Being a math teacher in The Netherlands I *always* tell my students about these properties (and make sure they get proven in class either by me or by the students), but only because I like them so much myself and think they're more important than a lot of the crap we otherwise spend time and energy on. These properties are not (anymore) part of the standard curriculum, unfortunately. And yes I agree, the educational system never misses an opportunity to miss an opportunity... which saddens me no end.
I live in the United States (in Wisconsin), and have also sadly watched the decline of the quality of mathematics education over the past forty years. Like @chaosmeister , I also teach these properties (as well as the length of the _latus rectum_ which isn't included in modern textbooks because the kids giggle at the name and sensitive teachers are embarrassed) in algebra 2, but the study of ellipses and hyperbolas is no longer included in the curriculum.
Every topic in every math class seems so diluted as compared to when I was in school, which leaves modern students inadequate for university study. Too many politicians and administrators emphasize students' feelings and self-esteem instead of their education; this has led textbook authors to spend more time writing review material to which students have already been exposed rather than expanding more deeply. They believe keeping students moving along to the next grade level has become more important than actually teaching them, and the result is a cascade of students (who are never allowed to fail) taking mandatory next-level classes for which they are woefully unprepared.
@McOinky ... which is a voluntary subject, chosen by less than 10% (or less?) of all math students. Right... [Edit: "less than 10% (or less)" is incredibly silly coming from a math teacher... sorry 'bout that... End of self-flagellation :-).]
@@mjones207 In my U.S. Pennsylvania school, we are taught the properties about hyperbolas and ellipses in Alg II, but nothing about the latus rectum (lol) or much of the details of parabolas other than converting to vertex form. We spend much more time on the other conic sections.
Your concerns are well founded. I was a math and science teacher at two private schools in the United States (each time for two years). The textbook selections never aligned well with my ideals of making math and science relevant, useful and entertaining. For 3 of the four years, I was able to accomplish that in spite of the books. During the 4th year the academic police state finally caught up with me, insisting that the reasons we teach anything are foremost to increase test scores and grow our market share through test-based reputation. That authority banned all non-standard curricula and forced me out of the profession I loved. All the texts from which I taught were ostensibly aligned with their goals. They were also filled with endless drills, BS examples, incomplete history, and frankly serpentine reasoning far more likely to confuse that to convey any valuable understanding. I’ve lived 4 years in Germany (which was a bit better) and 4 more in South Korea (which seemed much better). Unfortunately, the Korean kids were under immense pressure to perform, making almost all of them profoundly depressed and/or stressed from perhaps 10 years of age. Most of the great contributors to the betterment of our world recognize the importance of conquering fun challenges. Both mindless, meaningless repetition and idiotic complication turn beautifully curious and malleable children into miserable adults. When teaching became primarily the indoctrination of future workers, it necessarily ceased teaching young people to think critically and creatively in favor of teaching them to do what those in authority tell them. This bodes terribly for the future and causes me to be deeply concerned for our posterity. Thank you for making learning the entertaining challenge it’s meant to be!
I can't speak for the rest of my country (I live in the US), but I didn't learn any of this in school. Which is a large part of why I watch mathematical RUclips channels.
State of math affairs in southern Ontario Canada (grade 10, ~age 15/16) is we are taught the quadratic formula, sin law, and cosine law, we were allowed to try and derive it on our own if we wanted to. We weren't taught any of the applications for any of the math except in trigonometry and measuring the heights of hills, and we certainly never got as in depth as many mathematics teachers on the internet, which is part of why I love this channel so much. If I didn't have you, numberphile, Matt Parker, or a brilliant subscription, I'd probably be dragging my feet through the mnemonics like everyone else
I went to a high school for behaviorally challenged students in the United States. I was always an Honor Roll student, but the school didn't teach anything higher than Algebra 2. When I decided to go to college, I only placed into remedial math classes. Now, I'm 1 semester away from an Associates in Computer Science. I took 5 remedial math classes to work my way up to Calculus 1 and 2. Since I've been attending half-time while I work, my 2 year degree has taken me almost 6 years to complete.
Here in Brazil we just learn to memorize everything to be able to do tests. Our teachers even say that we “won’t need this in the future unless we become engineers or mathematicians”, instead of trying to show how mathematics can actually be interesting and beautiful. I’m glad I have the internet, because I’d never get this kind of stuff in high school.
Oh gosh. Now you've done it. Here in BC, Canada, the situation is equally dire. We spend months beating the grade 11's over the head with parabolas until they can hardly remember their own names, let alone any useful or interesting properties. Parabolic trajectories and parabolic bridge arches are as exciting as it gets. There's not any mention of focus and directrix anymore, just 'general form', 'vertex form' and 'factored form', and going back and forth between them ad nauseam. It puzzled me why we force everyone to 'study' 'mathematics' in this way until a fellow teacher explained it to me thus: our education system is really a sifting system whose purpose is to divide people into categories for later life. It's unfortunate that in the process it takes almost all the fun and certainly all of the wonder out of the world. Except for the valiant efforts of some exceptional teachers, the effect would be complete. Don't even get me started on 'rationalizing the denominator' in grade 10 ...
Lmao, the curriculum has gotten so bad that it almost seems some students (like me) know more about parabolas than the teachers. I showed one of my math teachers a proof of the quadratic formula by completing the square (which is where it comes from anyway), and he seemed completely lost. I also showed him other properties such as Vieta's formulas, and many of the the properties mentioned in the video, but he mentioned that he has never seen any of those before. I don't know whether or not these were never taught to him, or he has simply forgotten from never teaching it himself. Whatever the case may be, that was a deeply concerning experience.
In the Basque Country (northen Spain), we learnt a circle is the set of points equidistant to a given point, an ellipse is the set of points with constant distance to two points, and the parabola is the set of points equidistant to a point and a line. However that was just a definition, for problem solving we'd immediately would resort to algebra and solve things in equation-space, never geometrically. So basically, we knew about focus and directrix, but never used it in practice, as far as I can remember.
I live in France, and we never had those type of explanations when we learned about quadratics (and it was like last year) Great video by the way, love what you do!
I suppose it depends if Maths was a “major” and the time. I got a full course on comic sections in high school. Searching on Internet, now you get it later in Maths Preparatory Schools, it seems.
I've had one of these parabolic mirror hologram toys for years. It always causes a wow moment the first time someone experiences it. Good luck on your mission to put the fun and relevancy back into school Math worldwide. Understanding how to apply Math is way more important and enjoyable than just churning through calculations ad nauseam - that's what computers are for.
India : I being a student myself and having prepared for JEE advanced have came across almost all of these properties in our coaching classes but we were never taught the proofs since there were so many properties to mug up there was simply not enough time to do them in class. They didn't even encourage us to do it at home and ask our doubts if we were stuck with it. Luckily I had one great book of geometry by SL Loney which I solved to learn as well as prove some of these proofs myself but even that book had no such beautiful real world examples. It's sad to know what beautiful stuff there is which I missed during my school days and might never ever stumble across.
Interesting that this video would pop up in my recommended today because I just read in my book ( Charles Seife’s Zero) about how an ellipse with one of it’s foci extended to a point at infinity forms a parabola. This is the sort of maths I love. Great video Mathologer!
You're hell lotta lucky you didn't go to an Indian school so shut up and be happy with what you got. Btw, here teachers don't even teach the full syllabus, they are too lazy to do so.
Lol in America, they just kinda skipped from "ok so here's how you know these angles are the same" to "ok so that's why the integral is really just an antiderivative"
@Poo Guy In Germany, the school adminstration has come to the conclusion that learning math (i.e. this formal stuff like algebra, calculus, proofs, logic etc.) is obsolete, because modern calculators can do it. So they redefined "math" to mean "solving real world problems using calculators". Then they realised that this did not work, because problems as they appear in the real world are too complicated. So they replaced the real world problems with invented problems formulated in a very obscure language that makes them look like real world problems for anybody who doesn't understand this language. And then they programmed the calculators so that exactly these problems can be solved by pressing special keys. Now everything works fine. Teaching math has become teaching the special language that is used in official exams and which kind of question requires which command on the calculator to be executed. The students get good marks, because it is not too complicated to associate certain words with certain keys. And if anybody dares to ask questions, you can simply give them the official exam from the previous year. Even a old-school mathematician will not be able to solve it correctly. But modern students can, proving that the school system is much better than ever.
I was schooled in Italy and I didn't imagine the situation in Germany was so bad. In Italy all the educational system is still rooted in the fascist tradition of the cult of classical Rome and Greece. In high school you are forced to learn latin grammar and literature, even if you choose to attend the so called "Liceo Scientifico", which should be focused on science and math, and in some high schools even ancient greek. Really useful for preparing you to understand the world and get a job in the 21st century!
Here is another challenge for who everybody who makes it to the end of the video: If you set up the mirrors like the whisphering dishes, that is far apart and you put Leia at one of the focal points, will you see her hologram appear at the other focal point?
Well, due to the large distance I think whatever made it look like a hologram wouldn't be there as the only light which would come out would be the parallel light making it look like a normal everyday phenomenon.
My guess is: in that case you wouldn't see the object from everywhere (like from the footage you have shown the hologram appeared visible from different points of view outside the "ufo" ), but only if you had your eyes exactly in the focal point. is it correct?
Yes, but only if you were looking in the first mirror in a direction that showed the reflection of the other mirror, a very limited field of view, so your head would be in the way. So that's a no then.
I remember playing a game called Time Traveller by Sega on a school trip sometime in the 80's. It's a laserdisc game with a control system similar to Dragon's Lair and Space Ace. But this employs a very old principal using opposing mirrors to create the illusion of levitation rather than the flat tv screen. The original design from Victorian times had a ball inside which would roll around the inside on the bottom mirror, but would appear to be spinning whilst floating above the hole above the top mirror. Hope this makes sense. Great video as always. McWomble.
math at school in Belgium was always very abscart and something i just had cram into my head for exams, and forgotten a day later. i wish I had better teachers like you. only now in my forties i'm getting interested again.
At school they tell me these mirrors are spherical, not parabolic, and I'm so annoyed by that... I even argued with my teacher about it and she said that we learn the general regarded physics and not some weird new inventions but I can write an essay on it and maybe win a Nobel Prize as well xD In case you want to know, Poland.
RandoM_ 11 even not that really. My teacher insisted that they’re spherical and argued with me. Probably because she never wants to mess in people’s heads. But I ask - isn’t telling sb false messing in people’s heads?
I am a math tutor in Ontario, Canada. When I was in high school in the 1980's, the lessons about quadratics and parabolas included the focus and directrix. For the last couple of decades of tutoring high school math students, this focus and directrix and the associated properties are generally not covered. Introductory complex numbers seem to get taught (1 week of grade 11 at only one school in my city) more often than focus and directrix (enriched program by one teacher, sometimes).
So beautifully explained Mathologer. Here in the UK from what I remember of my sons math classes at school the same problems occurred as you described.
Just got through precal last semester here In America, have taken algebra and geometry courses but never heard of the focus or the directrix. I learned about the secrets of the parabola in a Brilliant course
here in spain we learn about them in "technical drawing", can't really translate it (dibujo técnico) making it look like an actual name of a subject. This is about 2 years before university/other advanced courses
Weird, I learned about the focus, directrix, and others, in my 10th grade algebra in 1963 (not mis-typed) in my one-horse high school in rural San Diego County. I suspect that Maths education in the states is, and was, highly dependent on the teacher. My daughter did NO constructions in her plane geometry course in her high school Maths. =:-O
New mathologer video! I haven't done all the _math_ yet but this is _adding_ up to be a great day. Does the fact that I get so excited for these uploads have anything to do with me being single? I would argue yes. Definitely yes.
High schools tend to be more varied in the quality of math education, but middle school and below feels just like Paul Lockhart described in "A Mathematician's Lament". For me personally, my math classes were nothing like your videos up until Calculus which felt like a breath of fresh air. Thanks for such amazing content, we need more math teachers like you!
Czechia: Somehow at the secondary school we never realized parabola is a single parameter shape, as teachers often mentioned shallow or steep one. I do not think they themselves realized they are talking about scaling of a single shape. At the university (math) no one mentioned it as it was obvious. I remember realizing it in the third year when explaining parabola to my cousin. There were no smart examples in math whatsoever.
In India we are never shown the practical applications of maths practically. It is really sad that we never went to a mathematical museum. But still looking at patterns of seeds of the sunflower makes me wonder how beautifully maths lies in the bottom of the heart of the nature. There can be no better museum of science than nature.🔥
When my son was in second grade, I taught him arithmetic the way I was taught (short example: subtract from most significant digit first, borrow from the answer). He picked it up easily and was bored with what the teacher taught. So I explained some basic algebra and trigonometry. He failed 2nd grade math. Solid 0. We had a meeting at school with the teacher, principal, education director and guidance counselor. Teacher says "he's deficient in math." I said "How about a demonstration?" So I brought the boy in the room and asked him "If we both start in this room, I walk 3 miles west and you walk 4 miles north, how far apart are we?" He drew the diagram on the chalk board, and then started guess-and-check to get the square root of 25. After about two minutes, he says "We're five miles apart." (Yeah, I pitched him an easy one. He was seven.) The teacher asks "Well, is he right?" The principal face-palmed. The education director groaned. The teacher says "What? Well, is he?" I looked around the room and said "I think we're done here. Fix the boys grade."....And then I took my son out for a hotdog and a slurpee. "Well, is he right?" explains everything you need to know about math education in the good ol' U-S-of-A.
Thats a great story. Too bad its bullshit. Elementary schools do not give abcdf grades. They give referrals if students show signs of learning disabilities this has been law since bush was in office. They make sure that the students are segregated based on iq in order to midigate problems with learning. By the time kids are in middle school everyone is put in their place and everyone has a lesson perfectly dumbed down to their level.
@@AdrianParsons We have "Common Core" now. 41 states+ have adopted it. Now, that's what's taught, not how it's graded or how students are evaluated. I don't know.
@@LeeClemmer 41+? That's more unity than just about anything I have seen in the US! (And I've heard nothing but complaints about common core & it makes me think of all the complaints about "the new math" I heard from my parents generation about synthetic division.)
I would never know if you don't tell you had now inspired me to see the beauty of maths most students didn't like maths because they don't know the beauty it has and knowing how this is beautiful is even more beautiful
I never got the love for math at school but with you, 3b1b and others I have. I never learned what I watch the nice way it's exposed and when I share that kind of video about the maths that I know friends do at school, they find it appealing and they always find a real use of it. A good way of approaching math is more beautiful but also easier, I think
The best math teacher I ever had just taught things through repetitive problem solving. I was one of those kids who got distracted really easily (which I think is the case with most kids), so back then I don't think I would have been all that fascinated with these applications. What stuck with me were just the basics of algebra and calculus, in a bit more of an abstract way than this. And learning these basics actually taught me to think mathematically. From there, understanding more complicated applications came naturally, even if I wasn't taught them directly. My point is that there is a fine balance between making a subject interesting, and cluttering it with too much information. Consider this just as a friendly reminder, and in a small way an attempt to balance out a bit of the criticism toward the education system in this comment section.
In high school in the Netherlands I've learned about parabola's but it wasn't until my physics bachelor when they finally revealed how interesting they actually are. Optics professors tend to be much more excited about this feature of the curve because it tends to make lenses with a perfect focal point, effectively annihilating the spherical aberration errors we have to deal with because parabolic lenses are expensive to mass-produce.
Here in Switzerland it depends on the teacher in my experience. The first math teacher I had in high school didn't mention anything about these properties. However, after two years, we got a new math teacher and when reviewing old stuff we had to learn how to compute the coordinates of the focus point and the equation of the directrix.
Mathloger, I am also unhappy for the same reasons! I am an undergraduate university physics student. I've lived in the U.S. all my life, and I have come to a disturbing conclusion: So-called "education" is sometimes (not always) more about being indoctrinated and controlled, rather than learning, thinking independently, or being innovative. The problem occurs not just in math education, but from what I can tell, there are similar problems in all subjects. Students are treated like programmable calculators, instead of human beings who should be free to question, think, and create. No single university professor or teacher is to blame for all this. The source of the problem is deeper and more systematic. I think it spreads throughout the human population. But why? How did this begin? Why does this problem propagate? Can we trace these problems to the first human civilization or is there some kind of conspiracy to turn us all into robots? I would very much like to know. Here are some interesting quotes from the famous book "Proof is the Pudding" "There is a grand tradition in mathematics of not leaving a trail of corn so that the reader may determine how the mathematical material was discovered or developed. Instead, the reader is supposed to figure it all out for himself. The result is a Darwinian world of survival of the fittest: only those with real mathematical talent can make their way through the rigors of the training procedure." "In this sense Bourbaki follows a grand tradition. The master mathematician Carl Friedrich Gauss used to boast that an architect did not leave up the scaffolding so that people could see how he constructed a building. Just so, a mathematician does not leave clues as to how he constructed or found a proof." So, there is a "grand tradition" of competitiveness. Instead of seeking to pave the way for all children to surpass us, we are to worried about competing against everyone else. Perhaps this is part of why our education system sucks. Just an idea.
Wow I was also thinking something of the sort but you put it into better words than me! My goal is to one day become a teacher and teach kids to question themselves, the teacher, and the knowledge itself. Is it too ambitious and maybe disruptive to the system? Some might fear it but I think it will be the beginning of a more open-minded, less fearful and more daring society.
2nd grade-undergraduate level math/science tutor here. I'm a generalist and differential equations, multivariable calculus, and linear algebra are *about* where I draw the line right now. Fascinating insights. Thank you for this. I ALWAYS tutor my students with the goal in mind of making sure they never need another math/science tutor again. Which means I am trying to put myself out of a job. Which only increases my competition in the future as more students are empowered to see how bull**** our system of competition really is. I don't care, though. I want our future generations to have the opportunities to BUILD and GROW on what I, PERSONALLY, have achieved. I don't need some artifice of "grand perfection". I want someone in the future that I've personally engaged with to smash down all of the supposed mathematics that I've taught them and to show me in great detail how utterly wrong I am. Because I'm always wrong when I teach. The concepts, connections, and applications are far more important than the grades and "mathematical rigor" that is focused on in Texas elementary-high school mathematics. Don't even get me started on university maths, because I don't actually have anything to say besides .
Coincidentally, I looked up and was reading about the focus/directrix definition and properties of parabolas and other conics this morning, and then saw your video this evening. I had run across the terms before, but I was not taught them, to my recollection, in my school mathematics, much less the properties that flow directly from this classical treatment. I found the video as delightful as it was fortunately timed. Thank you for such a clear and wonderful demonstration!
I always introduced parabolas with the paper folding activity. (Victoria - Australia). Next, students repeated with dynamic geometry (locus) and loved exploring the different forms of the parabola in this very interactive and visual manner. Lots of applications before we started graphing y = x^2. I tried to emphasis how beautiful mathematics can be ... such an amazing curve with so many properties with such an elegantly simple algebraic form. My methods, however, are not indicative of the typical introduction and certainly not 'reflective' of the standard textbook approach. Perhaps its about time we 'shifted the focus'. Well done Mathologer on another brilliant video.
here at my high school in new mexico I work as a scholar tutor (meaning the school pays me to tutor my peers) and while most of the kids I get haven't gotten to algebra yet (we're a combined middle and high school), I still try to incorporate intuition about the math we cover the same way your and others' videos do. I haven't taken a math class for a couple years but from what I remember nothing has ever been motivated or interesting-you just drift along a river of math as you solve endless problems to hopefully row to shore, though most end up drowning.
I'm from argentina, I think having a good explanation of things is up to the teachers, some will be super energetic and try 300 different ways of explaining to you while others just want you to do your homework, most times we dont get explanations for uses, I remember someone asking "What do we use complex numbers for?" and of course the answer was "Physicist use them for difficult things"... on the upside we had 4 classes more or less about fractals and their uses nearing the end of our last year but that's it
Super nice explanation for why the crease is a tangent to the parabola, requiring no equations. The key fact is that, as the parabola is the set of points equidistant to the focus and the directrix, it divides the plane into two parts, where points "inside" the parabola would be closer to the focus than the directrix, and points "outside" the parabola closer to the directrix than the focus. Consider a crease c folding a point P on the directrix d onto the focus F. Clearly, for all points A on c, there is AP = AF due to the folding. Now, erect a line through P perpendicular to d, intersecting c at T. Clearly, T is a point on the parabola, as TP = TF by T lying on c, and TP is perpendicular to d by construction. Also consider another point B on c, different from T. We still have BP = BF, but as BP is not perpendicular to d, BF would be larger than the distance from B to d, hence all these points are "outside" the parabola. Therefore, T is the only point on c that is on the parabola, and all other points on c are on the same side of the parabola. Quite Easily Done. Also, a nice extension: we know that, by folding all points on a circle towards a fixed point, the creases would form tangents to an ellipse (fixed point inside circle) or a hyperbola (fixed point outside circle), with the center of the circle and the fixed point as its foci. This fact can be proven by a very similar argument, considering the division of plane by the ellipse or hyperbola into regions in which the points satisfy certain distance inequalities, and noting that the unique shortest path between two points is the line segment joining them.
I've heard about this from a very application oriented calculus book (my first one couldn't find anything else). they are properties of different conic sections. the coolest for me is the ellipse which can concentrate waves at 2 special locations corresponding with the foci. so you could hear something very clearly far away in an elliptical room. it's also where I first learned that parabolas can diffuse and concentrate any waves because of their interesting shape. There being applications in the form of special mirrors and glasses. Really glad I read that chapter because that's not the kind of thing you learn in pure math or at least not without any emphasis. More of a side result/application but really cool.
ex-maths teacher here. I tried innovating maths education by having my kids learn with Khan academy. It was working great, especially for the weaker kids who could learn at their own pace. Soon enough parents started to complain because I was "experimenting" with their kids. The head teacher (dik-head I should add) quickly mandated I revert to the traditional pedagogy and boring book with rote exercises and homework. Only a new society will bring a new education.
As an aspiring teacher, this is one of the things I fear the most. Even if I want to do what I think is best for the students' education, I might not be allowed to.
youtube is a resource for learning new things. I wish more people would realize that instead of just relying on a boring textbook which might be teaching in a way that is difficult for that person to understand.
Dear Michele Bagaglio, I would advise teaching math(s) outside the U.S. at an international school - preferably IB. You get to do cool stuff, and use Mathologer videos because you will be the "wise American" bringing "American innovation" to the classroom.
6:44 In my hometown they have a pair of those! They are on the "Parque de las Ciencias" (literally "Park of Sciences"), which is a science museum with a lot of interesting things to do and to watch (as well as conferences given by lecturers from our University). If you ever visit Granada in Spain, don't forget going there, you'll enjoy it. And being a mathematician you will probably enjoy the historical architecture (the Alhambra, the "town or "casco histórico" -literally "historical bucket" in Spanish-) as it has a lot of interesting mathematical patterns. The Spanish muslim medieval arquitecture has a lot of mathematics on it, taken from the old Greek and Roman geometrical patterns.
Sydneysider here and my maths teachers weren't paid enough to try and supplement the syllabus with "real world examples". Some of them tried anyway, for which my classmates and I were grateful. I can honestly say I learned more from 3Blue1Brown, yourself, Matt Parker, Brady Haran, and Vi Hart than from the advanced mathematics my high school offered.
In the Greek book, there is a small reference on mirrors (a small paragraph), but there are mostly equations. Thank you very much for creating this video.
As a student from MWHS in Madison, WI I was blessed with an open minded public school system. Without a doubt some instructors just read the textbook but for most of my STEM field classes the teachers were enthusiastic and brought up new ways of looking at things.
Hi, I’m a Jr (11th grade) in high school, and last year my teacher actually told us the fundamental theorem of algebra WRONG. The explanation she gave us was that polynomials with odd number orders tended to infinity with ends pointing the opposite ways, and even orders eventually pointed the same way. It’s honestly awful that our education systems hire teachers who don’t even know their subjects to teach curriculums that completely miss the point. For instance, in trig classes, way too much emphasis is placed on knowing how to graph ANY transformation of a trigonometric function, rather than more important topics like polar coordinates. Videos like yours are honestly the only way to learn today so thank you so much for continuing to grow our ability to understand the world around us. Ps. You might be thinking “oh well his school is underfunded etc etc.” No. My school wins awards for being the best staffed and best test scores in our state almost every year. Makes you concerned for underfunded, inner-city schools right?
I'm studying to become a math teacher in California by the fall. Resources like Mathologer and Numberphile are so useful as I stockpile engaging lesson ideas that relate to curriculum. Thank you for this and the many great videos you've produced. I want nothing more than to stoke the flames of mathematical curiosity and excitement in my future students so that they don't feel that the standards they are learning are just arbitrary tedium they have to do in between studying history and reading the classics. Just because not everyone CONSCIOUSLY uses mathematics of this kind in their daily life or profession, does not mean that they aren't surrounded by exemplars of fascinating mathematical principles. Exciting kids about math shouldn't be any harder than exciting them about Physics and other sciences, as math is the basis of science and capable of explaining anything they might have an interest in: engineering, music, sports, games, language. It all can be viewed, quite usefully, through a mathematical lens. Thank you again!
For people who enjoyed the connection with origami, there's a lot of other cool math relating origami and the study of polynomials. There's two main ways I know of that polynomials and origami interact: Algebraic Geometry and "Galois Theory." I did a reading course of the Galois Theory of origami last year, and got to read a lot of cool resources that others might find interesting. In particular, with usual origami you can solve any cubic equations, which makes it more powerful than a compass and straight edge (1), and if you allow two folds at once you can solve all degree 7 equations! (2) In fact, if you allow an arbitrary number of simultaneous folds, you can solve polynomial equations of any degree. (3) This last paper is particularly interesting, because the proof uses a method call's "Lill's Method" for solving arbitrary polynomials. This method was discovered over 100 years ago, but hardly anyone knows it anymore, since it was never really useful for anything. So, the fact that its now being used in a paper alongside much more modern techniques like Gröbner Bases is pretty cool to me. That paper also defines "origami cubics" which are defined using exactly the trick proved at the end of the video, which is what made me think to go on this whole tangent in the first place. (I also think there's a lot more connections between the Galois Theory of origami and some more modern Algebraic Geometry that haven't been fully explored yet, but idk, I'm just an undergrad.) There's also a ton or origami math that has nothing to do with Galois Theory, like Erik Demaine's work on computation origami, the cut and fold theorem, etc; Robert Lang's work on Tree Maker, Reference Finder, and other math related to origami design; and a ton of work by various other mathematicians on geometric origami. (Disclaimer: When I say solve all polynomials, I really mean find all real roots or polynomials with integer coefficients, though I think there are some extensions of this, that are just much messier.) (1) Solving cubics with 1-fold origami www.researchgate.net/publication/233592288_Solving_Cubics_With_Creases_The_Work_of_Beloch_and_Lill (2) Solving septics with 2-fold origami forumgeom.fau.edu/FG2016volume16/FG201625.pdf (3) Solving arbitrary degree polynomials with arbitrarily many folds www.langorigami.com/wp-content/uploads/2015/09/o4_multifold_axioms.pdf (4) Lill's Method Explanation www.math.psu.edu/tabachni/prints/Polynomials.pdf (5) Lill's Method Original French eudml.org/doc/98167 (6) Erik Demaine's Papers erikdemaine.org/papers/ (7) Robert Lang's Papers langorigami.com/articles/mathematics/ (I really just had a bunch of links that I thought were cool, and wanted to share lol)
Hi, Prof Mathologer. I grew up in Melbourne and went to Camberwell High where we had some pretty damn good math and science teachers in the late 70's and early 80's. In later high school (years 9 and onward) we typically had various different math classes. I did pure and applied mathematics and the applied math was the most fun. In it we learned the typical things like solving differential eq'ns for flows into and out of tanks, applications of probability and statistics to practical problems and so forth... BUT in pure math we learned all of the usual stuff almost entirely devoid of real world applications. I guess that made sense with pure math and, although I understood applications for parabolic (and other conic section) curves, hyperbolae, etc,. I could not for the life of me understand, apart from rotating things, any extensive use for the equations for a circle nor the use of complex numbers.... until I went and did electronic engineering at Melb Uni. Then, circles and complex numbers were damn well everywhere. I am grateful to the high school teachers I had, they were very good and inspired a great interest in math and science in me resulting in a, thus far, 30+ year career in engineering. However, I cannot fathom what has happened to education in Victoria in this day and age. It is as if they are trying to make it as unappealing as possible, perhaps to stupefy everyone into some kind of non-STEM field career. In the 1970's, Australia had a growing industry in electronics but subsequent governments, whether left or right, have done all in their power to destroy all such industries. I know this sounds conspiratorial but it is as if some outside influences seek to make this nation into one of dumbed down obedient dullards willing to dig holes and wash cars for a few scraps of cash. I have nothing against those who do such work for a living, but something seems to be undermining high tech in this nation.
In South Africa, not much attention is given to the properties of mathematical objects in class. You're pretty much just expected to rote learn everything. But I really enjoy intuitive and visual explanations. Thank you Mathologer!
(Romania High school) The property at 2:03 we learnt in high school, but in our math class we only done exercises (derivative, complex numbers, quadratic equation, quadratic formula). So highly theoretical. There could be some examples given, but few. At exams you need to solve exercises and to memorize-paste the theory. The practical application or curiosities were reduced. In the explanation part all the graphs were drawn (at least in 2015), but a animation can not be drawn. For example the analogies of derivatives and integrals, you could seen more easy from an animation that the integral is the area under the graph, sum of a lot of tiny rectangles. In the actual lesson, the teacher draw the rectangles with which the function is approximated and give the formula of this ( with epsilon csi etc.). A lot of complains are the very high number of theory required to learn. At college, course Special Mathematics, we learn about differential equations, the professor presented only the methode of solving, the amount of theory was reduced. A lot of problems, but i like a lot the youtube educational community (all of it, mathematics, history etc.) that actually explains in a simplified manner a subject.
In Serbia, most high school students aren't thought the intuition behind anything. We are thought to calculate derivatives, without mentioning their essential geometric properties. Nothing is proven and we are thought to carry out calculations like mere computers.
By the way, I discovered the physical significance of the directrix for myself while doing ripple tank experiments while teaching physics. If a plane wave hits a parabolic reflector and deflects to the focus, the directrix is where that plane wave would have ended up at the same time if the reflector hadn't been there. So the parabola can be thought of as "transforming" a line (the directrix) into a point (the focus) through reflection. This gets to the equal time property you mentioned and accounts for why the focused light waves interfere constructively at the focus.
There was a screen saver that was drawing lines and often created parabola shapes. Some artwork made of a wood board with two row and two columns of nails and threads connecting between the nails was also creating parabolas with the added bonus of a 3d effect of threads intersecting. The code that draw this kind of shape is actually two bouncing balls with line draws between them as they move in discrete steps.
My experience in Texas is that we learned about the focus and directorex in Algebra II, which is in the 10th grade, and the cool properties of parabolas in Physics under the Optics and Wave Mechanics units, 11th grade (though the equals-x explanation was new 😁). Idk if it was just my school district, but the public education in Texas is surprisingly good in comparison with what I hear from other people online. Art, Music, History, Math, Science, English Language, Literature and Writing are all given a rather high priority, despite also getting three years dedicated to just Texas history. 😅 Honestly the biggest downside is that there is little to nothing about practical subjects. No personal finances like paying taxes, setting a budget, ect. Health is limited to "do drugs and you'll die" and "have sex and you'll die". And there is no philosophy or discussion of religion at all (which means you get one day of watching videos in Biology for the evolution unit). All this stuff you're just supposed to "get from home"
I can't say enough how much I wish this was true for the rest of the states. I noticed a dramatic drop in the quality of education when I moved from Texas to North Carolina. Without even trying (and I literally mean that I didn't have to try hard at all), I went from the dead middle of my class of 120 student class in Texas while barely passing to graduating 36th out of around 205 with a 4.2 or 4.3 weighted (I can't remember which) GPA from a fairly decent school here in North Carolina. topics like vectors, matrices, and actual probability weren't taught to me until precalculus my senior year, which was a good bit after I had taken the ACT and is one of the more common routes people take at my old high school. I'm just glad to be out of high school and finally in a place where people care more about education so I can finally learn more about calculus, linear algebra, and differential equations.
I remember from math class that my teacher told us to find a spot on the schoolyard which has an equal distance to a given point and line and we all "magically" lined up to form a parabola. I think this was a nice way to first try to spark some curiosity before he actually explained why. And something a little less related. Parabolic reflectors which can be used for example for wifi do use a specific focal distance, such that the direct path from the focal point is exactly one (or maybe a bigger integer) wave length(s) shorter then when reflected, such that both signals are in-phase and give a slightly stronger signal strength.
i just read a book by richard feynman where he described working on a committee tasked with selecting course books for elementary schools. he was distraught with the system and how underwhelming the material in the books were, and this was of course several decades ago. the issue was that the committee members were often not qualified to assess a range of subjects, and these sort of "text book lobbyists" would corrupt the system. at the end of the day, the quality of the lessons in the book weren't the determining factor in what was handed out in schools. i fear that system hasnt changed
The distance between a point on the directrix and the focus is always equidistant as the line between them is always perpendicular to the tangent of the parabola. By applying Pythagoras we can obtain that the point above the point on the directrix will be on the tangent which would make another right angle triangle if we join those points. That triangle would be a 45 45 90 as they are the half of a right angled triangle and henceforth the points lying on the non 90° will be along the same axis.
I'm currently in ap calc, and schooled in Florida. Of all the math classes I've taken the only thing you mentioned here that I've heard before are how to find focus and directrix, and that lines perpendicular to the directrix will reflect towards the focus.
Australian here from NSW. It has been quite a few years since I was in high school, but math classes back in the mid 2000s did include the focus/directrix definition of a parabola. Either in Math Extension 1 or Math Extension 2. Not sure where you would find it in the Victorian curriculum, but it's probably there somewhere.
American here from New Hampshire Math education in high school isn't that bad here, I actually had to write an essay about the applications of conic sections in construction for my pre-calculus class Though we didn't do anything quite as cool as this lol
Here in Upstate New York, we weren't ever taught any real or fun applications for math, except indirectly through the occasional word problem. The focus and directrix weren't ever mentioned when it came to parabolas. I think it mostly comes down to the teacher, rather than location. I don't think most grade school math teachers have a particular passion for the subject, and the pay levels aren't enough for them to go out of their way to find ways to make it fun or interesting. It's sad, really, and definitely needs to change. I hated math in school; it felt like tedious memorization of rules that'd be irrelevant in the real world; can only assume others felt the same way. I only got interested in it beyond its utilitarian uses well after high school, when I stumbled upon videos like yours. Thank you for them!
In an international school in the Netherlands, And we also never learn about this stuff... It’s so simple and clear and satisfying, why isn’t this taught everywhere?
yes, we studied this thing about parabola in our 5th grade. ellipse also has this property that you can whisper something standing in one focus of an elliptic room and the other person hears you well standing in the other focus.
This is super cool! At my old college, our quad has a parabolic shaped building and one day when I was leaving to the parking lot I heard someone talking right at my ear and freaked out. I looked back to see who it was and saw some people standing talking way far from me. I knew it had to do with the parabolic shape since I remembered learning about parabolic dishes in physics and diff eqs class but I never pinned down exactly what happened there that day. I'm tempted to go back with a range finder and see if I can take some measurements and calculate the locations of the foci!
In British Columbia, Canada, I tutor high school pre-calculus. The kids learn how to factor quadratics, but I haven't seen them studying much in the way of proofs, such as those behind the quadratic formula or Pythagoras. I *have* seen an intuitive derivation of the surface area of a cone based on that of a pyramid. No directrix in sight. I would like to see students given fewer problems and told to suck the marrow out of them. This is why I started a local MathTetminds group and RUclips channel. To make math more like finger painting. Enjoyed this video immensely. I like the Mathologer videos. They make me think. 😎
Your sermon goes down to the heart of all education. Curriculum seems to be giving us facts and formulas and it expects us to learn it and use it. I don’t know what their definition of learn is, but it doesn’t work. It takes thinking in order to learn something, not mindlessly plunging numbers into formulas. Even if you mention this problem for math, it happens in every subject from English, History, and sometimes Science. Learning is about understanding, questioning, and experimenting yet it is never taught in a way that is learnable. I have talked to a teacher about doing problem solving exercises, where the students take the time to figure out a proof for some important thing in math. Her response was that not all kids are good at problem solving. My thought is that it must start in elementary school, in order to build up those skills, something like finding the area of a triangle. “Given a triangle, could you find it’s area, could you find a formula that works for any triangle?” I would love to try and take this as far as it goes but it is a huge change that most teachers would not agree with. I hope that someday people could learn. Actually learn. Asking questions is the way that new math gets invented, and new research, new facts. Can we get kids to ask the same questions?
Just to add to this, I was given a false geometry theorem about circle radii, angles. and arc length, all relating because circles are similar. I noticed that I was ambiguous and proved it was false. I wonder if my math teacher knew any math or if she was just following the book.
American from southern California, Algebra 2 was about 10 years or so ago for me. I don't think the directrix was even introduced in our classes. And if it was, it surely wasn't utilized in proofs until precalculus, which ended up being in junior college for me. As a Math Major I really enjoy the simple elegant and visual geometric "proofs". Love the videos
100% agree to your little rant on how maths is taught in many places. I'm studying maths/physics for becoming a teacher in Austria and I'm regularly shocked by how little a significant portion of my soon-to-be-teachers colleagues knows about the actual practical applications of maths. And in almost any school textbook there are the same old boring pseudo-application problems - and yes, in one of them a bike path in the shape of a polynomial function is featured as well. It's really kind of depressing, that many students will leave school, thinking that the pinnacle of calculus is determining the volume of a liquor glass in the shape of a paraboloid...
The paper folding ellipse construction leads directly to a proof of the reflection property of ellipses, analogously to the parabola proof. BTW, there is a reflection property for hyperbolas analogous to both of the above. A ray heading toward one focus of a hyperbola will be deflected, by either branch of the hyperbola, to the other focus. This is used in Cassegrain telescopes. You may know this, but many people don't.
In California public school we found the foci and directrix (or its analog) of all the conic sections, especially as it pertains to drawing a graph of an arbitrary conic by hand. The parabola was defined in terms of the equidistant property and we learned about the reflective property. The relationship with the whispering dish-style setups was mentioned offhand but nothing about tangent property or the math of the whisper dish.
My high school (California) had examples related to planetary orbits for ellipses and their foci. They didn't bother with the application of parabolic foci, but the foci lessons were bundled together. I think physics classes took advantage of this more than math classes. Certainly my middle school algebra education did not dabble in this, but my math teacher then was not the most enthusiastic educator.
The little mathologer's have a very similar experience to mine in class. I did learn about some cool math in my math club. I personally always enjoyed math as a kid. I'm not sure why, but I always found solving math problems fun regardless if there wasn't good motivation for the problem. In high school and university I found that well motivated problems are even more rewarding than the problems I faced as a kid. In general it was very instructor dependent. I remember loving Calc 3, ChemE Math, Geometry, and combinatorics because the instructors had good examples and taught in an organized manner, but I hated Diff Eq and Number theory because the class was all over the place.
I am in the USA have two teenage boys. I was so excited when the oldest got to High School Geometry since for me that was Analytic Geometry for me with its Geometric Constructions and Geometric Proof - where the math had to move past memorization into analysis/understanding (from what to why). Sadly, I talked to the teacher and it was going to basically geometric shapes, formulas, and trigonometry. He too wished he could teach Analytic Geometry; but, it was not in the curriculum. I suspect some of the why is the standardized testing model. Memorization is much easier to test and teach to. What is always easier tor test than why. I think another reason is that many students struggle when you try to move them past memorization and so the short pole can sometimes set the height of the tent rather then the tall one. Well, I haven't let my boys get off with just that (at least not my oldest - who really enjoys math). I think that to fully grow, everyone must be able to more past memorization. You need to understand not only the what but also the why. No brain is large enough to memorize everything and not all results as as convenient as to be memorize-able. Understanding the why breaks the door to infinity.
I actually learned about this in middle school (7-9 grade) and it was explained very well but it was probably because we had a special program with a university professor. I live in Poland.
As a current chemistry graduate student, I remember learning about the parabola, focus, and directrix (idr how to spell that), but none of these particular properties. To be fair, I grew up in Michigan in the U.S. and attended public schools. For those who don't know, Michigan is ranked worst for public education in the U.S. (It's where Betsy Devos came from, i.e. our current secretary of education). The directrix and focus at that point felt like such transient topics. Their significance was not made clear to us as middle schoolers making our first real foray into algebra. Such a simple explanation would have helped, and probably would have led to higher engagement in maths across the years.
Thank you so much for your explanation on properties of the parabola....sans hyperbolic statements ;-) My children were educated that the local community college for their high School education'sbut did not receive such elegant and complete an explanation as yours. Cheers from Seattle!
Austria (that one without Kangaroos ;) ) - I am doing some private tution for age 14 to 18 and I found out (most) teachers still miss the point of getting young people interested in math and make it "easier" for them. e.g. it took me just two pens and a sheet of paper to explain how to measure the angle between a straight line and a plan (and I was not drawing any formulas or graphic ;) what has been never shown or discussed in classes. So no things pretty same around the world. School books are made by people who either hate kids or are completely ignorant about teaching - and poor dedicated minds have to get this fixed. Like you do, thanks a lot.
@@Saisai-uh8do Agreed I was not using the right words - With School books I do actually summarize all the rules, curriculas and material provide to teachers to be used with less and less freedom to really TEACH something. Instead they have to follow some "standardize quality" levels to produce uniform students. If the students don't fit into those standards they will me put aside. It is not only about what you learn but HOW you learn, and some people need different approaches to understand but in the end they might outperform others.
The equal-optical-path-length property goes hand and hand with the focusing property of parabolas and applies to any focusing device (lenses, mirrors, combinations, whatever...). It's a consequence of Fermat 's principle which is a consequence of Huygens principle.
I live in america. And the particular high school that i went to quite literally graduated people who did not numerically "pass" in order to meet a deadline so they could remain a school. Math class was taught by football and cheerleader coaches. My chemistry teacher did not know that water boils at different elevations
My chemistry class was taught by a football coach that was reading the textbook at the same time as us. He would read a chapter the week before us and then teach it... >_
A bit late for this, but in North Carolina high school classes, bits of conics were taught. We were taught about the directrix, focus, and how that light entering a parabola would always reflect into the focus of said parabola in precalculus class. We, however, were not taught why light reflects into the focus of the parabola (to my recollection). The only actual proofs of anything mathematical taught to me in high school was how to derive the distance formula, and how to find the lengths of the sides of a few angles on the unit circle. This though wasn't the most egregious thing though. During my senior year at high school (which was 2018 for me), I was given a wonderful opportunity to compare the precalculus textbook and the math 2 textbook (for reference, there is a year of math between these two classes). In it, I discovered that the two had nearly the same exact topics covered in them, including (if my memory serves me correctly) conics. Even though the two textbooks might have had varying levels of difficulty (which probably isn't much), nothing new was really being taught to students. Math was incredibly easy in high school for me, and one didn't have to put much effort in to make an A in the class. This saddeneds me. I had begun to develop a passion for maths in high school but felt cheated out of a decent math education because of the curriculum. It's nearly a joke with how easy it is. Precalculus in community college hasn't taught me anything new either, but I half expected that given that they have to make sure that everyone is caught up to the standard of the school.
I forgot to mention that most people are also not taught the intuition behind most anything taught in class (at least in the curriculum I was in). This is why teachers who go into why stuff works mean a lot to me. I learn best when I understand why something worked because I typically remember the process of deriving the formulas better than the formulas themselves. This also makes it easier to understand later topics because the intuition typically builds on itself. And yet, I still remember the teacher in my precalculus class in high school deducting points because I didn't follow her way instead of the way I learned by myself. Hell, I made my own factoring calculator using a method never taught to me that I still use today to check my math because I saw a simpler method.
Express proof: Every point on the crease line is equidistant from the focus and the target point (the line is a bisector). The point on the crease vertical to the target satisfies this property, making it on the parabola. The line is then tangent to the parabola as there can only be one common point, since only one point of the directrix is folded onto the focal point. Sorry if you don't understand, but I lack mathematical vocabulary in English.
Proof for the challenge: the first thing to note is that on one side of the parabola are those points that are closer to the focus than any point on the directrix, and vice versa for the other side. And the crease is the perpendicular bisector, which separates the plane into points closer to that particular directrix point and those closer to the focus. So the line being tangent to the parabola is the same as saying that if a point is closer to this one particular directrix point than the focus, it’s closer to some point on the directrix than the focus, which is obvious!
I have a different explanation for the property you proved about that line being tangent, the line you need to fold by is simply the perpendicular bisector of the focus and the point on the edge, which is all the points with equal distance from both points, when does intersect the parabola? The distance from the focus is just the distance from the line on the bottom, so we want the distance from a point on the parabola and a the point on the bottom to be the same as the distance between the point on the parabola and the line, and that happens only when the point on the parabola is right above the point on the line, so that process that it's tangent and that the point in the parabola is right above the point on the line
Hungary, nearly 40 years ago. Geometric properties of the parabola were taught in "math hardened" classes around the age of 16 of the students. More than 3/4 of the students were never hear about. There was a nice problem, I remember. Given two tangent of the parabola (the parabola is arbitrarily placed on a plane), determine the the curve itself! (That is: give the position of the focus and the directrix.)
I always had an intrinsic love for mathematics but the day I found your videos , I am more than happy. I always wanted to see mathematics comming out of paper and doing real things ( bcz in schools we have only solved the long equations . Our teachers would leave the chapters on graphs saying " these things will not be part of paper " ).
At school I never learned about the focus and the directrix when learning quadratics. But I read about them in books about geometry, which never explained how the quadratic equation arose from a point and a line. In physics I learned that a parabolic mirror can focus a beam onto a point, but it was never explained that it was a mathematical necessity that the mirror be parabolic.
A simpler proof of the end is just to say the line is the locus of points equidistant from focus and the specific point onthe directrix, which we know only touches the parabola in exactly one point since the distance from said point to directrix is minimal thus the line is tangent
2nd year Canadian Math and Physics major plus high school math and science tutor here and I've grown frustrated as I realized that so many of our students are not being taught these applications of functions and relations --heck I was doing a mindyourdecisions puzzle the other day and didn't know how to solve it because in sixteen years of math schooling, I had never been taught the Intersecting Chords Theorem and 90% of our Calc III class had never been introduced to conic sections. While this may not necessarily be horrible for people with a natural curiosity/intuition for math --it puts many students at a great detriment. My last tutoring session with a grade 10 girl, the poor kid was at a complete loss for how to solve for the equations of a triangle's median or a line's perpendicular bisector --because NO ONE had taken the time to explained the Geometric meaning of those words. Once we broke it down by definition and addressed a few of the more fundamental formulas, she basically came up with the correct procedure on her own. It just irritates me to no end that the majority of educators in our education system don't take the time to help our students develop the skills to interpret more complex problems or understand the properties that arise from them
Being a math teacher in The Netherlands I *always* tell my students about these properties (and make sure they get proven in class either by me or by the students), but only because I like them so much myself and think they're more important than a lot of the crap we otherwise spend time and energy on. These properties are not (anymore) part of the standard curriculum, unfortunately. And yes I agree, the educational system never misses an opportunity to miss an opportunity... which saddens me no end.
I live in the United States (in Wisconsin), and have also sadly watched the decline of the quality of mathematics education over the past forty years. Like @chaosmeister , I also teach these properties (as well as the length of the _latus rectum_ which isn't included in modern textbooks because the kids giggle at the name and sensitive teachers are embarrassed) in algebra 2, but the study of ellipses and hyperbolas is no longer included in the curriculum.
Every topic in every math class seems so diluted as compared to when I was in school, which leaves modern students inadequate for university study. Too many politicians and administrators emphasize students' feelings and self-esteem instead of their education; this has led textbook authors to spend more time writing review material to which students have already been exposed rather than expanding more deeply. They believe keeping students moving along to the next grade level has become more important than actually teaching them, and the result is a cascade of students (who are never allowed to fail) taking mandatory next-level classes for which they are woefully unprepared.
but we do get this in wiskunde D vwo
@McOinky ... which is a voluntary subject, chosen by less than 10% (or less?) of all math students. Right...
[Edit: "less than 10% (or less)" is incredibly silly coming from a math teacher... sorry 'bout that... End of self-flagellation :-).]
@@mjones207 Amen. Recognisable. Unfortunately.
@@mjones207 In my U.S. Pennsylvania school, we are taught the properties about hyperbolas and ellipses in Alg II, but nothing about the latus rectum (lol) or much of the details of parabolas other than converting to vertex form. We spend much more time on the other conic sections.
Your concerns are well founded. I was a math and science teacher at two private schools in the United States (each time for two years). The textbook selections never aligned well with my ideals of making math and science relevant, useful and entertaining. For 3 of the four years, I was able to accomplish that in spite of the books. During the 4th year the academic police state finally caught up with me, insisting that the reasons we teach anything are foremost to increase test scores and grow our market share through test-based reputation. That authority banned all non-standard curricula and forced me out of the profession I loved. All the texts from which I taught were ostensibly aligned with their goals. They were also filled with endless drills, BS examples, incomplete history, and frankly serpentine reasoning far more likely to confuse that to convey any valuable understanding.
I’ve lived 4 years in Germany (which was a bit better) and 4 more in South Korea (which seemed much better). Unfortunately, the Korean kids were under immense pressure to perform, making almost all of them profoundly depressed and/or stressed from perhaps 10 years of age.
Most of the great contributors to the betterment of our world recognize the importance of conquering fun challenges. Both mindless, meaningless repetition and idiotic complication turn beautifully curious and malleable children into miserable adults.
When teaching became primarily the indoctrination of future workers, it necessarily ceased teaching young people to think critically and creatively in favor of teaching them to do what those in authority tell them. This bodes terribly for the future and causes me to be deeply concerned for our posterity.
Thank you for making learning the entertaining challenge it’s meant to be!
I can't speak for the rest of my country (I live in the US), but I didn't learn any of this in school. Which is a large part of why I watch mathematical RUclips channels.
State of math affairs in southern Ontario Canada (grade 10, ~age 15/16) is we are taught the quadratic formula, sin law, and cosine law, we were allowed to try and derive it on our own if we wanted to. We weren't taught any of the applications for any of the math except in trigonometry and measuring the heights of hills, and we certainly never got as in depth as many mathematics teachers on the internet, which is part of why I love this channel so much. If I didn't have you, numberphile, Matt Parker, or a brilliant subscription, I'd probably be dragging my feet through the mnemonics like everyone else
I went to a high school for behaviorally challenged students in the United States. I was always an Honor Roll student, but the school didn't teach anything higher than Algebra 2. When I decided to go to college, I only placed into remedial math classes. Now, I'm 1 semester away from an Associates in Computer Science. I took 5 remedial math classes to work my way up to Calculus 1 and 2. Since I've been attending half-time while I work, my 2 year degree has taken me almost 6 years to complete.
I went to public school in nyc all my life, i didn't learn any of this until my junior year of college.
Congratulations. Seriously.
Here in Brazil we just learn to memorize everything to be able to do tests. Our teachers even say that we “won’t need this in the future unless we become engineers or mathematicians”, instead of trying to show how mathematics can actually be interesting and beautiful. I’m glad I have the internet, because I’d never get this kind of stuff in high school.
Oh gosh. Now you've done it.
Here in BC, Canada, the situation is equally dire. We spend months beating the grade 11's over the head with parabolas until they can hardly remember their own names, let alone any useful or interesting properties. Parabolic trajectories and parabolic bridge arches are as exciting as it gets. There's not any mention of focus and directrix anymore, just 'general form', 'vertex form' and 'factored form', and going back and forth between them ad nauseam.
It puzzled me why we force everyone to 'study' 'mathematics' in this way until a fellow teacher explained it to me thus: our education system is really a sifting system whose purpose is to divide people into categories for later life. It's unfortunate that in the process it takes almost all the fun and certainly all of the wonder out of the world. Except for the valiant efforts of some exceptional teachers, the effect would be complete.
Don't even get me started on 'rationalizing the denominator' in grade 10 ...
Lmao, the curriculum has gotten so bad that it almost seems some students (like me) know more about parabolas than the teachers. I showed one of my math teachers a proof of the quadratic formula by completing the square (which is where it comes from anyway), and he seemed completely lost. I also showed him other properties such as Vieta's formulas, and many of the the properties mentioned in the video, but he mentioned that he has never seen any of those before. I don't know whether or not these were never taught to him, or he has simply forgotten from never teaching it himself. Whatever the case may be, that was a deeply concerning experience.
In the Basque Country (northen Spain), we learnt a circle is the set of points equidistant to a given point, an ellipse is the set of points with constant distance to two points, and the parabola is the set of points equidistant to a point and a line. However that was just a definition, for problem solving we'd immediately would resort to algebra and solve things in equation-space, never geometrically. So basically, we knew about focus and directrix, but never used it in practice, as far as I can remember.
I live in France, and we never had those type of explanations when we learned about quadratics (and it was like last year)
Great video by the way, love what you do!
Same in germany
I suppose it depends if Maths was a “major” and the time. I got a full course on comic sections in high school. Searching on Internet, now you get it later in Maths Preparatory Schools, it seems.
It's because the conics aren't taught anymore in France
I've had one of these parabolic mirror hologram toys for years. It always causes a wow moment the first time someone experiences it. Good luck on your mission to put the fun and relevancy back into school Math worldwide. Understanding how to apply Math is way more important and enjoyable than just churning through calculations ad nauseam - that's what computers are for.
In Italy, we study all of these properties (and relative proofs) among 14 and 16 years old (the beginning of secondary school). Loving your videos!!!!
India : I being a student myself and having prepared for JEE advanced have came across almost all of these properties in our coaching classes but we were never taught the proofs since there were so many properties to mug up there was simply not enough time to do them in class. They didn't even encourage us to do it at home and ask our doubts if we were stuck with it. Luckily I had one great book of geometry by SL Loney which I solved to learn as well as prove some of these proofs myself but even that book had no such beautiful real world examples. It's sad to know what beautiful stuff there is which I missed during my school days and might never ever stumble across.
Interesting that this video would pop up in my recommended today because I just read in my book ( Charles Seife’s Zero) about how an ellipse with one of it’s foci extended to a point at infinity forms a parabola.
This is the sort of maths I love. Great video Mathologer!
Germany: Here in 10th grade most people fail at finding the intersection of two lines.
We learn this in 8th grade or so, our schools really aren't that good...
You're hell lotta lucky you didn't go to an Indian school so shut up and be happy with what you got.
Btw, here teachers don't even teach the full syllabus, they are too lazy to do so.
Lol in America, they just kinda skipped from "ok so here's how you know these angles are the same" to "ok so that's why the integral is really just an antiderivative"
@Poo Guy In Germany, the school adminstration has come to the conclusion that learning math (i.e. this formal stuff like algebra, calculus, proofs, logic etc.) is obsolete, because modern calculators can do it. So they redefined "math" to mean "solving real world problems using calculators". Then they realised that this did not work, because problems as they appear in the real world are too complicated. So they replaced the real world problems with invented problems formulated in a very obscure language that makes them look like real world problems for anybody who doesn't understand this language. And then they programmed the calculators so that exactly these problems can be solved by pressing special keys. Now everything works fine. Teaching math has become teaching the special language that is used in official exams and which kind of question requires which command on the calculator to be executed. The students get good marks, because it is not too complicated to associate certain words with certain keys. And if anybody dares to ask questions, you can simply give them the official exam from the previous year. Even a old-school mathematician will not be able to solve it correctly. But modern students can, proving that the school system is much better than ever.
I was schooled in Italy and I didn't imagine the situation in Germany was so bad.
In Italy all the educational system is still rooted in the fascist tradition of the cult of classical Rome and Greece. In high school you are forced to learn latin grammar and literature, even if you choose to attend the so called "Liceo Scientifico", which should be focused on science and math, and in some high schools even ancient greek.
Really useful for preparing you to understand the world and get a job in the 21st century!
Here is another challenge for who everybody who makes it to the end of the video: If you set up the mirrors like the whisphering dishes, that is far apart and you put Leia at one of the focal points, will you see her hologram appear at the other focal point?
Well, due to the large distance I think whatever made it look like a hologram wouldn't be there as the only light which would come out would be the parallel light making it look like a normal everyday phenomenon.
I think you would see it like through a magnifying glass. (distance between lenses and so forth) Would also work a lot better in the dark.
My guess is: in that case you wouldn't see the object from everywhere (like from the footage you have shown the hologram appeared visible from different points of view outside the "ufo" ), but only if you had your eyes exactly in the focal point. is it correct?
Yeah, why not
Yes, but only if you were looking in the first mirror in a direction that showed the reflection of the other mirror, a very limited field of view, so your head would be in the way. So that's a no then.
I remember playing a game called Time Traveller by Sega on a school trip sometime in the 80's. It's a laserdisc game with a control system similar to Dragon's Lair and Space Ace. But this employs a very old principal using opposing mirrors to create the illusion of levitation rather than the flat tv screen. The original design from Victorian times had a ball inside which would roll around the inside on the bottom mirror, but would appear to be spinning whilst floating above the hole above the top mirror. Hope this makes sense.
Great video as always.
McWomble.
math at school in Belgium was always very abscart and something i just had cram into my head for exams, and forgotten a day later. i wish I had better teachers like you. only now in my forties i'm getting interested again.
At school they tell me these mirrors are spherical, not parabolic, and I'm so annoyed by that... I even argued with my teacher about it and she said that we learn the general regarded physics and not some weird new inventions but I can write an essay on it and maybe win a Nobel Prize as well xD In case you want to know, Poland.
And then they would say that it would only work for "small angles" and never bother to explain the real maths
Same from Turkey here
same in Brazil
RandoM_ 11 even not that really. My teacher insisted that they’re spherical and argued with me. Probably because she never wants to mess in people’s heads. But I ask - isn’t telling sb false messing in people’s heads?
Cemalettin Cem Belentepe wow has’t Serbia removed you yet???
I am a math tutor in Ontario, Canada. When I was in high school in the 1980's, the lessons about quadratics and parabolas included the focus and directrix. For the last couple of decades of tutoring high school math students, this focus and directrix and the associated properties are generally not covered. Introductory complex numbers seem to get taught (1 week of grade 11 at only one school in my city) more often than focus and directrix (enriched program by one teacher, sometimes).
So beautifully explained Mathologer. Here in the UK from what I remember of my sons math classes at school the same problems occurred as you described.
Just got through precal last semester here In America, have taken algebra and geometry courses but never heard of the focus or the directrix. I learned about the secrets of the parabola in a Brilliant course
here in spain we learn about them in "technical drawing", can't really translate it (dibujo técnico) making it look like an actual name of a subject. This is about 2 years before university/other advanced courses
Weird, I learned about the focus, directrix, and others, in my 10th grade algebra in 1963 (not mis-typed) in my one-horse high school in rural San Diego County. I suspect that Maths education in the states is, and was, highly dependent on the teacher. My daughter did NO constructions in her plane geometry course in her high school Maths. =:-O
You are amazing as usual, how things became so clear with you.
I am extremely thankful to you just for existing there :)
New mathologer video!
I haven't done all the _math_ yet but this is _adding_ up to be a great day.
Does the fact that I get so excited for these uploads have anything to do with me being single? I would argue yes. Definitely yes.
High schools tend to be more varied in the quality of math education, but middle school and below feels just like Paul Lockhart described in "A Mathematician's Lament". For me personally, my math classes were nothing like your videos up until Calculus which felt like a breath of fresh air. Thanks for such amazing content, we need more math teachers like you!
Czechia: Somehow at the secondary school we never realized parabola is a single parameter shape, as teachers often mentioned shallow or steep one. I do not think they themselves realized they are talking about scaling of a single shape. At the university (math) no one mentioned it as it was obvious. I remember realizing it in the third year when explaining parabola to my cousin.
There were no smart examples in math whatsoever.
In India we are never shown the practical applications of maths practically. It is really sad that we never went to a mathematical museum.
But still looking at patterns of seeds of the sunflower makes me wonder how beautifully maths lies in the bottom of the heart of the nature. There can be no better museum of science than nature.🔥
When my son was in second grade, I taught him arithmetic the way I was taught (short example: subtract from most significant digit first, borrow from the answer). He picked it up easily and was bored with what the teacher taught. So I explained some basic algebra and trigonometry. He failed 2nd grade math. Solid 0. We had a meeting at school with the teacher, principal, education director and guidance counselor. Teacher says "he's deficient in math." I said "How about a demonstration?" So I brought the boy in the room and asked him "If we both start in this room, I walk 3 miles west and you walk 4 miles north, how far apart are we?" He drew the diagram on the chalk board, and then started guess-and-check to get the square root of 25. After about two minutes, he says "We're five miles apart." (Yeah, I pitched him an easy one. He was seven.)
The teacher asks "Well, is he right?"
The principal face-palmed. The education director groaned. The teacher says "What? Well, is he?" I looked around the room and said "I think we're done here. Fix the boys grade."....And then I took my son out for a hotdog and a slurpee.
"Well, is he right?" explains everything you need to know about math education in the good ol' U-S-of-A.
Thats a great story. Too bad its bullshit. Elementary schools do not give abcdf grades. They give referrals if students show signs of learning disabilities this has been law since bush was in office. They make sure that the students are segregated based on iq in order to midigate problems with learning. By the time kids are in middle school everyone is put in their place and everyone has a lesson perfectly dumbed down to their level.
@@baileymoody3916 isn't the school system in the US by state? If it is how can you make a country wide statement about specifics of the law?
@@AdrianParsons We have "Common Core" now. 41 states+ have adopted it. Now, that's what's taught, not how it's graded or how students are evaluated. I don't know.
@@LeeClemmer 41+? That's more unity than just about anything I have seen in the US!
(And I've heard nothing but complaints about common core & it makes me think of all the complaints about "the new math" I heard from my parents generation about synthetic division.)
@@AdrianParsons "no child left behind" was the term given to the legislation. It was federal and every state ratified it when bush was in office
I would never know if you don't tell you had now inspired me to see the beauty of maths most students didn't like maths because they don't know the beauty it has and knowing how this is beautiful is even more beautiful
I never got the love for math at school but with you, 3b1b and others I have. I never learned what I watch the nice way it's exposed and when I share that kind of video about the maths that I know friends do at school, they find it appealing and they always find a real use of it. A good way of approaching math is more beautiful but also easier, I think
The best math teacher I ever had just taught things through repetitive problem solving. I was one of those kids who got distracted really easily (which I think is the case with most kids), so back then I don't think I would have been all that fascinated with these applications. What stuck with me were just the basics of algebra and calculus, in a bit more of an abstract way than this. And learning these basics actually taught me to think mathematically. From there, understanding more complicated applications came naturally, even if I wasn't taught them directly. My point is that there is a fine balance between making a subject interesting, and cluttering it with too much information.
Consider this just as a friendly reminder, and in a small way an attempt to balance out a bit of the criticism toward the education system in this comment section.
Fair enough :)
In high school in the Netherlands I've learned about parabola's but it wasn't until my physics bachelor when they finally revealed how interesting they actually are. Optics professors tend to be much more excited about this feature of the curve because it tends to make lenses with a perfect focal point, effectively annihilating the spherical aberration errors we have to deal with because parabolic lenses are expensive to mass-produce.
Here in Switzerland it depends on the teacher in my experience. The first math teacher I had in high school didn't mention anything about these properties. However, after two years, we got a new math teacher and when reviewing old stuff we had to learn how to compute the coordinates of the focus point and the equation of the directrix.
Mathloger, I am also unhappy for the same reasons! I am an undergraduate university physics student. I've lived in the U.S. all my life, and I have come to a disturbing conclusion: So-called "education" is sometimes (not always) more about being indoctrinated and controlled, rather than learning, thinking independently, or being innovative. The problem occurs not just in math education, but from what I can tell, there are similar problems in all subjects. Students are treated like programmable calculators, instead of human beings who should be free to question, think, and create. No single university professor or teacher is to blame for all this. The source of the problem is deeper and more systematic. I think it spreads throughout the human population. But why? How did this begin? Why does this problem propagate? Can we trace these problems to the first human civilization or is there some kind of conspiracy to turn us all into robots? I would very much like to know.
Here are some interesting quotes from the famous book "Proof is the Pudding"
"There is a grand tradition in mathematics of not leaving a trail of corn so that the
reader may determine how the mathematical material was discovered or developed. Instead,
the reader is supposed to figure it all out for himself. The result is a Darwinian
world of survival of the fittest: only those with real mathematical talent can make their
way through the rigors of the training procedure."
"In this sense Bourbaki follows a grand tradition. The master mathematician Carl
Friedrich Gauss used to boast that an architect did not leave up the scaffolding so that
people could see how he constructed a building. Just so, a mathematician does not leave
clues as to how he constructed or found a proof."
So, there is a "grand tradition" of competitiveness. Instead of seeking to pave the way for all children to surpass us, we are to worried about competing against everyone else. Perhaps this is part of why our education system sucks. Just an idea.
Wow I was also thinking something of the sort but you put it into better words than me! My goal is to one day become a teacher and teach kids to question themselves, the teacher, and the knowledge itself. Is it too ambitious and maybe disruptive to the system? Some might fear it but I think it will be the beginning of a more open-minded, less fearful and more daring society.
2nd grade-undergraduate level math/science tutor here. I'm a generalist and differential equations, multivariable calculus, and linear algebra are *about* where I draw the line right now.
Fascinating insights. Thank you for this. I ALWAYS tutor my students with the goal in mind of making sure they never need another math/science tutor again. Which means I am trying to put myself out of a job. Which only increases my competition in the future as more students are empowered to see how bull**** our system of competition really is. I don't care, though. I want our future generations to have the opportunities to BUILD and GROW on what I, PERSONALLY, have achieved. I don't need some artifice of "grand perfection". I want someone in the future that I've personally engaged with to smash down all of the supposed mathematics that I've taught them and to show me in great detail how utterly wrong I am. Because I'm always wrong when I teach. The concepts, connections, and applications are far more important than the grades and "mathematical rigor" that is focused on in Texas elementary-high school mathematics. Don't even get me started on university maths, because I don't actually have anything to say besides .
love
Coincidentally, I looked up and was reading about the focus/directrix definition and properties of parabolas and other conics this morning, and then saw your video this evening. I had run across the terms before, but I was not taught them, to my recollection, in my school mathematics, much less the properties that flow directly from this classical treatment. I found the video as delightful as it was fortunately timed. Thank you for such a clear and wonderful demonstration!
I always introduced parabolas with the paper folding activity. (Victoria - Australia). Next, students repeated with dynamic geometry (locus) and loved exploring the different forms of the parabola in this very interactive and visual manner. Lots of applications before we started graphing y = x^2. I tried to emphasis how beautiful mathematics can be ... such an amazing curve with so many properties with such an elegantly simple algebraic form. My methods, however, are not indicative of the typical introduction and certainly not 'reflective' of the standard textbook approach. Perhaps its about time we 'shifted the focus'. Well done Mathologer on another brilliant video.
You could augument it by showing how cuberoots can be extracted by folding. Something you can't do with ruler and compass.
here at my high school in new mexico I work as a scholar tutor (meaning the school pays me to tutor my peers) and while most of the kids I get haven't gotten to algebra yet (we're a combined middle and high school), I still try to incorporate intuition about the math we cover the same way your and others' videos do. I haven't taken a math class for a couple years but from what I remember nothing has ever been motivated or interesting-you just drift along a river of math as you solve endless problems to hopefully row to shore, though most end up drowning.
I'm from argentina, I think having a good explanation of things is up to the teachers, some will be super energetic and try 300 different ways of explaining to you while others just want you to do your homework, most times we dont get explanations for uses, I remember someone asking "What do we use complex numbers for?" and of course the answer was "Physicist use them for difficult things"... on the upside we had 4 classes more or less about fractals and their uses nearing the end of our last year but that's it
Super nice explanation for why the crease is a tangent to the parabola, requiring no equations.
The key fact is that, as the parabola is the set of points equidistant to the focus and the directrix, it divides the plane into two parts, where points "inside" the parabola would be closer to the focus than the directrix, and points "outside" the parabola closer to the directrix than the focus.
Consider a crease c folding a point P on the directrix d onto the focus F. Clearly, for all points A on c, there is AP = AF due to the folding.
Now, erect a line through P perpendicular to d, intersecting c at T. Clearly, T is a point on the parabola, as TP = TF by T lying on c, and TP is perpendicular to d by construction.
Also consider another point B on c, different from T. We still have BP = BF, but as BP is not perpendicular to d, BF would be larger than the distance from B to d, hence all these points are "outside" the parabola. Therefore, T is the only point on c that is on the parabola, and all other points on c are on the same side of the parabola.
Quite Easily Done.
Also, a nice extension: we know that, by folding all points on a circle towards a fixed point, the creases would form tangents to an ellipse (fixed point inside circle) or a hyperbola (fixed point outside circle), with the center of the circle and the fixed point as its foci. This fact can be proven by a very similar argument, considering the division of plane by the ellipse or hyperbola into regions in which the points satisfy certain distance inequalities, and noting that the unique shortest path between two points is the line segment joining them.
I loved the T-Shirt with the quadratic formula!
Yeah, it's a nice one. But it would be even better if they'd used the same owl for the coefficient a and also the same owl for the coefficient b :)
@@Mathologer I was wondering why they didn't. More maths should have owls for variables.
@@Mathologer I noticed that howler too.
@@PopeGoliath it's not math, it's "owlgebra" :D
@@Mathologer : it's owlful... :D
I've heard about this from a very application oriented calculus book (my first one couldn't find anything else). they are properties of different conic sections. the coolest for me is the ellipse which can concentrate waves at 2 special locations corresponding with the foci. so you could hear something very clearly far away in an elliptical room. it's also where I first learned that parabolas can diffuse and concentrate any waves because of their interesting shape. There being applications in the form of special mirrors and glasses. Really glad I read that chapter because that's not the kind of thing you learn in pure math or at least not without any emphasis. More of a side result/application but really cool.
ex-maths teacher here. I tried innovating maths education by having my kids learn with Khan academy. It was working great, especially for the weaker kids who could learn at their own pace. Soon enough parents started to complain because I was "experimenting" with their kids. The head teacher (dik-head I should add) quickly mandated I revert to the traditional pedagogy and boring book with rote exercises and homework. Only a new society will bring a new education.
two independent thought alarms in one day! willy remove all the colored chalk from the classrooms
@@azathoth00 You allow colored chalk (markers) in your room! Heresy!
As an aspiring teacher, this is one of the things I fear the most. Even if I want to do what I think is best for the students' education, I might not be allowed to.
youtube is a resource for learning new things. I wish more people would realize that instead of just relying on a boring textbook which might be teaching in a way that is difficult for that person to understand.
Dear Michele Bagaglio, I would advise teaching math(s) outside the U.S. at an international school - preferably IB. You get to do cool stuff, and use Mathologer videos because you will be the "wise American" bringing "American innovation" to the classroom.
6:44 In my hometown they have a pair of those! They are on the "Parque de las Ciencias" (literally "Park of Sciences"), which is a science museum with a lot of interesting things to do and to watch (as well as conferences given by lecturers from our University). If you ever visit Granada in Spain, don't forget going there, you'll enjoy it. And being a mathematician you will probably enjoy the historical architecture (the Alhambra, the "town or "casco histórico" -literally "historical bucket" in Spanish-) as it has a lot of interesting mathematical patterns. The Spanish muslim medieval arquitecture has a lot of mathematics on it, taken from the old Greek and Roman geometrical patterns.
Sydneysider here and my maths teachers weren't paid enough to try and supplement the syllabus with "real world examples". Some of them tried anyway, for which my classmates and I were grateful. I can honestly say I learned more from 3Blue1Brown, yourself, Matt Parker, Brady Haran, and Vi Hart than from the advanced mathematics my high school offered.
In the Greek book, there is a small reference on mirrors (a small paragraph), but there are mostly equations. Thank you very much for creating this video.
As a student from MWHS in Madison, WI I was blessed with an open minded public school system. Without a doubt some instructors just read the textbook but for most of my STEM field classes the teachers were enthusiastic and brought up new ways of looking at things.
Hi, I’m a Jr (11th grade) in high school, and last year my teacher actually told us the fundamental theorem of algebra WRONG. The explanation she gave us was that polynomials with odd number orders tended to infinity with ends pointing the opposite ways, and even orders eventually pointed the same way. It’s honestly awful that our education systems hire teachers who don’t even know their subjects to teach curriculums that completely miss the point. For instance, in trig classes, way too much emphasis is placed on knowing how to graph ANY transformation of a trigonometric function, rather than more important topics like polar coordinates. Videos like yours are honestly the only way to learn today so thank you so much for continuing to grow our ability to understand the world around us.
Ps. You might be thinking “oh well his school is underfunded etc etc.” No. My school wins awards for being the best staffed and best test scores in our state almost every year. Makes you concerned for underfunded, inner-city schools right?
I'm studying to become a math teacher in California by the fall. Resources like Mathologer and Numberphile are so useful as I stockpile engaging lesson ideas that relate to curriculum. Thank you for this and the many great videos you've produced. I want nothing more than to stoke the flames of mathematical curiosity and excitement in my future students so that they don't feel that the standards they are learning are just arbitrary tedium they have to do in between studying history and reading the classics. Just because not everyone CONSCIOUSLY uses mathematics of this kind in their daily life or profession, does not mean that they aren't surrounded by exemplars of fascinating mathematical principles. Exciting kids about math shouldn't be any harder than exciting them about Physics and other sciences, as math is the basis of science and capable of explaining anything they might have an interest in: engineering, music, sports, games, language. It all can be viewed, quite usefully, through a mathematical lens.
Thank you again!
Have you succeeded in achieving this ideal?
For people who enjoyed the connection with origami, there's a lot of other cool math relating origami and the study of polynomials. There's two main ways I know of that polynomials and origami interact: Algebraic Geometry and "Galois Theory." I did a reading course of the Galois Theory of origami last year, and got to read a lot of cool resources that others might find interesting. In particular, with usual origami you can solve any cubic equations, which makes it more powerful than a compass and straight edge (1), and if you allow two folds at once you can solve all degree 7 equations! (2) In fact, if you allow an arbitrary number of simultaneous folds, you can solve polynomial equations of any degree. (3) This last paper is particularly interesting, because the proof uses a method call's "Lill's Method" for solving arbitrary polynomials. This method was discovered over 100 years ago, but hardly anyone knows it anymore, since it was never really useful for anything. So, the fact that its now being used in a paper alongside much more modern techniques like Gröbner Bases is pretty cool to me. That paper also defines "origami cubics" which are defined using exactly the trick proved at the end of the video, which is what made me think to go on this whole tangent in the first place. (I also think there's a lot more connections between the Galois Theory of origami and some more modern Algebraic Geometry that haven't been fully explored yet, but idk, I'm just an undergrad.)
There's also a ton or origami math that has nothing to do with Galois Theory, like Erik Demaine's work on computation origami, the cut and fold theorem, etc; Robert Lang's work on Tree Maker, Reference Finder, and other math related to origami design; and a ton of work by various other mathematicians on geometric origami.
(Disclaimer: When I say solve all polynomials, I really mean find all real roots or polynomials with integer coefficients, though I think there are some extensions of this, that are just much messier.)
(1) Solving cubics with 1-fold origami
www.researchgate.net/publication/233592288_Solving_Cubics_With_Creases_The_Work_of_Beloch_and_Lill
(2) Solving septics with 2-fold origami
forumgeom.fau.edu/FG2016volume16/FG201625.pdf
(3) Solving arbitrary degree polynomials with arbitrarily many folds
www.langorigami.com/wp-content/uploads/2015/09/o4_multifold_axioms.pdf
(4) Lill's Method Explanation
www.math.psu.edu/tabachni/prints/Polynomials.pdf
(5) Lill's Method Original French
eudml.org/doc/98167
(6) Erik Demaine's Papers
erikdemaine.org/papers/
(7) Robert Lang's Papers
langorigami.com/articles/mathematics/
(I really just had a bunch of links that I thought were cool, and wanted to share lol)
Thank you very much for sharing all this :)
This is a good math channel. The stuff you teach here makes nonsense sensible. Classes should make the teaching materials more like this.
Hi, Prof Mathologer.
I grew up in Melbourne and went to Camberwell High where we had some pretty damn good math and science teachers in the late 70's and early 80's. In later high school (years 9 and onward) we typically had various different math classes. I did pure and applied mathematics and the applied math was the most fun. In it we learned the typical things like solving differential eq'ns for flows into and out of tanks, applications of probability and statistics to practical problems and so forth... BUT in pure math we learned all of the usual stuff almost entirely devoid of real world applications. I guess that made sense with pure math and, although I understood applications for parabolic (and other conic section) curves, hyperbolae, etc,. I could not for the life of me understand, apart from rotating things, any extensive use for the equations for a circle nor the use of complex numbers.... until I went and did electronic engineering at Melb Uni. Then, circles and complex numbers were damn well everywhere. I am grateful to the high school teachers I had, they were very good and inspired a great interest in math and science in me resulting in a, thus far, 30+ year career in engineering. However, I cannot fathom what has happened to education in Victoria in this day and age. It is as if they are trying to make it as unappealing as possible, perhaps to stupefy everyone into some kind of non-STEM field career. In the 1970's, Australia had a growing industry in electronics but subsequent governments, whether left or right, have done all in their power to destroy all such industries. I know this sounds conspiratorial but it is as if some outside influences seek to make this nation into one of dumbed down obedient dullards willing to dig holes and wash cars for a few scraps of cash. I have nothing against those who do such work for a living, but something seems to be undermining high tech in this nation.
In South Africa, not much attention is given to the properties of mathematical objects in class. You're pretty much just expected to rote learn everything. But I really enjoy intuitive and visual explanations. Thank you Mathologer!
(Romania High school)
The property at 2:03 we learnt in high school, but in our math class we only done exercises (derivative, complex numbers, quadratic equation, quadratic formula). So highly theoretical.
There could be some examples given, but few. At exams you need to solve exercises and to memorize-paste the theory.
The practical application or curiosities were reduced. In the explanation part all the graphs were drawn (at least in 2015), but a animation can not be drawn.
For example the analogies of derivatives and integrals, you could seen more easy from an animation that the integral is the area under the graph, sum of a lot of tiny rectangles.
In the actual lesson, the teacher draw the rectangles with which the function is approximated and give the formula of this ( with epsilon csi etc.).
A lot of complains are the very high number of theory required to learn.
At college, course Special Mathematics, we learn about differential equations, the professor presented only the methode of solving, the amount of theory was reduced.
A lot of problems, but i like a lot the youtube educational community (all of it, mathematics, history etc.) that actually explains in a simplified manner a subject.
In Serbia, most high school students aren't thought the intuition behind anything.
We are thought to calculate derivatives, without mentioning their essential geometric properties.
Nothing is proven and we are thought to carry out calculations like mere computers.
So basically you have education identical to everyone else.
@@galbatrollix5125 Yeah, shame ...
*taught
What is the value of one? Tje question vs the answer.
They obviously didnt teach you grammar
By the way, I discovered the physical significance of the directrix for myself while doing ripple tank experiments while teaching physics. If a plane wave hits a parabolic reflector and deflects to the focus, the directrix is where that plane wave would have ended up at the same time if the reflector hadn't been there. So the parabola can be thought of as "transforming" a line (the directrix) into a point (the focus) through reflection. This gets to the equal time property you mentioned and accounts for why the focused light waves interfere constructively at the focus.
There was a screen saver that was drawing lines and often created parabola shapes.
Some artwork made of a wood board with two row and two columns of nails and threads connecting between the nails was also creating parabolas with the added bonus of a 3d effect of threads intersecting.
The code that draw this kind of shape is actually two bouncing balls with line draws between them as they move in discrete steps.
I was struggling with your t-shirt joke until you revealed the point. Owl-gebra, really smooth.😂😂
You are the only one who reminds me that maths can make me smile!
8:40 "... Don't believe me, just watch..." I started to dance like that Boston Dynamics robot )))
11:50 "Our educational authorities never miss an opportunity to miss an opportunity ". I can't be more agreeing.
My experience in Texas is that we learned about the focus and directorex in Algebra II, which is in the 10th grade, and the cool properties of parabolas in Physics under the Optics and Wave Mechanics units, 11th grade (though the equals-x explanation was new 😁).
Idk if it was just my school district, but the public education in Texas is surprisingly good in comparison with what I hear from other people online. Art, Music, History, Math, Science, English Language, Literature and Writing are all given a rather high priority, despite also getting three years dedicated to just Texas history. 😅
Honestly the biggest downside is that there is little to nothing about practical subjects. No personal finances like paying taxes, setting a budget, ect. Health is limited to "do drugs and you'll die" and "have sex and you'll die". And there is no philosophy or discussion of religion at all (which means you get one day of watching videos in Biology for the evolution unit).
All this stuff you're just supposed to "get from home"
I can't say enough how much I wish this was true for the rest of the states. I noticed a dramatic drop in the quality of education when I moved from Texas to North Carolina. Without even trying (and I literally mean that I didn't have to try hard at all), I went from the dead middle of my class of 120 student class in Texas while barely passing to graduating 36th out of around 205 with a 4.2 or 4.3 weighted (I can't remember which) GPA from a fairly decent school here in North Carolina. topics like vectors, matrices, and actual probability weren't taught to me until precalculus my senior year, which was a good bit after I had taken the ACT and is one of the more common routes people take at my old high school. I'm just glad to be out of high school and finally in a place where people care more about education so I can finally learn more about calculus, linear algebra, and differential equations.
I remember from math class that my teacher told us to find a spot on the schoolyard which has an equal distance to a given point and line and we all "magically" lined up to form a parabola. I think this was a nice way to first try to spark some curiosity before he actually explained why.
And something a little less related. Parabolic reflectors which can be used for example for wifi do use a specific focal distance, such that the direct path from the focal point is exactly one (or maybe a bigger integer) wave length(s) shorter then when reflected, such that both signals are in-phase and give a slightly stronger signal strength.
i just read a book by richard feynman where he described working on a committee tasked with selecting course books for elementary schools. he was distraught with the system and how underwhelming the material in the books were, and this was of course several decades ago. the issue was that the committee members were often not qualified to assess a range of subjects, and these sort of "text book lobbyists" would corrupt the system. at the end of the day, the quality of the lessons in the book weren't the determining factor in what was handed out in schools. i fear that system hasnt changed
The distance between a point on the directrix and the focus is always equidistant as the line between them is always perpendicular to the tangent of the parabola. By applying Pythagoras we can obtain that the point above the point on the directrix will be on the tangent which would make another right angle triangle if we join those points. That triangle would be a 45 45 90 as they are the half of a right angled triangle and henceforth the points lying on the non 90° will be along the same axis.
I'm currently in ap calc, and schooled in Florida. Of all the math classes I've taken the only thing you mentioned here that I've heard before are how to find focus and directrix, and that lines perpendicular to the directrix will reflect towards the focus.
Australian here from NSW. It has been quite a few years since I was in high school, but math classes back in the mid 2000s did include the focus/directrix definition of a parabola. Either in Math Extension 1 or Math Extension 2. Not sure where you would find it in the Victorian curriculum, but it's probably there somewhere.
American here from New Hampshire
Math education in high school isn't that bad here, I actually had to write an essay about the applications of conic sections in construction for my pre-calculus class
Though we didn't do anything quite as cool as this lol
Here in Upstate New York, we weren't ever taught any real or fun applications for math, except indirectly through the occasional word problem. The focus and directrix weren't ever mentioned when it came to parabolas.
I think it mostly comes down to the teacher, rather than location. I don't think most grade school math teachers have a particular passion for the subject, and the pay levels aren't enough for them to go out of their way to find ways to make it fun or interesting.
It's sad, really, and definitely needs to change. I hated math in school; it felt like tedious memorization of rules that'd be irrelevant in the real world; can only assume others felt the same way. I only got interested in it beyond its utilitarian uses well after high school, when I stumbled upon videos like yours. Thank you for them!
You are the best math "loger" on the internet... thank you for all your great videos!
In an international school in the Netherlands, And we also never learn about this stuff...
It’s so simple and clear and satisfying, why isn’t this taught everywhere?
yes, we studied this thing about parabola in our 5th grade.
ellipse also has this property that you can whisper something standing in one focus of an elliptic room and the other person hears you well standing in the other focus.
This is super cool! At my old college, our quad has a parabolic shaped building and one day when I was leaving to the parking lot I heard someone talking right at my ear and freaked out. I looked back to see who it was and saw some people standing talking way far from me. I knew it had to do with the parabolic shape since I remembered learning about parabolic dishes in physics and diff eqs class but I never pinned down exactly what happened there that day. I'm tempted to go back with a range finder and see if I can take some measurements and calculate the locations of the foci!
In British Columbia, Canada, I tutor high school pre-calculus. The kids learn how to factor quadratics, but I haven't seen them studying much in the way of proofs, such as those behind the quadratic formula or Pythagoras. I *have* seen an intuitive derivation of the surface area of a cone based on that of a pyramid. No directrix in sight. I would like to see students given fewer problems and told to suck the marrow out of them. This is why I started a local MathTetminds group and RUclips channel. To make math more like finger painting.
Enjoyed this video immensely. I like the Mathologer videos. They make me think. 😎
I didn't learn about this until i took a community college pre-calc class! interesting how many ways there are to describe familiar shapes and curves
Your sermon goes down to the heart of all education. Curriculum seems to be giving us facts and formulas and it expects us to learn it and use it. I don’t know what their definition of learn is, but it doesn’t work. It takes thinking in order to learn something, not mindlessly plunging numbers into formulas. Even if you mention this problem for math, it happens in every subject from English, History, and sometimes Science. Learning is about understanding, questioning, and experimenting yet it is never taught in a way that is learnable. I have talked to a teacher about doing problem solving exercises, where the students take the time to figure out a proof for some important thing in math. Her response was that not all kids are good at problem solving. My thought is that it must start in elementary school, in order to build up those skills, something like finding the area of a triangle. “Given a triangle, could you find it’s area, could you find a formula that works for any triangle?” I would love to try and take this as far as it goes but it is a huge change that most teachers would not agree with. I hope that someday people could learn. Actually learn. Asking questions is the way that new math gets invented, and new research, new facts. Can we get kids to ask the same questions?
Just to add to this, I was given a false geometry theorem about circle radii, angles. and arc length, all relating because circles are similar. I noticed that I was ambiguous and proved it was false. I wonder if my math teacher knew any math or if she was just following the book.
American from southern California, Algebra 2 was about 10 years or so ago for me. I don't think the directrix was even introduced in our classes. And if it was, it surely wasn't utilized in proofs until precalculus, which ended up being in junior college for me. As a Math Major I really enjoy the simple elegant and visual geometric "proofs". Love the videos
100% agree to your little rant on how maths is taught in many places. I'm studying maths/physics for becoming a teacher in Austria and I'm regularly shocked by how little a significant portion of my soon-to-be-teachers colleagues knows about the actual practical applications of maths. And in almost any school textbook there are the same old boring pseudo-application problems - and yes, in one of them a bike path in the shape of a polynomial function is featured as well. It's really kind of depressing, that many students will leave school, thinking that the pinnacle of calculus is determining the volume of a liquor glass in the shape of a paraboloid...
Great semi-obscure topic selection, awesome visual and math explanations that connect to reality!
I live for these videos. Thank you so much for your awesome content!
The paper folding ellipse construction leads directly to a proof of the reflection property of ellipses, analogously to the parabola proof. BTW, there is a reflection property for hyperbolas analogous to both of the above. A ray heading toward one focus of a hyperbola will be deflected, by either branch of the hyperbola, to the other focus. This is used in Cassegrain telescopes. You may know this, but many people don't.
All very good points you are making in your comments :)
In California public school we found the foci and directrix (or its analog) of all the conic sections, especially as it pertains to drawing a graph of an arbitrary conic by hand. The parabola was defined in terms of the equidistant property and we learned about the reflective property. The relationship with the whispering dish-style setups was mentioned offhand but nothing about tangent property or the math of the whisper dish.
My high school (California) had examples related to planetary orbits for ellipses and their foci. They didn't bother with the application of parabolic foci, but the foci lessons were bundled together. I think physics classes took advantage of this more than math classes. Certainly my middle school algebra education did not dabble in this, but my math teacher then was not the most enthusiastic educator.
The little mathologer's have a very similar experience to mine in class. I did learn about some cool math in my math club. I personally always enjoyed math as a kid. I'm not sure why, but I always found solving math problems fun regardless if there wasn't good motivation for the problem. In high school and university I found that well motivated problems are even more rewarding than the problems I faced as a kid. In general it was very instructor dependent. I remember loving Calc 3, ChemE Math, Geometry, and combinatorics because the instructors had good examples and taught in an organized manner, but I hated Diff Eq and Number theory because the class was all over the place.
I am in the USA have two teenage boys. I was so excited when the oldest got to High School Geometry since for me that was Analytic Geometry for me with its Geometric Constructions and Geometric Proof - where the math had to move past memorization into analysis/understanding (from what to why). Sadly, I talked to the teacher and it was going to basically geometric shapes, formulas, and trigonometry. He too wished he could teach Analytic Geometry; but, it was not in the curriculum. I suspect some of the why is the standardized testing model. Memorization is much easier to test and teach to. What is always easier tor test than why. I think another reason is that many students struggle when you try to move them past memorization and so the short pole can sometimes set the height of the tent rather then the tall one.
Well, I haven't let my boys get off with just that (at least not my oldest - who really enjoys math). I think that to fully grow, everyone must be able to more past memorization. You need to understand not only the what but also the why. No brain is large enough to memorize everything and not all results as as convenient as to be memorize-able. Understanding the why breaks the door to infinity.
Good luck with your boys :)
All I have to say is... I love this channel
I actually learned about this in middle school (7-9 grade) and it was explained very well but it was probably because we had a special program with a university professor. I live in Poland.
As a current chemistry graduate student, I remember learning about the parabola, focus, and directrix (idr how to spell that), but none of these particular properties. To be fair, I grew up in Michigan in the U.S. and attended public schools. For those who don't know, Michigan is ranked worst for public education in the U.S. (It's where Betsy Devos came from, i.e. our current secretary of education). The directrix and focus at that point felt like such transient topics. Their significance was not made clear to us as middle schoolers making our first real foray into algebra. Such a simple explanation would have helped, and probably would have led to higher engagement in maths across the years.
Thank you so much for your explanation on properties of the parabola....sans hyperbolic statements ;-)
My children were educated that the local community college for their high School education'sbut did not receive such elegant and complete an explanation as yours.
Cheers from Seattle!
Austria (that one without Kangaroos ;) ) - I am doing some private tution for age 14 to 18 and I found out (most) teachers still miss the point of getting young people interested in math and make it "easier" for them.
e.g. it took me just two pens and a sheet of paper to explain how to measure the angle between a straight line and a plan (and I was not drawing any formulas or graphic ;) what has been never shown or discussed in classes.
So no things pretty same around the world. School books are made by people who either hate kids or are completely ignorant about teaching - and poor dedicated minds have to get this fixed. Like you do, thanks a lot.
Bro.. I don't think this is about books. These things should be taught by teachers in schools because books cannot have everything.
@@Saisai-uh8do Agreed I was not using the right words - With School books I do actually summarize all the rules, curriculas and material provide to teachers to be used with less and less freedom to really TEACH something. Instead they have to follow some "standardize quality" levels to produce uniform students. If the students don't fit into those standards they will me put aside. It is not only about what you learn but HOW you learn, and some people need different approaches to understand but in the end they might outperform others.
The equal-optical-path-length property goes hand and hand with the focusing property of parabolas and applies to any focusing device (lenses, mirrors, combinations, whatever...). It's a consequence of Fermat 's principle which is a consequence of Huygens principle.
I live in america. And the particular high school that i went to quite literally graduated people who did not numerically "pass" in order to meet a deadline so they could remain a school. Math class was taught by football and cheerleader coaches. My chemistry teacher did not know that water boils at different elevations
My chemistry class was taught by a football coach that was reading the textbook at the same time as us. He would read a chapter the week before us and then teach it... >_
A bit late for this, but in North Carolina high school classes, bits of conics were taught. We were taught about the directrix, focus, and how that light entering a parabola would always reflect into the focus of said parabola in precalculus class. We, however, were not taught why light reflects into the focus of the parabola (to my recollection). The only actual proofs of anything mathematical taught to me in high school was how to derive the distance formula, and how to find the lengths of the sides of a few angles on the unit circle. This though wasn't the most egregious thing though. During my senior year at high school (which was 2018 for me), I was given a wonderful opportunity to compare the precalculus textbook and the math 2 textbook (for reference, there is a year of math between these two classes). In it, I discovered that the two had nearly the same exact topics covered in them, including (if my memory serves me correctly) conics. Even though the two textbooks might have had varying levels of difficulty (which probably isn't much), nothing new was really being taught to students. Math was incredibly easy in high school for me, and one didn't have to put much effort in to make an A in the class. This saddeneds me. I had begun to develop a passion for maths in high school but felt cheated out of a decent math education because of the curriculum. It's nearly a joke with how easy it is. Precalculus in community college hasn't taught me anything new either, but I half expected that given that they have to make sure that everyone is caught up to the standard of the school.
I forgot to mention that most people are also not taught the intuition behind most anything taught in class (at least in the curriculum I was in). This is why teachers who go into why stuff works mean a lot to me. I learn best when I understand why something worked because I typically remember the process of deriving the formulas better than the formulas themselves. This also makes it easier to understand later topics because the intuition typically builds on itself. And yet, I still remember the teacher in my precalculus class in high school deducting points because I didn't follow her way instead of the way I learned by myself. Hell, I made my own factoring calculator using a method never taught to me that I still use today to check my math because I saw a simpler method.
Express proof:
Every point on the crease line is equidistant from the focus and the target point (the line is a bisector). The point on the crease vertical to the target satisfies this property, making it on the parabola. The line is then tangent to the parabola as there can only be one common point, since only one point of the directrix is folded onto the focal point.
Sorry if you don't understand, but I lack mathematical vocabulary in English.
Nice :)
Proof for the challenge: the first thing to note is that on one side of the parabola are those points that are closer to the focus than any point on the directrix, and vice versa for the other side.
And the crease is the perpendicular bisector, which separates the plane into points closer to that particular directrix point and those closer to the focus.
So the line being tangent to the parabola is the same as saying that if a point is closer to this one particular directrix point than the focus, it’s closer to some point on the directrix than the focus, which is obvious!
I have a different explanation for the property you proved about that line being tangent, the line you need to fold by is simply the perpendicular bisector of the focus and the point on the edge, which is all the points with equal distance from both points, when does intersect the parabola? The distance from the focus is just the distance from the line on the bottom, so we want the distance from a point on the parabola and a the point on the bottom to be the same as the distance between the point on the parabola and the line, and that happens only when the point on the parabola is right above the point on the line, so that process that it's tangent and that the point in the parabola is right above the point on the line
Hungary, nearly 40 years ago. Geometric properties of the parabola were taught in "math hardened" classes around the age of 16 of the students. More than 3/4 of the students were never hear about. There was a nice problem, I remember. Given two tangent of the parabola (the parabola is arbitrarily placed on a plane), determine the the curve itself! (That is: give the position of the focus and the directrix.)
I always had an intrinsic love for mathematics but the day I found your videos , I am more than happy. I always wanted to see mathematics comming out of paper and doing real things ( bcz in schools we have only solved the long equations . Our teachers would leave the chapters on graphs saying " these things will not be part of paper " ).
At school I never learned about the focus and the directrix when learning quadratics. But I read about them in books about geometry, which never explained how the quadratic equation arose from a point and a line.
In physics I learned that a parabolic mirror can focus a beam onto a point, but it was never explained that it was a mathematical necessity that the mirror be parabolic.
A simpler proof of the end is just to say the line is the locus of points equidistant from focus and the specific point onthe directrix, which we know only touches the parabola in exactly one point since the distance from said point to directrix is minimal thus the line is tangent
2nd year Canadian Math and Physics major plus high school math and science tutor here and I've grown frustrated as I realized that so many of our students are not being taught these applications of functions and relations --heck I was doing a mindyourdecisions puzzle the other day and didn't know how to solve it because in sixteen years of math schooling, I had never been taught the Intersecting Chords Theorem and 90% of our Calc III class had never been introduced to conic sections. While this may not necessarily be horrible for people with a natural curiosity/intuition for math --it puts many students at a great detriment.
My last tutoring session with a grade 10 girl, the poor kid was at a complete loss for how to solve for the equations of a triangle's median or a line's perpendicular bisector --because NO ONE had taken the time to explained the Geometric meaning of those words. Once we broke it down by definition and addressed a few of the more fundamental formulas, she basically came up with the correct procedure on her own. It just irritates me to no end that the majority of educators in our education system don't take the time to help our students develop the skills to interpret more complex problems or understand the properties that arise from them