I have a use for the pythagorean theorem in real life application. I’m told a TV’s screen size always as side C and I know it is a 16:9 aspect ratio. I can find the height and width of that screen when the site doesn’t list the dimensions.
@@chesterotontop unless you're an engineer, architect, scientist, mathematician, programmer, city planner, game developer, digital artist, etc etc etc.
Well, you'll be able to find out the height and width of the LCD panel; but that screen size doesn't include the plastic frame around the LCD panel. So it can be useful for comparing monitors to one another, but not for knowing exactly whether or not the monitor/tv can fit in a given area. It wouldn't be a bad ballpark for it, though.
I highly recommend the essay “A Mathematician’s Lament” for anyone who wants to go deeply into the way we teach math and how poorly it’s taught that most students find math boring and frustrating in most math classes (I know mine classes were definitely not taught well). Jordan has the energy and love of mathematics that would make him an excellent teacher, and I wish I had someone like him while I was crying over my algebra 2 homework.
I hated math in school with the exception of geometry in 10th grade. That was a blast. But algebra was always a nightmare. Then I signed up for an algebra class in college with a specific math teacher everyone recommended. She taught math on colorful handouts and in true layman’s terms to where it all connected and made sense. It was like learning math where every lesson is “explain to me like I’m 5”, and her method of teaching was extremely effective and fun! So many students needed a total refresh of some basic math concepts just because of how poorly they were taught in the public school system, and she helped so many students, including me, to be unafraid of math. I wish there were more teachers like her around.
Same goes for language and literature tbh. People tried to make language more interesting to learn by including pop culture and popular fiction, but it still doesnt capture the incredible nuance and thought/philosophy that goes into writing, speaking, and communication in general.
I'm a fan of this series, but Jordan was a particularly strong communicator. Thank you for bringing him on, and thank you to Jordan for being a fantastic ambassador for geometry and math writ large.
@lucasm4299 because they understand language better than you 😊. "But" can be used to contrast with (e.g. I'm a fan, but this was trash), or to add to (I'm a fan, but this was excellent).
@@lucasm4299 'But' is used to contrast or add something. I have to agree with Alan, they used the word because they have a better understanding of English. 'But' is very commonly used to mean 'as an aside', 'also', or 'additionally'. 'This was a wonderful meal, but I have to say the chicken was particularly excellent!'
As we all know, hexagons are the bestagons, but it was nice to hear an explanation about it being incidental in the case of hive cells. Never heard that before in explanations of the subject.
Jordan gave a bad explanation here though. What is special about the regular hexagon is that among all regular polygons (i.e. whose sides are all the same length and the angles between adjacent sides are the same), it's the one with the most sides, such that you can fill a plane with them without gaps. So this uniquely satisfies the goals of maximizing the space for larvae with round cross-section, packing as many compartments into a given space, and minimizing material (wax) cost while having uniform wall strength. No other possible shape is as good as that. You can build a honeycomb out of regular triangles of squares and you'll fill the space with compartments and maintain uniform wall thickness, but it's a bad use of space because you need to make the triangle or squares rather big to fit the round-cross section larvae; if you take regular polygons with 7, 8, 9, or any large number of sides, you will leave unused gaps or waste wax.
I think is simpler than that, you can really only put around 6 circles of the same diameter as the central one. In any circle diameter that happens. And what happens when you smash those sides equally together? You create an hexagon. Just check a box of straws and you will see that happen. Alas you can create a honeycomb filter for a light (that focus light) with straws inside a tube
@@PauloSousa86100% this. Hexagon is the byproduct of the design, not the design itself. I think color pencil is another good example, by holding 7 of them together, 6 will be surrounding 1 of them regardless of the pencil shape. One things people also tend to miss is that most of those insects(bees and wasps) uses a sort of liquid mixture of saliva and other stuff to build their hive. So by starting with liquid, what making the hexagon shape could have something to do with surface tension where it started with circle, and the surface of those circles then stick to one another stretching themselves out before hardening into hexagonal shape. If somehow we can drop 7 honey droplets on a flat surface in a rubber band at the same time, with all of them have different dyed, we might be able to see those honey spread out to be shaped like hexagon, considering if the droplets is equal in size and have the same consistency.
As a quilter I use the Pythagorean theorem to figure out how many triangles I can get out of my fabric and how big to measure them. Once I had a pattern for a skirt that wanted right triangles of a certain length on the "c" side so I used it to calculate the "a" and "b" sides
As someone who's played around trying to program a pool game, balls in the game have known X and Y coordinates, I've used the Pythagorean theorem to find the distance between balls to check when the balls hit each other.
I once had to use the Pythagorean theorem as a web developer to calculate the size of a resizable widget when you clicked and dragged the corner! I was like "huh I guess knowing that actually was useful after all"
Contractors can also use it to make sure a corner is actually 90 degrees. Measure 3 going one way, 4 going the other, and adjust the corner until the hypotenuse is 5.
I love how unapologetic Jordan is about drawing crappy circles! 😂 On a more serious note, I was impressed by how well you pronounce the German names (Einstein and Möbius) in such a casual manner.
As a regular Dungeons & Dragons DM, I have sometimes used the Pythagorean theorem to calculate the distance of flying creatures moving diagonally to the ground to attack players. I'm just glad online calculators exist so I don't have to do the math myself. 🤣
I created a Collatz Dungeon for a party that was testing the Constable's patience. They would get dropped in Room 3,505,346, and they would be connected to two others, one double the number, and the other half as much. Eventually, they would hit an odd-numbered room, n, which would connect to Room '3n + 1'. All numbers, eventually, will connect to Room 1, where the exit would be.
Its very helpfull in vidéo games too ! I use it all the time to calculate distances between two objects in a plane in small personnal game projects :) Every object has x and y coordonnates, calculating the distance between the two is one of the most important things in a game. For détection, colisions etc... and Pythagore is always used.
I think the issue with the "Does a straw have one hole or two?" question is that everyone treats it as a geometry problem when it's more of a language problem.
@@hugomondoloni9808Because literally everything you just said uses language. It's a language game. Every single person has different definitions of when it is 2 or 1 hole. Math is still predicated on language.
6:58 The A paper sizes (A4, A3 etc) have a similar property, but it uses sqrt(2) instead of the golden ratio. When you fold it in half the ratio between the long and short side remains sqrt(2).
As a math teacher, this brings back memories of my college geometry and math history courses! Love it! It’s awesome to see somebody love their profession so much! 😊
The Pythagorean theorum has a lot of real world applications in architecture. For example, it's useful for designing staircases, since if you know the height of the upper floor, you can calculate the length of the staircase for any given footprint.
I used it recently to calculate the bill of materials on the roof of a shed I was building. Of all the mathematical / geometrical rules, this one is one of the more applicable ones to the real world... of course, if you google 'Trigonometry calculator', that's even more precise, and takes away the actual need to do the math...
@@FHL-DevilsI did something similar to turn the old, flat, but too short driveway into a new, longer driveway that would have a steep slope. Needed to make sure the rise on the slope wouldn't scrape the car
i was able to use phytagorean theorem on how much we need to extend our roofing for us not to have side-hitting rain hit our wall (which can weaken the concrete overtime due to accumulation of moisture). i was actually surprised when the calculations worked!! i felt like a wizard
TV screens are all measured in the diagonal dimension. So if you have a space on your wall that is X inches wide, you can use the Pythagorean theorem (and the fact that most TVs have a 16:9 aspect ratio) to determine the largest screen size you can put there.
Honey combs is 100% a packing efficiency problem. If you take any circular object, beer bottle, golf ball, whatever. Any circle, and more circles of the same size. You can wrap 6 more circles around the original.
That’s wrapping around to make another circle. So yea. But there’s still negative space that’s not being utilized. With strait lines you can take away that negative space. Hence Tetris etc.
For sure. I was a bit disappointed at his glib and not particularly knowledgeable or intuitive answer. Makes me wonder about some of his other answers.
It's not only packing efficiency, but also structural support and area. Squares tile just as well as Hexagons do, and they use less sides to do it, but they aren't as strong. Triangles tile just as easily as Hexagons do, and use even fewer sides than squares, but they have a smaller area. Hexagons are the perfect middleground for largest area (for storing more honey), fewest sides (easier to make), and ability to tile.
@@xkinsey3831while that is logically true, nature doesn’t calculate those stuff up. Bees just make circles, the same size as each others next to each others and those circle edge or surface tension or whatever it is just pull them to make a Hexagonal shape, that’s all. Hence why the original comment has every explanation needed to explain why it’s hexagon, because you can surround a circle with 6 more circles, and those circles surface, just like bubbles, they stick to each others and stretch out to form hexagon. Imagine this, you have 7 syringes of honey hold together, 6 surrounding 1 of them. Let’s say you dyed the honey in the middle syringe red. If you squeeze these 7 syringes on a table at the same time to form 7 circles, the honey will slowly spread out and the middle red honey one will spread out to look like, you guest it, hexagon. BOOM, mystery solved.
I used to use Pythagoras to mark out an accurate filed when laying out our clubs field hockey lines at the start of each season. Mark the baseline and then use a 3,4,5 triangle to make 90 degree corners for each sideline.
That’s the one. The Pythagorean theorem’s most useful real world application is to mark out exact right angles when the biggest square you have is still far too small: you can do it with a tape measure.
I feel that now I have a greater respect for straws. Thank you Mr. Jordan Ellenberg, it is incredible how you manage so much information to filter it into something simple, direct and with great humor
I had to give this a watch. I just used the Pythagorean theorem about two minutes ago. Creating miters for a picture frame and I needs to determine what the third side is going to be!
On the Pythagorean theorem : when I was a little boy, on my usual path to school, I had to around two sides of a square, as to not walk on a bit of lawn. I wondered how much distance I would spare every day if I just crossed that lawn across a diagonal. Well... One day I learned how to get that answer. You just have to be curious and you will need math in your everyday life.
When he mentioned the super hero movie not inventing the tesseract, I angry-scrolled to make sure "A Wrinkle in Time" was mentioned, just as he said it.
6:10 if you pinch the bottom, it has zero holes. A bowl or a plate don't have a hole, and an open-topped bottle is the same shape as a bowl or a plate.
@@fulanoide718 correct, it does not. A balloon that is not tied is a deformed disc. Holes cannot be created nor destroyed merely by stretching a shape. Balloons are just bowls with small necks.
"[Geometry] is the only part of math where you're asked to prove something..." Number theorists: "Am I a joke to you?" *war flashbacks to Abstract Algebra* (To be clear, it's fun, but hard)
I like to define a “Hole” as a region of Entrance, and Passage. For the purpose of my point an Entrance acts as a doorway you can enter and exit via a “doorway”, A passage is always accessed via an Entrance or Entrances. The straw and closed straw both have 1 hole Because both has At least 1 Entrance and a Passage via an Entrance. And a donut also has one hole via my definition.
Fun fact: the four dimensional tesseract was the central plot feature of Robert Heinlein's short story 'And He Built a Crooked House' published in 1941, twenty one years before 'A Wrinkle in Time' came out. Though I loved a Wrinkle in Time, Heinlein did a far better job describing it.
I just finished Stranger in a Strange Land, and his description of Mike sending things Away was so good! I'll have to check out that short story sometime soon!
I absolutely love the "how many holes are there in a straw" question. As someone who routinely does CAD as a design engineer, my perspective is in the way I'd create it (two steps): 1) Create a cylinder. 2) Cut extrude a hole through the entire body. Done. One hole.
Math finance PhD student here, just a comment about the random walk question. The Bachelier model in finance is a terrible model for stock movements and this was known at the time they published their model. A better model nowadays is models of the form e^(X(t)) where X(t) is some stochastic process (see something like geometric Brownian motion, the vasicek model, or more exotic models like the Heston model or general jump diffusion models). I bring up this detail because people get really silly and paranoid with stocks and it's important to note that these modeling problems are remarkably complex and nuanced. They require much more than just a random walk to be useful.
5:52 You are changing the goalposts. If you pinch the bottom of the straw, it is no longer the straw as we understand it to be in your question, or what we commonly recognize to be a straw.
The Pythagorean theorem is good for calculating straight distances on a map with grid lines. You count how many vertical and horizontal lines you're crossing and then use Pythagoras to calculate the distance.
5:04 There is one hole on the straw. When you cover the bottom, then the straw has no holes (a water bottle can be deformed into a bowl or a plate, for example)
@@xraygamer9895Z. It's either homeomorphic to a solid torus (if you assume it has width) or a cylinder (assuming no width). Either case the fundamental group is Z. It could still have higher dimensional holes but the homology groups are of course all trivial except for dim 0 and 1.
That's a lie it still has one hole in the top. A hole doesn't have to go completely through an object, if you have a hole punched into your car door on the outside, but not the inside you still have a hole in your car door
@@RCG3. I'm not sure if I understood, but the car door has many layers and if you count each layer as an object you can have a hole that doesn't go through the whole door. The equivalent to the straw example would be a dent on the car.
Arithmetic - Here's a number line Algebra - The formal logical rules and language of math Geometry - Can you draw it in some logical way and then deconstruct that drawing into an algebraic expression? Trigonometry - Geometry is cool, but we're going to spend a year talking about just Triangles but secretly it's about Circles. Calculus - Turns out rates of change are related to measured values are related to total accumulation is related to rates of change. Hope you really learned that Trig stuff. Linear Algebra - Box of Numbers Means Everything Differential Equations - How well did you understand Calculus? Statistics - Look at this bell curve for a full semester Set Theory - You will understand Set Theory so well that it will hinder your understanding of everything else forever. Real Analysis - Nobody really knows
The straw answer was confusing. Topologically, the straw clearly has ONE hole, like a bagel. And a bottle has NO holes. Think about it: A bottle is basically just a deformed bowl, and a bowl is just a plate with an higher edge. A plate has no holes.
There are people who view a straw, rather than as continuous surface, as a set of two spaces: an inside and an outside. The argument for a straw having two holes is that there are two clear connection points between the spaces, at the top and at the bottom. Depends on if you view the straw as a topographical surface or as a household object.
If you dig a hole in the ground, you would call it a hole. Does it go through the Earth? No, but we still call it a hole. Maybe we need better definitions of what is a hole that goes through an object vs a hole that is subtracted volume.
@@k-herseyperhaps more precisely, one can view a straw as a 2-manifold-with-boundary, and the boundary consists of two disjoint circles. those two circles constitute the holes of the straw. however, i do agree that it was confusingly worded; in his effort to avoid jargon, he ended up watering down the discussion and making it seem less certain than it is.
@@averynicebean I agree, a clear definition would help. The definition in everyday live is not rigorous. It will heavily depend on the actual shape of the subtracted volume, not the total amount, what we call a hole. No one looks at a valley and says "That's a hole!"
Literally started reading "How Not to Be Wrong- The Power of Mathematical Thinking" 2 days ago and this is the first thing that popped up when I Googled him. Highly recommend the book!
When you pinch one end of the straw it has 0 holes. A hole is only when you go in one end and come out of another. Otherwise it's a dent. So a(n unpinched) straw has one hole, like a bagle, but unlike a water bottle that's not broken, that has a dent out of which you can drink. You also wouldn't say a bowl has a hole (although it sounds good).
A slight variant of the Pythagorean theorem is very useful in the real world: for a triangle, a^2+b^2=c^2 precisely when the angle opposite c is 90 degrees. This can be used, for example, when pouring house foundations, to ensure the corners are (very close to) right angles. It translates the accuracy of length measurements to accuracy of angle measurement.
@@NandrewNordrew Usually I see Pythagorean Theorem presented as "For a right triangle: a^2 +b^2 = c^2", he seems to be saying "If a^2 + b^2 = c^2 then you know the angle opposite c is 90 degrees", which is a slight variation.
I always had a much easier time with geometry than algebra. At least with geometry I could get a mental picture of what I was trying to do, whereas algebra was just letters on a piece of paper. Of course, I still didn't do very well in geometry because I wasn't that good with the mathematics portion, but at least I knew when I got the wrong answer even if I wasn't sure why!
For me it was quite the opposite. In algebra I was always top of my class but then we moved on to geometry. I sucked at geometry because I don't have the "mental picture" that all the other kids claimed to have. When doing algebra I just had to look at the equation and I would be able to write the correct answer almost immediately. Geometry wasn't like that though.
Pythagoras' theorem is incredibly useful when you are trying to make right-angle triangles. Since you generally want a house to have walls at right angles to each other, you can achieve that by just building a decently sized triangle that you can place into the corners. Apparently, not every mason knows this considering the ones that built our house screwed up and built the wall of angle to each other.
There is one hole at regular straw. If other side is plugged then there are no holes since if you start cut the straw shorter you end up plane. Also you can tie a string throught the hole of regular straw, but not plugged one.
If you've ever been walking down the side of an empty street, and you jaywalked diagonally to the other side instead of going straight across and down because it made for less walking overall to your destination... guess what, you used the Pythagorean theorem
no, you used the triangle inequality theorem if you walked diagonally for less walking distance. you would use the pythagorean theorem if you wanna know how much distance you walked and would have walked
Pythagoras is very handy for figuring out neat ways to build Lego in an interesting angle and still keep to the required strict dimensons of a piece. The recent Tranquil Garden set uses this to place the supports of a building five studs apart.
If you imagine time as the fourth dimension that works to my brain. The cube exists now and in a second and in two seconds. You can kind of imagine it moving through time.
given all the other better UA-cam mathematecians, this person is now below average. 5 yrs ago, this person would be the best math teacher I would have come across.
2:50 How to use the Pythagorean theorem to solve a problem in your life? Simple: Build something. A shed maybe. Then put a corner beam as support. How long does that corner beam need to be? sqrt(a^2+b^2), where a and b are the 2 side it's connecting to. If your problem isn't solved by a shed, maybe you can build something else that might be able to use some kind of corner bracing.
11:31 - My German brain perked up when he pronounced Einstein the German way. For a second, it felt unreal; I had expected the American "ine stine" variant and had to rewind to check if I was bullshitting myself. Strange how that works.
The straw has one hole by definition of topology. There is no wiggle room here. The bottle example is really interesting and if you used it right it would have proofed the point. The bottle has ZERO holes (when you remove the cap). It is the same if punch a "hole" in a baloon. what you get is a surface with STILL zero holes. The bottle can be put flat with the rim of the opening to become the OUTER edge of a disc. So when you punch a hole in the bottom of the bottle you add ONE hole - and this bottle has ONE hole - as the straw has. Fun side view: A trouser has TWO holes. If you remove the height of the trousers to "zero" you have the two holes of the legs and the upper part of the trousers become the outer edge of a disc. Topology is precise in these definitions. I recomend checking out Matt Parker on this topic.
"by definition of topology" that's the problem. Even though when talking about topology a straw undeniably has 1 hole, people aren't usually talking about topology when talking about straws. for example, if you asked most people how many holes a bottle has, they would probably say one.
Enlightening! Someone recently asked how many holes a T-shirt has, and I said 6. My reasoning was: 1 - neck to waist 2 - neck to left arm 3 - neck to right arm 4 - waist to left arm 5 - waist to right arm 6 - left arm to right arm Thus, the number of holes would be the number of openings (n): (n-1) + (n-2) ... + (n-n) But with your stretching out revelation, I can see that it is openings (n): n-1. A T-shirt has 3 holes.
Pythagoras's theorem manifests most simply in how we navigate. You can see it in action when you have a block that's an open field and most people, rather than walking two sides of the block, cut diagonally across it to meet the other side, because we intuitively know that is a shorter distance. And if you extrapolate that in wider space, what you realise is that we intuitively recognise that walking a series of hypotenuses (i.e. radial lines plotted against imaginary right angles) is actually the most efficient way to navigate, which is why, as much as rectangular city blocks seem like the most efficient use of space for building, neat squares and right angles are actually very unnatural to us and a radial city plan is the most efficient for travel.
I live in eastern Europe, I had a friend over, and he asked how many inches big my new monitor is. I could not remember it, but then I remembered the Pythagorean theorem, and that 1 inch is roughly about 2.5 cm-s. So I took my measuring tape, measured the sides, did the quick math, and could tell him it's 27". Could I have just measured the distance across? Yes Would that have been fun? No
Hypercubes are used "for real" in many applications. In computer graphics we have chips that connect to four other chips...so if you connect up a lot of these chips to get more and more performance - you place them at the vertices of a hypercube.
I am an engineer, and yes I use the pythagorean theroem to make the odd calculation but I use it qute more often in a much simpler form: If want to reach the other corner on a block and you may have 2 options: 1. Walk the sides of the square (block), or if possible 2. Walk straight diagonally If you call the sides of the square a, b and the diagonal c you can do some pretty simple math to prove that (a+b)^2 > a^2+b^2, or (a+b)^2 > c^2, or that if you have the chance to take the diagonal, you can walk less, saving time.
I'm still proud of the time I used the Pythagoran theorem to drill a hole through a wall, it was for a fiberoptics duct. My co-worker just said sure go ahead, thinking it was just a waste of time. So I did my measurements of the wall (thickness and drop to the target), did the math and marked of a point on drill (the drill was the hypotenuse) and then used a ruler to measure the distance of that point to the wall so that I got the right angle. The triangle that I created with the drill and ruler was a smaller version of the triangle that the desired path was taking through the wall. I nailed the target exactly, my co-worker just looked at me and said that we'll be using my method going forward 😅
Loved this episode! I didn't take geometry in high school; Ellenberg's knowledge, insight and enthusiasm make me want to take an online course to see what I missed.
Great video!! Two points… 1. Isn’t your bright green “square” really just a very flat rectangle? 😃 2. Would love to have seen a discussion about Penrose Rectangles.
As an artist, I never realized how much geometry I would have to incorporate into composition and form, it has to be proportional to real life, the fibbonaci fuckin square is a game changer
3:27 An example is if you wanted to "square" a square or rectangle, think laying out the foundation of a building, you can use whats called the "3-4-5 rule" which is an implementation of the Pythagorean theorem. Probably not something everyone does in real life but it's one example that can be applied to "squaring" a rectangle or square.
Another way to think of the straw is it has no holes, if you think of holes as something that isn't naturally there. A straw is a particular shape that allows you to use it for drinking water or other things. If it had a hole in it, it would leak out the sides. Pants can be thought of as not having holes unless they are ripped. Or having holes if you think of them as the place you put your legs.
In topology, whether a hole is natural or not is irrelevant. A straw has 1 hole, which is how it transmits the liquid. If you try to drink through something with no holes, like a baseball, you would fail.
The answer to the straw problem is it is no longer a straw if pinched and a bottle is no longer a bottle with a hole in the bottom. And thank you for mentioning honeycombs are actually circular when created. They settle into hexagonal shapes because of how tightly the bees pack them in and how flexible the material is initially.
A straw with a pinched bottom has completely lost it's function. Is it still a straw if it can't do what a straw is supposed to do? At this point, it becomes a philosophical question. Another interesting question regarding this: of you hang up a spinning disk and the shadow of the disk is exactly under it, is the shadow also spinning or is it stationary?
5:09 The straw thing just depends on your definition of a hole. If you define it as an opening through something, like a hole in your shirt, then a straw has only one hole. If it doesn't need to perforate, if a pit in the ground is a hole, then a straw has two holes. This probably rifts off the word 'hole' having confusingly similar but strictly different meanings in language and in maths.
I disagree. According to the dictionary, a hole is 'an opening through something" or "an area where something is missing". Since a straw is simply a tube (a bent plane whose edges meet), the continuous surface of it has no openings or areas where anything is missing. If it did, you wouldn't be able to create a vacuum. Therefore it has zero holes.
I used Pythagoras in the army. It made going through bncoc incredibly easy because i didnt have to manually gind the distance between points. I had it exact everytime.
During a debate with a debunker, a flat earther was asked, "If a triangle has sides 1, 1, and 1, what are its angles?" The flat earther said, "One what?"
I have a use for the pythagorean theorem in real life application. I’m told a TV’s screen size always as side C and I know it is a 16:9 aspect ratio. I can find the height and width of that screen when the site doesn’t list the dimensions.
@@chesterotontop unless you're an engineer, architect, scientist, mathematician, programmer, city planner, game developer, digital artist, etc etc etc.
Didn't expect to see AustinJohnPlays here but cool to see!
@@RunstarHomer He’s being sarcastic
Well, you'll be able to find out the height and width of the LCD panel; but that screen size doesn't include the plastic frame around the LCD panel. So it can be useful for comparing monitors to one another, but not for knowing exactly whether or not the monitor/tv can fit in a given area. It wouldn't be a bad ballpark for it, though.
@@darkfoxfurre 1/t = screen border where t is time because as time goes on, the border decreases XD
I highly recommend the essay “A Mathematician’s Lament” for anyone who wants to go deeply into the way we teach math and how poorly it’s taught that most students find math boring and frustrating in most math classes (I know mine classes were definitely not taught well). Jordan has the energy and love of mathematics that would make him an excellent teacher, and I wish I had someone like him while I was crying over my algebra 2 homework.
OMG THANK YOU SO MUCH FOR THIS RECOMMENDATION....One page in and I absolutely love this premise. It's so perfect.
I hated math in school with the exception of geometry in 10th grade. That was a blast. But algebra was always a nightmare. Then I signed up for an algebra class in college with a specific math teacher everyone recommended. She taught math on colorful handouts and in true layman’s terms to where it all connected and made sense. It was like learning math where every lesson is “explain to me like I’m 5”, and her method of teaching was extremely effective and fun! So many students needed a total refresh of some basic math concepts just because of how poorly they were taught in the public school system, and she helped so many students, including me, to be unafraid of math. I wish there were more teachers like her around.
Same goes for language and literature tbh. People tried to make language more interesting to learn by including pop culture and popular fiction, but it still doesnt capture the incredible nuance and thought/philosophy that goes into writing, speaking, and communication in general.
@@nessamillikan6247do you perhaps remember the name of said teacher? I'd love to check out more about her
I loved math and science, I hate my psrents for hitting me for not answering correctly, but I always knew it was them, I've always loved math.
I'm a fan of this series, but Jordan was a particularly strong communicator. Thank you for bringing him on, and thank you to Jordan for being a fantastic ambassador for geometry and math writ large.
To be fair, he did miss the opportunity to proclaim that hexagons are the bestagons
thats my uncle lmao
Why did you use “but” as though you were going to say something bad?
@lucasm4299 because they understand language better than you 😊.
"But" can be used to contrast with (e.g. I'm a fan, but this was trash), or to add to (I'm a fan, but this was excellent).
@@lucasm4299 'But' is used to contrast or add something. I have to agree with Alan, they used the word because they have a better understanding of English. 'But' is very commonly used to mean 'as an aside', 'also', or 'additionally'.
'This was a wonderful meal, but I have to say the chicken was particularly excellent!'
As we all know, hexagons are the bestagons, but it was nice to hear an explanation about it being incidental in the case of hive cells. Never heard that before in explanations of the subject.
actually, triangles are the divine shape
Don't make me call RCE. lol
Jordan gave a bad explanation here though. What is special about the regular hexagon is that among all regular polygons (i.e. whose sides are all the same length and the angles between adjacent sides are the same), it's the one with the most sides, such that you can fill a plane with them without gaps. So this uniquely satisfies the goals of maximizing the space for larvae with round cross-section, packing as many compartments into a given space, and minimizing material (wax) cost while having uniform wall strength. No other possible shape is as good as that. You can build a honeycomb out of regular triangles of squares and you'll fill the space with compartments and maintain uniform wall thickness, but it's a bad use of space because you need to make the triangle or squares rather big to fit the round-cross section larvae; if you take regular polygons with 7, 8, 9, or any large number of sides, you will leave unused gaps or waste wax.
I think is simpler than that, you can really only put around 6 circles of the same diameter as the central one. In any circle diameter that happens.
And what happens when you smash those sides equally together? You create an hexagon. Just check a box of straws and you will see that happen.
Alas you can create a honeycomb filter for a light (that focus light) with straws inside a tube
@@PauloSousa86100% this. Hexagon is the byproduct of the design, not the design itself.
I think color pencil is another good example, by holding 7 of them together, 6 will be surrounding 1 of them regardless of the pencil shape.
One things people also tend to miss is that most of those insects(bees and wasps) uses a sort of liquid mixture of saliva and other stuff to build their hive. So by starting with liquid, what making the hexagon shape could have something to do with surface tension where it started with circle, and the surface of those circles then stick to one another stretching themselves out before hardening into hexagonal shape.
If somehow we can drop 7 honey droplets on a flat surface in a rubber band at the same time, with all of them have different dyed, we might be able to see those honey spread out to be shaped like hexagon, considering if the droplets is equal in size and have the same consistency.
As a quilter I use the Pythagorean theorem to figure out how many triangles I can get out of my fabric and how big to measure them. Once I had a pattern for a skirt that wanted right triangles of a certain length on the "c" side so I used it to calculate the "a" and "b" sides
As someone who's played around trying to program a pool game, balls in the game have known X and Y coordinates, I've used the Pythagorean theorem to find the distance between balls to check when the balls hit each other.
I once had to use the Pythagorean theorem as a web developer to calculate the size of a resizable widget when you clicked and dragged the corner! I was like "huh I guess knowing that actually was useful after all"
I came here to say I use the Pythagorean theorem for sewing too! For me it's to make zero waste flaired skirts. 💃🏻
I use it for plastic wrap 😂 the wrap is normally always the c, so if you go all the way down to a, it will always be enough wrap
Contractors can also use it to make sure a corner is actually 90 degrees. Measure 3 going one way, 4 going the other, and adjust the corner until the hypotenuse is 5.
I love how unapologetic Jordan is about drawing crappy circles! 😂
On a more serious note, I was impressed by how well you pronounce the German names (Einstein and Möbius) in such a casual manner.
As a regular Dungeons & Dragons DM, I have sometimes used the Pythagorean theorem to calculate the distance of flying creatures moving diagonally to the ground to attack players. I'm just glad online calculators exist so I don't have to do the math myself. 🤣
This is precisely the only way I've used the theorem in the last 25 years hahahahaha. And quite often, I must say.
saaame lol. til the dm reminds me that diagonals technically don't exist in dnd lol
I created a Collatz Dungeon for a party that was testing the Constable's patience. They would get dropped in Room 3,505,346, and they would be connected to two others, one double the number, and the other half as much. Eventually, they would hit an odd-numbered room, n, which would connect to Room '3n + 1'.
All numbers, eventually, will connect to Room 1, where the exit would be.
Its very helpfull in vidéo games too ! I use it all the time to calculate distances between two objects in a plane in small personnal game projects :)
Every object has x and y coordonnates, calculating the distance between the two is one of the most important things in a game. For détection, colisions etc... and Pythagore is always used.
And that's why the rules say not to worry about diagonals for flying creatures
I think the issue with the "Does a straw have one hole or two?" question is that everyone treats it as a geometry problem when it's more of a language problem.
Wittgenstein says hello!
I mean, more of a topology question than geometry.
I feel the same way about the question of “are we living in a simulation?”
Y it s a question of définition
In maths (topology) the straw has one hole cause it s fondamental group is isomorphic to Z
@@hugomondoloni9808Because literally everything you just said uses language. It's a language game. Every single person has different definitions of when it is 2 or 1 hole. Math is still predicated on language.
6:58 The A paper sizes (A4, A3 etc) have a similar property, but it uses sqrt(2) instead of the golden ratio. When you fold it in half the ratio between the long and short side remains sqrt(2).
As a math teacher, this brings back memories of my college geometry and math history courses! Love it! It’s awesome to see somebody love their profession so much! 😊
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The Pythagorean theorum has a lot of real world applications in architecture. For example, it's useful for designing staircases, since if you know the height of the upper floor, you can calculate the length of the staircase for any given footprint.
I used it recently to calculate the bill of materials on the roof of a shed I was building. Of all the mathematical / geometrical rules, this one is one of the more applicable ones to the real world... of course, if you google 'Trigonometry calculator', that's even more precise, and takes away the actual need to do the math...
@@FHL-DevilsI did something similar to turn the old, flat, but too short driveway into a new, longer driveway that would have a steep slope. Needed to make sure the rise on the slope wouldn't scrape the car
i was able to use phytagorean theorem on how much we need to extend our roofing for us not to have side-hitting rain hit our wall (which can weaken the concrete overtime due to accumulation of moisture). i was actually surprised when the calculations worked!! i felt like a wizard
TV screens are all measured in the diagonal dimension. So if you have a space on your wall that is X inches wide, you can use the Pythagorean theorem (and the fact that most TVs have a 16:9 aspect ratio) to determine the largest screen size you can put there.
I qm an engineer and i use it all the time
Honey combs is 100% a packing efficiency problem. If you take any circular object, beer bottle, golf ball, whatever. Any circle, and more circles of the same size. You can wrap 6 more circles around the original.
That’s wrapping around to make another circle. So yea. But there’s still negative space that’s not being utilized. With strait lines you can take away that negative space. Hence Tetris etc.
For sure. I was a bit disappointed at his glib and not particularly knowledgeable or intuitive answer. Makes me wonder about some of his other answers.
Hexagons are bestagons
It's not only packing efficiency, but also structural support and area. Squares tile just as well as Hexagons do, and they use less sides to do it, but they aren't as strong. Triangles tile just as easily as Hexagons do, and use even fewer sides than squares, but they have a smaller area. Hexagons are the perfect middleground for largest area (for storing more honey), fewest sides (easier to make), and ability to tile.
@@xkinsey3831while that is logically true, nature doesn’t calculate those stuff up.
Bees just make circles, the same size as each others next to each others and those circle edge or surface tension or whatever it is just pull them to make a Hexagonal shape, that’s all. Hence why the original comment has every explanation needed to explain why it’s hexagon, because you can surround a circle with 6 more circles, and those circles surface, just like bubbles, they stick to each others and stretch out to form hexagon.
Imagine this, you have 7 syringes of honey hold together, 6 surrounding 1 of them.
Let’s say you dyed the honey in the middle syringe red. If you squeeze these 7 syringes on a table at the same time to form 7 circles, the honey will slowly spread out and the middle red honey one will spread out to look like, you guest it, hexagon. BOOM, mystery solved.
I used to use Pythagoras to mark out an accurate filed when laying out our clubs field hockey lines at the start of each season. Mark the baseline and then use a 3,4,5 triangle to make 90 degree corners for each sideline.
That’s the one. The Pythagorean theorem’s most useful real world application is to mark out exact right angles when the biggest square you have is still far too small: you can do it with a tape measure.
A straw with one end closed, or a bottle, is just a disc with bent up sides, and NO holes.
I just watched a 17 minute video about math of all things, and was entirely entertained by the presenter. Incredible.
I feel that now I have a greater respect for straws. Thank you Mr. Jordan Ellenberg, it is incredible how you manage so much information to filter it into something simple, direct and with great humor
Love his enthusiasm for math and geometry!
This dude needs his own UA-cam channel where he teaches math. So much more charismatic than any teacher I ever had.
I had to give this a watch. I just used the Pythagorean theorem about two minutes ago. Creating miters for a picture frame and I needs to determine what the third side is going to be!
I’ve never liked math but I love this man’s enthusiasm.
Pythagorean theorem is really handy for figuring out distances in D&D where all battles are on a grid
Nice one lol I'll be using that now
This is like the sixth comment about D&D i'm reading, amazing
On the Pythagorean theorem : when I was a little boy, on my usual path to school, I had to around two sides of a square, as to not walk on a bit of lawn. I wondered how much distance I would spare every day if I just crossed that lawn across a diagonal. Well... One day I learned how to get that answer.
You just have to be curious and you will need math in your everyday life.
When he mentioned the super hero movie not inventing the tesseract, I angry-scrolled to make sure "A Wrinkle in Time" was mentioned, just as he said it.
6:10 if you pinch the bottom, it has zero holes. A bowl or a plate don't have a hole, and an open-topped bottle is the same shape as a bowl or a plate.
is a bottle a bowl?
@@theastuteangler sure, seems a reasonable grouping
A balloon doesn't have a hole either?
@@fulanoide718 correct, it does not. A balloon that is not tied is a deformed disc. Holes cannot be created nor destroyed merely by stretching a shape. Balloons are just bowls with small necks.
@@marshallc6215
You should definitely look into topology, it's basically the study of these "groupings"
"[Geometry] is the only part of math where you're asked to prove something..."
Number theorists: "Am I a joke to you?"
*war flashbacks to Abstract Algebra*
(To be clear, it's fun, but hard)
I'm pretty sure that proofs are common in exercises and tests for any undergrad level math courses lol
I like to define a “Hole” as a region of Entrance, and Passage. For the purpose of my point an Entrance acts as a doorway you can enter and exit via a “doorway”, A passage is always accessed via an Entrance or Entrances. The straw and closed straw both have 1 hole Because both has At least 1 Entrance and a Passage via an Entrance. And a donut also has one hole via my definition.
Fun fact: the four dimensional tesseract was the central plot feature of Robert Heinlein's short story 'And He Built a Crooked House' published in 1941, twenty one years before 'A Wrinkle in Time' came out. Though I loved a Wrinkle in Time, Heinlein did a far better job describing it.
I just finished Stranger in a Strange Land, and his description of Mike sending things Away was so good! I'll have to check out that short story sometime soon!
Agree. And this wasn't the only suboptimal description in this video.
11:31 that was a PERFECT "Einstein"!
What's fun about this guy is he's clearly talking to the people in the room, not necessarily to the camera. Looks like they were eating it up.
I absolutely love the "how many holes are there in a straw" question. As someone who routinely does CAD as a design engineer, my perspective is in the way I'd create it (two steps):
1) Create a cylinder.
2) Cut extrude a hole through the entire body.
Done. One hole.
Math finance PhD student here, just a comment about the random walk question. The Bachelier model in finance is a terrible model for stock movements and this was known at the time they published their model. A better model nowadays is models of the form e^(X(t)) where X(t) is some stochastic process (see something like geometric Brownian motion, the vasicek model, or more exotic models like the Heston model or general jump diffusion models). I bring up this detail because people get really silly and paranoid with stocks and it's important to note that these modeling problems are remarkably complex and nuanced. They require much more than just a random walk to be useful.
5:52 You are changing the goalposts. If you pinch the bottom of the straw, it is no longer the straw as we understand it to be in your question, or what we commonly recognize to be a straw.
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As an Army Sniper I used to do a brief/lecture called "How the Pythagorean Theorem Saved My Life." We use it in ballistics.
Send me the power point cuh
Yea aight
That was the most clear and succinct explanation of gerrymandering I have ever heard. Incredible
As someone who plays a lot of D&D we use the Pythagorean theorem all the time to figure out spell distances with flying creatures lol..
5:24 the straw theory makes my brain short circuit!!
Took me almost 10min to realise I own on of this guys' books. "How not to be wrong". Great read.
Such a great episode. You should film a few more with this guy!
The Pythagorean theorem is good for calculating straight distances on a map with grid lines. You count how many vertical and horizontal lines you're crossing and then use Pythagoras to calculate the distance.
In defense of the Pythagorean Theorem, if you wan to build anything, like a house or a cornhole game, you need to use this rule.
5:04 There is one hole on the straw. When you cover the bottom, then the straw has no holes (a water bottle can be deformed into a bowl or a plate, for example)
What is the fundamental group of the straw?
if you bend a straw too much, then it will have more holes and you won't be able to use it.
@@xraygamer9895Z. It's either homeomorphic to a solid torus (if you assume it has width) or a cylinder (assuming no width). Either case the fundamental group is Z. It could still have higher dimensional holes but the homology groups are of course all trivial except for dim 0 and 1.
That's a lie it still has one hole in the top. A hole doesn't have to go completely through an object, if you have a hole punched into your car door on the outside, but not the inside you still have a hole in your car door
@@RCG3. I'm not sure if I understood, but the car door has many layers and if you count each layer as an object you can have a hole that doesn't go through the whole door. The equivalent to the straw example would be a dent on the car.
Arithmetic - Here's a number line
Algebra - The formal logical rules and language of math
Geometry - Can you draw it in some logical way and then deconstruct that drawing into an algebraic expression?
Trigonometry - Geometry is cool, but we're going to spend a year talking about just Triangles but secretly it's about Circles.
Calculus - Turns out rates of change are related to measured values are related to total accumulation is related to rates of change. Hope you really learned that Trig stuff.
Linear Algebra - Box of Numbers Means Everything
Differential Equations - How well did you understand Calculus?
Statistics - Look at this bell curve for a full semester
Set Theory - You will understand Set Theory so well that it will hinder your understanding of everything else forever.
Real Analysis - Nobody really knows
The straw answer was confusing. Topologically, the straw clearly has ONE hole, like a bagel. And a bottle has NO holes. Think about it: A bottle is basically just a deformed bowl, and a bowl is just a plate with an higher edge. A plate has no holes.
There are people who view a straw, rather than as continuous surface, as a set of two spaces: an inside and an outside. The argument for a straw having two holes is that there are two clear connection points between the spaces, at the top and at the bottom. Depends on if you view the straw as a topographical surface or as a household object.
wait you might be right
If you dig a hole in the ground, you would call it a hole. Does it go through the Earth? No, but we still call it a hole. Maybe we need better definitions of what is a hole that goes through an object vs a hole that is subtracted volume.
@@k-herseyperhaps more precisely, one can view a straw as a 2-manifold-with-boundary, and the boundary consists of two disjoint circles. those two circles constitute the holes of the straw. however, i do agree that it was confusingly worded; in his effort to avoid jargon, he ended up watering down the discussion and making it seem less certain than it is.
@@averynicebean I agree, a clear definition would help. The definition in everyday live is not rigorous. It will heavily depend on the actual shape of the subtracted volume, not the total amount, what we call a hole. No one looks at a valley and says "That's a hole!"
3:29 the absolute mathematician sheldonesque sarcasm makes it for me ❤
Literally started reading "How Not to Be Wrong- The Power of Mathematical Thinking" 2 days ago and this is the first thing that popped up when I Googled him. Highly recommend the book!
When you pinch one end of the straw it has 0 holes. A hole is only when you go in one end and come out of another. Otherwise it's a dent.
So a(n unpinched) straw has one hole, like a bagle, but unlike a water bottle that's not broken, that has a dent out of which you can drink. You also wouldn't say a bowl has a hole (although it sounds good).
A slight variant of the Pythagorean theorem is very useful in the real world: for a triangle, a^2+b^2=c^2 precisely when the angle opposite c is 90 degrees. This can be used, for example, when pouring house foundations, to ensure the corners are (very close to) right angles. It translates the accuracy of length measurements to accuracy of angle measurement.
Thats exactly what the normal theorem is
@@NandrewNordrew Usually I see Pythagorean Theorem presented as "For a right triangle: a^2 +b^2 = c^2", he seems to be saying "If a^2 + b^2 = c^2 then you know the angle opposite c is 90 degrees", which is a slight variation.
wired messed up not giving this poor mathematician his chalk and board 😭
on a serious note, what delightful communication skills this guy has
I'm a quilter. I use the Pythagorean theorem almost every day
You also are an expert on knot theory, which is much deeper math than the Pythagorean Theorem.
The way I think about the number of holes in the straw: Shrink the straw down to a two-dimensional circle. One hole.
I always had a much easier time with geometry than algebra. At least with geometry I could get a mental picture of what I was trying to do, whereas algebra was just letters on a piece of paper. Of course, I still didn't do very well in geometry because I wasn't that good with the mathematics portion, but at least I knew when I got the wrong answer even if I wasn't sure why!
For me it was quite the opposite. In algebra I was always top of my class but then we moved on to geometry. I sucked at geometry because I don't have the "mental picture" that all the other kids claimed to have. When doing algebra I just had to look at the equation and I would be able to write the correct answer almost immediately. Geometry wasn't like that though.
I loved your straw answer! It shows that there are multiple ways to look at anything.
This was really good he made geometry sound pretty dope
Pythagoras' theorem is incredibly useful when you are trying to make right-angle triangles. Since you generally want a house to have walls at right angles to each other, you can achieve that by just building a decently sized triangle that you can place into the corners. Apparently, not every mason knows this considering the ones that built our house screwed up and built the wall of angle to each other.
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"Imagine someone with no sense of purpose."
Me: Of course I know him, he's me
There is one hole at regular straw. If other side is plugged then there are no holes since if you start cut the straw shorter you end up plane. Also you can tie a string throught the hole of regular straw, but not plugged one.
If you've ever been walking down the side of an empty street, and you jaywalked diagonally to the other side instead of going straight across and down because it made for less walking overall to your destination... guess what, you used the Pythagorean theorem
No you didn’t. You just walked across the street. You didn’t use any theorem at all.
not really. you're just using the fact that the shortest distance between two points is a straight line (in euclidean space)
@@ttmfndng201 phblttbtt Euclidean who? You'll never catch me using THAT daily :P
bro is onto nothing😭
no, you used the triangle inequality theorem if you walked diagonally for less walking distance. you would use the pythagorean theorem if you wanna know how much distance you walked and would have walked
3:06 "oh man the wife threw me out again"
a few calculations later: a squared plus b squared, carry the one
"oh shoot I forgot our anniversary again"
Pythagoras is very handy for figuring out neat ways to build Lego in an interesting angle and still keep to the required strict dimensons of a piece. The recent Tranquil Garden set uses this to place the supports of a building five studs apart.
pythagorean theorem is real clutch in d&d when you have to calculate distance with an enemy who's in the air.
The straw hole one is crazy. My answer was two holes tho. Also I wish I could understand shapes in a 4 dimension. It makes no sense to me.
If you imagine time as the fourth dimension that works to my brain. The cube exists now and in a second and in two seconds. You can kind of imagine it moving through time.
@@travelsandbooks I still don't understand it :(
How I think about shapes in 4 dimensions is by thinking about shapes in 3 dimensions, and hoping that similar reasoning carry over.
First time I've understood a hypercube!! Excellent presenter.
Why hexagons? Why hexagons??? Well, because hexagon is the bestagon!
The other ones just go on and on and onagon.
given all the other better UA-cam mathematecians, this person is now below average. 5 yrs ago, this person would be the best math teacher I would have come across.
Pretty sure he is the first non-german Person, that i've ever heard to pronounce the name "Einstein" 100% correctly. Neat!
"Ein-shtein"?
@@rebeccamcnutt5142 Yes.
2:50 How to use the Pythagorean theorem to solve a problem in your life?
Simple: Build something. A shed maybe. Then put a corner beam as support. How long does that corner beam need to be? sqrt(a^2+b^2), where a and b are the 2 side it's connecting to.
If your problem isn't solved by a shed, maybe you can build something else that might be able to use some kind of corner bracing.
This was a good one!! He's an excellent communicator and super engaging! Loved this ❤️😊
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@@redredred8408 pi-ish?
The Arithmetic of Holes sounds like something Lisa Ann would star in.
11:31 - My German brain perked up when he pronounced Einstein the German way. For a second, it felt unreal; I had expected the American "ine stine" variant and had to rewind to check if I was bullshitting myself.
Strange how that works.
haha totally the same. I had to check the comments to see if somebody else noticed it. Refreshing pronunciation. I hope it was intentional :D
2:33 is so real, I'm horrible at everything above basic geometry but really good at seeing the solutions for algebra
The straw has one hole by definition of topology. There is no wiggle room here.
The bottle example is really interesting and if you used it right it would have proofed the point. The bottle has ZERO holes (when you remove the cap). It is the same if punch a "hole" in a baloon. what you get is a surface with STILL zero holes. The bottle can be put flat with the rim of the opening to become the OUTER edge of a disc. So when you punch a hole in the bottom of the bottle you add ONE hole - and this bottle has ONE hole - as the straw has.
Fun side view: A trouser has TWO holes. If you remove the height of the trousers to "zero" you have the two holes of the legs and the upper part of the trousers become the outer edge of a disc. Topology is precise in these definitions. I recomend checking out Matt Parker on this topic.
"by definition of topology" that's the problem.
Even though when talking about topology a straw undeniably has 1 hole, people aren't usually talking about topology when talking about straws.
for example, if you asked most people how many holes a bottle has, they would probably say one.
Enlightening! Someone recently asked how many holes a T-shirt has, and I said 6. My reasoning was:
1 - neck to waist
2 - neck to left arm
3 - neck to right arm
4 - waist to left arm
5 - waist to right arm
6 - left arm to right arm
Thus, the number of holes would be the number of openings (n):
(n-1) + (n-2) ... + (n-n)
But with your stretching out revelation, I can see that it is openings (n):
n-1. A T-shirt has 3 holes.
Pythagoras's theorem manifests most simply in how we navigate. You can see it in action when you have a block that's an open field and most people, rather than walking two sides of the block, cut diagonally across it to meet the other side, because we intuitively know that is a shorter distance. And if you extrapolate that in wider space, what you realise is that we intuitively recognise that walking a series of hypotenuses (i.e. radial lines plotted against imaginary right angles) is actually the most efficient way to navigate, which is why, as much as rectangular city blocks seem like the most efficient use of space for building, neat squares and right angles are actually very unnatural to us and a radial city plan is the most efficient for travel.
I live in eastern Europe, I had a friend over, and he asked how many inches big my new monitor is. I could not remember it, but then I remembered the Pythagorean theorem, and that 1 inch is roughly about 2.5 cm-s.
So I took my measuring tape, measured the sides, did the quick math, and could tell him it's 27".
Could I have just measured the distance across? Yes
Would that have been fun? No
Hypercubes are used "for real" in many applications. In computer graphics we have chips that connect to four other chips...so if you connect up a lot of these chips to get more and more performance - you place them at the vertices of a hypercube.
One hole, two openings.
A donut has 2 openings?
New application for Pascal's triangle learned. Cool. Only one I knew was coefficients for binomial exponential expansion
6:35 I find it pretty funny that we call it the golden "ratio" despite the fact that it is, almost by definition, *irrational*.
pi is also said as a ratio too (unless im missing something)
I am an engineer, and yes I use the pythagorean theroem to make the odd calculation but I use it qute more often in a much simpler form:
If want to reach the other corner on a block and you may have 2 options:
1. Walk the sides of the square (block), or if possible
2. Walk straight diagonally
If you call the sides of the square a, b and the diagonal c you can do some pretty simple math to prove that (a+b)^2 > a^2+b^2, or (a+b)^2 > c^2, or that if you have the chance to take the diagonal, you can walk less, saving time.
A straw has 0 holes, its just a warped plane
a warped plane that formed a hole by definition
The dictionary says a hole is a small and unpleasant place 😂😂
@@Bangin0utWest 🤣🤣🤣
@@arablues4142 so a straw IS a hole 😂😂
A straw isnt a plane though. It is in 3 dimensions.
Pythagorean theorum is used to build houses.
3,4,5 triangles. I don't want a house that's not built with them.
I actually used the pythagorean theorem in real life! We built a shed in my backyard. Did this for each corner paver to make sure they were straight.
I'm still proud of the time I used the Pythagoran theorem to drill a hole through a wall, it was for a fiberoptics duct.
My co-worker just said sure go ahead, thinking it was just a waste of time.
So I did my measurements of the wall (thickness and drop to the target), did the math and marked of a point on drill (the drill was the hypotenuse) and then used a ruler to measure the distance of that point to the wall so that I got the right angle.
The triangle that I created with the drill and ruler was a smaller version of the triangle that the desired path was taking through the wall.
I nailed the target exactly, my co-worker just looked at me and said that we'll be using my method going forward 😅
Loved this episode! I didn't take geometry in high school; Ellenberg's knowledge, insight and enthusiasm make me want to take an online course to see what I missed.
Yay geometry!! The only math class that made sense to me!!
How about differential equations???
Great video!! Two points…
1. Isn’t your bright green “square” really just a very flat rectangle? 😃
2. Would love to have seen a discussion about Penrose Rectangles.
I’ve never heard anyone describe Euclid as “a guy who lived in North Africa” …
maybe Greece was part of north africa?
He was born in Egypt
I have severe discalculia and understood almost nothing but still somehow enjoyed this, thanks Jordan!
Jordan is type of guy to make easy exams and hard homeworks
As an artist, I never realized how much geometry I would have to incorporate into composition and form, it has to be proportional to real life, the fibbonaci fuckin square is a game changer
7:50
Because hexagons are the bestagons
3:27 An example is if you wanted to "square" a square or rectangle, think laying out the foundation of a building, you can use whats called the "3-4-5 rule" which is an implementation of the Pythagorean theorem. Probably not something everyone does in real life but it's one example that can be applied to "squaring" a rectangle or square.
hated doing geometry proofs in high school 😅😢😂
Another way to think of the straw is it has no holes, if you think of holes as something that isn't naturally there. A straw is a particular shape that allows you to use it for drinking water or other things. If it had a hole in it, it would leak out the sides. Pants can be thought of as not having holes unless they are ripped. Or having holes if you think of them as the place you put your legs.
In topology, whether a hole is natural or not is irrelevant. A straw has 1 hole, which is how it transmits the liquid. If you try to drink through something with no holes, like a baseball, you would fail.
The answer to the straw problem is it is no longer a straw if pinched and a bottle is no longer a bottle with a hole in the bottom.
And thank you for mentioning honeycombs are actually circular when created. They settle into hexagonal shapes because of how tightly the bees pack them in and how flexible the material is initially.
A straw with a pinched bottom has completely lost it's function. Is it still a straw if it can't do what a straw is supposed to do? At this point, it becomes a philosophical question.
Another interesting question regarding this: of you hang up a spinning disk and the shadow of the disk is exactly under it, is the shadow also spinning or is it stationary?
I'm in love with this guy and his passion and his demeanor
5:09 The straw thing just depends on your definition of a hole. If you define it as an opening through something, like a hole in your shirt, then a straw has only one hole. If it doesn't need to perforate, if a pit in the ground is a hole, then a straw has two holes.
This probably rifts off the word 'hole' having confusingly similar but strictly different meanings in language and in maths.
I disagree. According to the dictionary, a hole is 'an opening through something" or "an area where something is missing". Since a straw is simply a tube (a bent plane whose edges meet), the continuous surface of it has no openings or areas where anything is missing. If it did, you wouldn't be able to create a vacuum. Therefore it has zero holes.
I used Pythagoras in the army. It made going through bncoc incredibly easy because i didnt have to manually gind the distance between points. I had it exact everytime.
During a debate with a debunker, a flat earther was asked, "If a triangle has sides 1, 1, and 1, what are its angles?" The flat earther said, "One what?"
60° but I’m not understanding the joke or the ppint
@@kvonation8852the triangle cannot exist in Euclidean geometry (flat surface)
@@duckymomo7935 I mean it's just an equilateral triangle?
He actually makes it sound interesting. His passion for it shows.
6:00 How many holes in a bottle? Topologically speaking there are 0 holes.