are actually connected.

Rebecca 57219 asks anyone who is currently in a position using Pascal's triangle. She definitely used Pascal's triangle and the numbers it contains all the time. I have one here with me. There are these numbers. written in the form of a triangle and the ruler, if you wanted to make one of these yourself, it's just that each number is the sum of the two numbers that are above it, so look how this six is the sum of three and three and then, yes I didn't know what was here. I could look up and see a four and a six, oh, those add up to 10, so I have to put a 10 in there, but the cool thing is that these numbers actually mean something, they actually mean a lot. of different things, but one of my favorite things they mean is that they record the probability of various outcomes in a random scenario, like flipping coins, so how do you convert these numbers into probabilities?

If you added the six numbers together you would get you get 32 so you should 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 are six things that can happen I can get zero heads one head two head three head head four head okay, I ran out of fingers but or five heads is a sixth possibility and they correspond exactly to these six numbers in the fifth row of Pascal's triangle if you did an experiment and threw 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 the shape of a district matters and I'm going to assume 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 quite advanced mathematical

tech

niques are used to effectively explore that geometric space to find. the most partisan advantage you can get from a map is that legislators choose their voters instead of voters choosing their legislators, that's why we care pw11 one1 okay, I don't know how many there are, there are many, why GPS systems? I need to use geometry based on a sphere to work. What GPS essentially does is that there are a bunch of satellites that are in positions that we know can tell you what your distance is when you are somewhere on Earth from each of them. satellites and knowing those numbers is enough to specify your exact location.

Let's say I know that I am exactly 5,342 km from a given satellite. The set of all points that are exactly at that distance from the satellite is a sphere whose center is that satellite. What is the definition of a sphere? It is the set of all points at a fixed distance from a given center. If I have two satellites, I am at the intersection of two spheres. Once you have four or more of those spheres, they will never be. They have more than one. common point that is exactly the geometry underlying GPS Quantum stat asked what the geometry of deep learning networks can tell us about their inner workings.

I'm going to tell you the strategy he uses. It's basically a very intensive form of trial and error, we make some kind of modest change in our behaviors and see if it gives us better results and if so, we continue doing that. I think it's a kind of exploration of a spatial geometry in the modern sense. any context in which we can talk about things near and far we know what it means for two people to be close to each other geographically in a similar way the space of all the strategies to recognize a face those also have geometries there are some strategies that are close to each other another and some who are far away any context in which we can talk about near and far, whether it is the surface of the Earth or a social network or your family where you can talk about close relatives or distant relatives.

I know I'm sounding like I'm just saying geometry like everything, but I'm going to be honest, that's what I think, okay, so that's all the