The Secret of Parabolic Ghosts

The Secret of Parabolic Ghosts

Welcome to another Mathologer video. Have a look at this. What do you see? Yes, some cool Mathologer toys in the background but ignore those. Yes, I know, it’s hard but ignore the toys 🙂 Focus on the black UFO
at the bottom. On top of the UFO is a circular mirror and on top of the mirror
is a coin featuring pretty Princess Leia and her trusty robot r2d2. Now let me
pick up the coin. What!? My fingers are passing right through the
coin. Is this some kind of Jedi mind trick? What I want to do today is to explain
our ghostly Princess Leia as well as a closely related way to conjure up
ghostly voices. It’s really cool to be able to create these ghosts but today’s
maths is also super applicable. In fact, if this maths ceased to exist from one day to
the next you wouldn’t recognize the world you live in anymore. Well let’s get
to it. It’s all got to do with our high school friend
good old x-squared. Sticking with movie references, x squared is a little bit
like Clark Kent. Most of Clark’s friend think they know
all as to know about him but only a select few are aware that he’s actually
Superman. Same with x squared which also has some hidden superpowers that hardly
anybody knows about. Okay, here we go. Did you know that the point (0,1/4)
and the horizontal at y=-1/4 are super special for the
parabola y=x^2. The point is called the focus and the line
is called directrix of the parabola. What’s special is that every point on
the parabola is exactly the same distance from the focus and the
directrix. So these two distances there are always the same.
That looks tricky but to show it we just need help from our other school friend
Mr. Pythagoras. Here we go. If our parabola point has coordinates x and x
squared, then Pythagoras tells us the square of the green distance. And going
straight down to the directrix the square of the yellow distance is this.
Now it’s just a matter of going on algebra autopilot to check that these
two expressions are equal. And that shows the two distances are the same.
Easy-peasy. The directrix is the secret ingredient for lots of parabolic magic.
For the first magic trick, let’s position the parabola on a piece of paper so that
the red directrix coincides with the bottom edge of the paper. Now look at any point on the bottom edge, that one there. Fold the paper so that the black point ends up on the focus. So there fold, fold, fold. Right
on top. And unfold again. Okay it looks as if the paper crease is a tangent of the
parabola and that the touching point is right above the black point. And looks
are not deceiving. Starting with any point at the bottom, folding results in a
crease that is tangent to the parabola. If you do this for all the points of the
directrix, you get all the tangents of the parabola. So why does this work?
All those tangents suggests calculus but you really don’t need it. All you need is
a little middle school OWL-gebra 🙂 Anyway I’ll leave it as a challenge for
you to give a proof in the comments and if you’re desperate
I’ll give one possible proof at the very end of this video. Anyway for the record
let’s note that proving the second super property of the parabola is also easy
peasy. The second property gives a really pretty way to create a parabola from
scratch without having to calculate anything. Start with a piece of paper,
mark a point close to the middle of one of the sides and perform the folding
action for a bunch of points on the side. Then the parabola materializes as if by
magic. Super super nice 🙂 Okay after this little piece of paper magic we’re almost
ready to conjure our ghosts. We just need one more super property of the parabola
and actually you probably all know this one, although I’m guessing that only a
few of you will have seen a proper explanation. Looking again at our paper
folding notice that this green triangle there
gets folded smack on top of this identical pink triangle and that means
that the green angle and the pink angle are the same. Right?
Then the angle opposite the green is also of the same size. Now for what we
are after we just need that these two angles here are the same. We also don’t
need the directrix anymore, so let’s get rid of that too. There’s not much left of
our picture but it tells us something super interesting.
Imagine the parabola is a mirror and the vertical line is a ray of light hitting
the mirror. Then this ray of light will be reflected like this and the reflected
ray will pass through the focus. But of course the same is true for any vertical
ray of light and so all the vertical rays get focused on the, well, focus 🙂 Lucky
that that’s what we chose to call it. Of course, this also works in the opposite
direction: any ray emanating from the focus will be reflected into a vertical
ray. I’m sure that many of you are aware of this focusing property of the
parabola and it’s myriad applications in the guise of parabolic reflectors and
mirrors. Now finally we are ready to conjure some ghosts. To begin let’s add
another parabola to this picture like so. Then it’s clear that a whole sector of
rays emanating from the red focus will end up passing through the green focus.
Chances are you’ve seen the setup before in the guise of the mysterious
whispering dishes at science museums. A whispering dish is exactly a circular
paraboloid, the shape you get when you spin a parabola around its axis of
symmetry. As such the paraboloid inherits all the nice reflective properties of
the parabola. The whispering dish in the picture is located at Scienceworks the
Science Museum in my hometown of Melbourne. The focus of the dish is
located inside the ring I’m pointing at. Okay, now take two of these dishes and
place them 50 meters apart. Then if the junior mathologeress Lara
whispers at the green focus Mathologer junior Karl will hear her disembodied voice at the read focus. Really quite a stunning
effect. It’s a great experiment but what’s not
so great is the explanation on the whispering dish. What it says there is:
“The other person hears you clearly because the curved shape of the dish
focuses the sound into the ring at their end.” Pretty damn nothingy, isn’t it?
Well most science museums try a little harder and at least feature this
suggestive drawing here but there’s one very obvious question about this effect
and it’s a question that is seldom asked: Why isn’t a sound muffled? Specifically,
why doesn’t a sound wave leaving the green focus in different directions, then
arrive at the red focus at different times. Well that amounts to asking
whether all the yellow parts in this diagram are the same length.
They don’t look it but surprisingly they are. And there’s an easy explanation just
using our focal-directrix super property. Don’t believe me? Just watch! Let’s bring
back the two directrices of the two paraboli and let’s take a careful look
at one of the yellow paths. What can we say about the length of this path? Well,
let’s see. Because of the focus-directrix super property the red distance from the
focus to the reflecting point on the parabola is the same as the distance
from the reflecting point to the directrix. And, of course, the same is true
up on top. But this means that the length of the path from the red focus to the
green focus is exactly the distance between the two directrix lines. And
since this is the case for all paths all paths have the same lengths. How easy and how pretty a proof is that. This equal length property is also important for
many other really significant applications of parabolic reflectors, but
strangely unlike the focusing property the equal lengths property is rarely
mentioned by anybody. Okay, now what about ghostly Princess Leia? How do we conjure
her? For this we use proper parabolic mirrors
and instead of moving them apart, we move them close together. We
then place Leia and her robot friend in the middle of the bottom mirror, at the
green focus. Then we cut a hole in the middle of the upper mirror just above
the red focus. Then a hologram of Leia materializes at the second focus. Now, this is real mathematical magic 🙂 !! And on that happy note I will declare I am NOT happy. It’s time for a Mathologer
sermon. These days here in Melbourne my junior Mathologers Karl and Lara seem to
spend half their time in maths class torturing quadratics but they never get
to see any of the beautiful maths I’ve shown you today. Much less figure out why it works. If they didn’t happen to have an annoying Mathologer for a father
they’d never find out about any of this. Well except for the science museum’s
explanation which turns out to be a masterpiece of explaining nothing. This
is especially puzzling and especially annoying because the simple maths that
you need to explain all this super important and super applicable stuff
properly is exactly the school maths that it’s done ad nauseam. At the same
time Victoria’s maths textbooks are
chock-a-block with pseudo applications like parabolic bike paths, quadratic
types of cheese, and so on (I’m not making this up 🙂 And this is just the tip of a
parabolic iceberg. As my colleague Marty likes to say: our educational authorities
never miss an opportunity to miss an opportunity. I’d be very interested in
finding out from you guys what’s the state of educational affairs where you
are. Do kids learn about the things I talked about today in maths class.
Properly? At all? Let us know in the comments. And that’s it from me for today.
Okay, let’s end on a satisfying mathematical note and tie up a loose end.
Here’s a simple explanation for why our paper crease from before just touches
the parabola. If you want to figure this out for yourself now is the time to
avert your eyes and ears. Well, it’s easy to read the equation of the yellow
creased line off this diagram. For this we just need its slope and its y-intercept.
The slope of the creased line is 2a, the negative reciprocal of the slope of the
thin segment and it’s green y-intercept is minus a squared (may take you a second or two to convince yourself of this.) And so we get this for the equation of
the line. And then where does this crease line hit the parabola? Well we just
equate the line with x squared and solve for x. Okay, so there’s exactly one
solution at x=a and that means that the crease touches the
parabola at exactly the point we predicted. All is good, no loose ends,
I’m happy again and we’ll all be able to sleep tonight. And that’s really it for
today 🙂

100 thoughts on “The Secret of Parabolic Ghosts

  1. I live in the United States and they never went over these things. I found them out for myself when I took an Astrophysics class in my university… and it wasn't the focus of the assignment. I found it out by looking deeper into it. (We were told it was okay to just use a computer, but I used a bunch of old math textbooks and drew things by hand)

  2. Thanks for yet another cool video! If I were a math teacher, I'd show this in my class – any class, from say 4th or 5th grade on. Younger kids may not have the algebra, but it's the concepts that I'd want them to learn. Now, here in America the Once Great, as far as I can tell (as both a former student AND a father AND a grandfather!), the purpose of the curriculum is to get just enough kids to pass just enough tests… to justify the curriculum. Sad but true. We love to point out that x% of high school grads can't read a bus schedule, or point to Mexico on a map, or give correct change, or figure 7% sales tax, etc. – but we (unfortunately) don't have nearly enough will to DO anything about it. Homeschooling is increasing every year (at least by absolute number – I don't know the percentages.) [Hey, that's math stuff! Look at me! 😉 ] I CAN say from personal experience that although I wasn't home-schooled, I did go to a "Christian" school for 2 years (6th & 7th grades.) The difference was incredible! The worst "religion" part was simply going to 'chapel' on-campus (it was a church / school in one) 30min a day.; not awful. The educational difference? We wrote our own poems, drew our own cartoons; everybody had phys-ed every day; I took typing 1 semester & information storage (microfiche, etc.) another; we learned how to diagram whole complex, compound sentences (every word); we learned binary, decimal (our 'normal' system!), hexadecimal, octal, etc. (& how they're created!); we learned Roman numerals & Egyptian numerals; we learned names of numbers up to 999 nevtillion, 999 octillion, etc. When I got back into public school, it was literally like going to remedial class. If I'd known I could, I'd have asked for some kind of harder classes! In English class, we diagrammed 2 things: subject & verb. That's all!!! I got extra-credit for using stuff I'd learned in 2 'lower' grades!!! We read (out loud) "To Build a Fire" – about which the teacher's ACTUAL CRITIQUE was, "That's a sad story. It's SO sad." ?!?!?!? What a joke! Now, testing is EVERYTHING. AND it's largely controlled by 2 or 3 companies. You can literally go from kindergarten to teacher & never leave ONE company's purview! [John Oliver has a GREAT & scary video on this on YouTube.] It is – as my Christian school teachers might well have said, "A shame before God!" Well, it IS a shame – before ALL OF AMERICA. We're raising IDIOTS! Anyway, thanks for doing these videos! Hopefully some teachers (& home-schoolers!) will show your videos. Now, THAT's teaching! Just earlier today, I was watching a video about some historically odd events, like how America started a 'fake' war with Guatemala – & got the president to resign – JUST so we could keep the land necessary to ship CHEAP BANANAS! And how Pepsi-Co took possession of a whole lot of Soviet military equipment in exchange for 3 BILLION DOLLARS worth of Pepsi sent to Russia [and a couple of other stories.] I commented that if they taught THIS stuff in school, it might just hold the attention of kids in history class; AND they might actually learn some politics, geography, social studies & human nature! Why not? We've got very little to lose at this point. The state of our education is abysmal. So: all hail the Magic Mathologer! You teachers out there in the trenches: find a way to show these videos (and others like them.) Then we might stand a chance. Rikki Tikki.

  3. This video is fantastic.
    I have been well aware that vertical rays coming into a parabolic mirror focus onto a point, but seeing 6:15 I realized that we could use these parabolic mirrors to turn a source of light that emits in ALL DIRECTIONS into a SINGLE DIRECTION! That is crazy! It's like you're taking something that is totally disordered and turning it into perfect order. I would have thought that the only ways you can get all light to go into a single direction would be to have lasers or something but man. I'm guessing that this is probably a property that lighthouses or spotlights take advantage of. Absolutely insane.

    But is there anything interesting of consequence when want to change some properties about these curves?
    Like for instance, we can construct a parabola by using a directrix and a focus. But what if we say… have the directrix be a curve as well instead of a line? What about two directrix's instead of a focus, each of which could be some curve? Could curves constructed from these have interesting properties?

  4. Yeah, I was never taught the equal lengths property or the property of the directrix and focus in high school here in Romania. A damn shame, especially given that they are, as you said, the properties that make this math applicable in daily life.

  5. In Russia, in the 90s, we were taught maths only by formulas and numbers, by solving repetitive equations and problems. The only practical applications we learned about were motion and acceleration. Maybe some more advanced and innovative schools were better than this. Now the curriculum has improved a lot, they say, but I don't have any real info on this.

  6. 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.

  7. Nope, never a why. It's why after the first year my A's for math and physics dropped to C's and I went to study chemistry instead, which eventually is all maths and physics, but you then you do learn the why.

  8. Dear Mr Mathologer,
    do you think you could do a video on, or involving, the mathematics behind the twisted and untwisted states of bandsaws, i.e. the bandsaw mechanism? Thanks!

  9. Wonderful! I had not appreciated the significance of the directrix, nor the constant-distance property of the whispering mirrors (and we have one at our local museum). I remember reading that some telescopes used spinning pools of mercury to create a paraboloid reflector – might that be something to explore in a future video? Also angle of repose?

  10. what was the point of the quadratic-ish formulation on your tee-shirt ?
    from the comments it seems that nobody else really gets it either.
    i do have a little theory though.
    owls are supposed to say "WHO" – right ?
    so the tee-shirt reads
    "(minus WHO plus or minus the square root of
    (WHO squared minus 4 times WHO times WHO ))
    all divide by 2 times WHO"
    where each WHO definitely stands for an A (?) or a B (?) or a C (?).
    and since it's all easily-forgotten rote anyway –
    the nagging ambiguity is very real in the mind of the student.
    yeah – i'm not sure i believe it either – but WHO knows ?
    as Buffy Sainte-Marie sings
    "i've spent some time in study-oh
    i've taken my degrees.
    and memorized my formulae
    my As and Bs and Cs."

  11. well we had a good teacher at school but he mentioned the reflective property superficially without any proper explanation .

  12. 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!

  13. 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.🔥

  14. 95% of the people in my class fail to find the intersection from a parabola and straight line. this is 11th grade in germany btw. edit: not the normal math course. the advanced math course.

  15. Well, I’m a student of one of the most famous universities at Moscow, and in order to enter it, of cause it was also necessary to get relatively high results in mathematics. Fairly speaking, in the end I’ve decided to go to economy and eastern studies faculty, but still math was always a compulsory subject.. And I never happened to hear ANY of all that nice topics you’re talking about! Highly appreciate your and some other channels, that makes us believe in miracles again. Had literally no idea of existence this kind of noise transfer possibilities. Mind-blowing.

  16. Mathologer, while wrapping up my Masters Degree in 2000 in Dallas, Texas, one student, a math major in undergrad school and now a math teacher, could not calculate her height in inches. Another person was given a distractor task of counting backwards from 100 by 7. He got to 93, asked if he could do it by 5 instead and then gave up. Sad state of affairs.

  17. The focus and directrix are still occasionally covered in Alg2 in the US but like all of Alg2 the content varies widely district to district, And I suspect the interesting applications are often not reached even after that point.

  18. In 7th and 8th Grade (in the US), I moved to a new school that stuck everyone in remedial math for both years, then in 9th grade threw everyone straight into Algebra I. At my old school, I was on track to take Pre-Algebra in 7th grade and Algebra I in 8th so that I'd have taken Calculus by the time I graduated. But it turns out that in the new school, the mathematics teacher wasn't qualified to teach Calculus and that's why they held back everyone in 7th and 8th grade.

    Because of being in remedial math for two years, I started to hate math and didn't even bother to turn in my homework assignments. In fact, in 8th grade I had a D in the class and my teacher said, "If you turn in your missing homework assignments I'll give you 90% credit on your scores" because he knew I knew the material and it was just boring busywork. So that weekend I did all my missing homework and turned it in and instantly raised my grade to a B. In any case, by the time we all got thrown into Algebra I, I had a steep learning curve and only managed a C. But by Algebra II, it started making sense again so I had a B by the end of that course. Then an A+ in Geometry (my favorite math) and Trig. It wasn't that the harder the math got the better my grades got so much as the more INTERESTING the math got the more I wanted to do it.

    There were a lot of my classmates that took the bare minimum of math because of this, though. In fact, my sister (one year ahead of me) became convinced she couldn't do math even though I can say hands down that she's a much better student than I am. So I'll say this: Public School nearly killed off my desire for mathematics, but thankfully I enjoyed it enough that when I was able to get through the boring busywork and into the interesting theories it recaptured my attention. But given how hard they tried to kill my love of math, I can't credit my schooling with any of my mathematical successes.

  19. I actually did learn about the focus and directrix of parabolas. And the equivalent relations for other conic sections. The textbooks still managed to suck the life out of the topic though, although I had a teacher who did their best to counteract that.

  20. Back when I was learning about conic sections, the parabola was the last conic section to learn about, and yes, the directrix was mentioned and the equal length property was also mentioned. No proof, though. Similarly:
    – Construction of an ellipse using a string and two thumb tacks was mentioned somewhere
    – Construction of a circle is simple enough; use a compass or throw out one of the thumbtacks from the ellipse construction
    – I may have stumbled upon construction of a hyperbola by hand; find the corner of a piece of graph paper and trace some lines (I don't know how to describe it but do it right and you get a hyperbola but with its twin missing)
    – No construction of a parabola was ever covered; the only way I thought it would be constructed was using a T-square and a ruler, with the T-square on the directrix and the ruler pivoted on the focus.

    Can we just ban all current math curricula forever and replace it with something, I dunno, intuitive?

    Conversely, I personally enjoyed differential calc; I was more a number-cruncher so I could handle never seeing a visual proof of whatever the hell I was doing, but seeing a visual proof is still super satisfying.
    Statistics, on the other hand, is boring as fuck. It's basically more number-crunching except the units make no sense (squared… dolllars…?) and having to use a lookup table feels antiquated.
    I'm fine with combinatorics; it's basically learning how to count all over again except you're given a name to whatever the hell you're doing.
    Don't ask me to do induction proofs; I'm bound to screw things up.
    Took me a while to understand logarithms as being a way of converting multiplication and division into addition and subtraction, and I still don't know all the log properties.
    It's only a few months ago that I finally found the difference between a polynomial function (x^2) and an exponential function (2^x).

  21. I teach grade 6 math (11 yr olds) and am trying to shift the way maths is taught through using inquiry and visual explanations – not letting students get by with an answer or step that they don't know WHY it works. But it takes SO much time, and I am constantly in conflict between developing the attitudes and thinking skills and discovering concepts and being able to cover the content the higher grades say need to be covered. It is frustrating. I know the rote drudgery is not worth it – it turns people off math, makes them misunderstand the subject completely, and does not actually lead to lasting understandings anyway – you just have to re-teach it the next year, but in the end I don't have complete autonomy over my class. If anyone has good resources for teaching more basic math understandings with simulations like we see here I would love to get my hands on them. I'm already keyed in to youcubed and nrich. What else do you all use?

  22. Sadly I keep hearing from people, that math and science is the white man's science, and it oppresses people. So I doubt we'll improve this problem

  23. 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.

  24. In the UK it's a mixed economy (particularly in state schools). There is a shortage of maths teachers, so often non-mathematical people are seconded into teaching roles. Of course they are bright and (mostly) enthusiastic, but to really be inspirational about maths you eed to be able to go deep and show the beauty of it all (as you very nicely do in your videos). Too much UK teaching is about 'getting through the exam'. It is a fault of the education system, not the teachers in the main – who must deliver on pass grades or else face being sacked.

    There is also a very sad backdrop of 'oh, maths isn't needed' in the UK. I have heard examples of parents and even teachers saying, "oh you don't need to know that – you'll never use it when you leave school." This is so illogical as well as being untrue. Who uses a poem when they leave school? But it is still a window to truth and beauty.

    So – thank you for your videos, your very own window onto truth and beauty.

  25. at 8:24 ..arrive at red focus at differen time? different time? sound must travel with a velocity. i am not sure why you mentioned different time?

  26. University in Germany is not much better… the question is if most professors here really want to teach anything or if they are just bored… or if they just don't know it better…

  27. Hello mathologer! I love your videos. I have one request for you. Can you watch this video and explain what this guy is talking about because I am doing statistics and it sounds very wrong. His video did not convince me. Please math help me with this concept I cannot grasp.

  28. 7:30 – I should point out that a similar "contraption" also exists outside Questacon in Canberra too (it's the national science museum of Australia).

  29. Numberphile
    Mind your decesion

    🙏share something else if you know in comment section below.

  30. Your shirt doesn't seem right. It's treating all three owls interchangeably; surely it should use three different owls in different places?

  31. 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.

  32. In NYS, I don't think I learned any of this. The only time I encounter this math was the year between Algebra 2 and Calculus where I took a PreCalc course at a university. Also, math in NYS follows a very strict path. Algebra 1, Geometry, Algebra 2, PreCalculus, Calculus 1, and the rest are either AP or optional classes like Probability and Statistics. In fact, you only need to do Algebra 1 to graduate but you need 3 math courses of any variety to graduate.

  33. Question

    I understand your proof that the crease intersects the parabola at just one point (A,A²), but I have not yet proved that the crease does not cross the parabola at this point (just as the y-axis crosses the parabola at just one point, namely the origin) and so is indeed tangent to the parabola

  34. I was searching for a ghost video to scare my little sister …..but ended up getting into another maths video 🙄🙄😑

  35. 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.

  36. Unrelated but an interesting finding:
    The sum to the left surprisingly converges (I think: no proof so far)!

  37. Mathloger or any other good math guy:
    x = 1-1+1-1+1-1+1….
    it was supposed to be 1/2 🙂
    1/x = 1/(1-1+1-1+1-1+1…)
    x is 0 or 1
    but it cant be 0 because dividing by 0 doesn't give you 0
    so x = 1
    am i right?

  38. I find you videos –along with the humor and graphics—to be a wonderful way to get some fascinating concepts across to people. As an engineer I have a love of math, which was instilled in me by a Jesuit teacher who had a very similar approach as you: relating to real life examples with humor and easy to understand terminology. I wish I had the time to do this. (I have a compound miter explanation on my website, and I have always wanted to do a 3d animation instead of the simple 2D graphics.: it's been 20-years and counting. Hey, I am approaching retirement then, perhaps,…)

  39. I have a theory on bad math teachers, based on my lives experience:

    math is often taught by people that used to achieve a-levels in math over the course of their entire school education. They then usually get very frustrated, on finding out at college that memorizing the steps of evaluation on a couple of hundred examples encountered at school is unequal to understanding math. later in live, after choosing to become a teacher, they unintentionally ensure to replicate that experience for the one, or two a-level grade math pupils encountered each season, by making a huuuge fuzz about how math is always generalizing and one should never confound 'real math' with 'just solving the equations', which results in the types of convoluted explanations, math teachers tend to give and are so known and hated for.

    this ensures loosing anyone but those one, or two 'busy bees' that happen to have the capability of memorizing hundreds of possible chains of evaluation steps.

    not only do they replicate their own frustrating experience for those 'busy bees', should any of them chose to study math (usually encouraged by that same teacher), they also reliably discourage everyone else, entirely independent of their possible math skills.

    I explicitly didn't dare to study a natural science, like I have wanted to, after having been discouraged very intensively from even trying to understand math during school time. My first impression, I might have missed something vital dawned on me, when I did an apprenticeship to become a carpenter and realized, that it doesn't need anything but some straight etch, a nail and a piece of string to do all the things I've been taught to do using a pocket calculator. Many more such impressions followed over the years and while I flunked my college education by trying to study something I wasn't really interested in, but thought came closest to a natural science without the math part. I now work as devop, usually in a position that was dedicated to be filled by an academic. I regularly find myself explaining math to people with degrees in computer sciences. math they are not only supposed of having understood, but also own a degree to show for it. I'm currently writing a functional programming library involving a great deal of category theory. something many cs-majors seem to be terrified of, since it's one of those courses, that often end the attempt to get a cs-degree.

    Turns out most mathematicians just totally suck at explaining what they are doing in 'proper terms' before generalizing on it. the theory itself seems almost intuitive, given enough programming experience in any typed language.

  40. I learned almost everything I know from the internet or on the job… education is not very effective, imo. without a physical objective or some real application, it just gets forgotten — and rightfully so; the internet is a really great starting spot for people who want to learn. teachers are incentivised to make content they are passionate about, and teachers usually usually don't have much incentive to teach things.

    also, classrooms are miserable and lack very little visual stimulation, and compared to todays world of hyper visual stimulation, it's near to impossible to hold someone's attention. I caused problems early on in my school years because I was bored. I found a way to go to the university at 12, and since then finally busy and happy

  41. 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?

  42. 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.

  43. Alas, though our teachers did plenty of triangle geometry stuff they never applied it to a parabola, or in generall didn't show us the cross-connections between the topics so the proofs, when they happened came out of nowhere and the isolated portrayal of the topics made it seem like its all random made up stuff. It's really the interconnections that make you appreciate the inherent power & beauty of logic.
    I moved alot as a teen so I saw how its done in different federal states.
    Though the schools in Baden Würtemberg were generally speaking much better funded and competently staffed, they just made us memorize and use the formula on your shirt ad nasuseam without ever explaining how it worked.
    In the Saarland they actually taught the 'completing the square' method which was much better because it is so simple and logical and you can see right away how it works. If you forget any part of it you can re-deduce it in mid maths test.

    They even taught us basic logic and linea algebra which has proven invaluable in university. Still I think we would have been more motivated if they told us that this stuff can help you understand artificial intelligence, computer graphics and even quantum physics. Basic lin alc is ridiculously powerful.
    So yeah generally BW had better schools but in maths the Saarland wins.
    I kinda wish we'd done more "trigonometry applied to complex functions" bc thats so useful for proofs and I think I'd think of it faster if I had it engrained as a kid.
    – that said im 25 already, two of my siblings are studying to work with kids and they say the schools are much improved, though neither of them is a math teacher specifically.

  44. Here in California the only time they teach you about this sort of thing is in undergrad physics. When they teach you about quadratics the problems are just as pointless as the ones your children have been getting.

  45. Two "golden oldies" for treatment of curves:
    – Protter&Morrey's Analytical Geometry (2nd ed. 1975) and
    – E.H. Lockwood's "A book of curves"
    There you'll find many things related to curves that are barely, if at all, treated in standard high school math nowadays.
    If you like to see some original math (partly) related to curves, check out "New Horizons in Geometry" by Apostol & Mamikon.

  46. 3:23
    So you can find derivatives by folding…
    Does this work with the focus and directrix of any parabola?

  47. Ukraine here

    Had these properties explained with real life examples, except the bit that all paths travelled through the focus points are equal

  48. 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

  49. IMHO, one of the roots of the education problem is that it was originally conceived as a way to educate proletariate people to run the factories and technical works that the industry demanded in the XVIII century. That's why students are taught in batches per year and the school works like a sequential manufacturing: one feature at a time with one worker in every step of the chain until you finish the product. People too smart can vote for their own interests.

  50. To mathologers :

    Is anything wrong in this argument?
    If yes, do let me know.

    From Wallis product
    Suppose pi is rational. So, pi/2=(a/b)
    where a and b are relatively prime non-zero integers.

    Using Wallis product, we can write as follows :
    We see that RHS has infinite no. of factors of 2. Therefore, LHS should also have infinite no. of factors of 2.
    But, (…) don't have any factor of 2. a can also not have infinite no. of factors of 2 because a is a finite non-zero integer. So, we get a contradiction. Thus, pi is irrational.

  51. Geometric proof that the folding line is tangent to the parabola: When one point is folded onto another, the resulting crease is contained in the perpendicular bisector of the segment joining the two points. So, Let F be the focus, P be the point on the parabola, and let N be the projection of P onto the directrix. Since P is equidistant from F and N, it lies on the perpendicular bisector of segment FN, which is the line containing the crease. To see that the crease is indeed tangent to the parabola, suppose the crease were to meet the parabola at another point, Q, and let M be the projection of Q onto the directrix. Since Q is on the perpendicular bisector of FN, the length of segment QF is equal to the length of segment QN. On the other hand, Q is on the parabola, so QF must also equal QM. But QN is the hypotenuse of triangle QNM, so QN is greater than QM, a contradiction. Thus, the crease only meets the parabola at P, i.e. it is tangent to the parabola at P. (Diagram: )
    I used to teach middle school math and this problem is in Phillip's Exeter Academy's Math 2 problem set. Very good problem set, available for free online. Just google PEA Math 2.

  52. 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 " ).

  53. Hi I went to a selective school in NSW (NBHS), and we did almost all the examples in this video (we didnt do the equal lengths one). We had really good teaching, however my understanding is that this is not the norm for the comprehensive schools in NSW

  54. Maths teaching in the UK is abysmal. I'm not a Maths teacher but I've seen the interest in Maths crushed out of my 13 year old daughter. I talk to Maths teachers who are in despair, they have to keep strictly to the curriculum. Anything remotely interesting, such as the Fibonacci series, is not in the curriculum. Teachers are not allowed to mention it!
    You grab a child's interest and you've 90% won, from then on children teach themselves. Oh: Apologies for being so stupid!
    The only consolation I have is that Maths teaching has always been abysmal, and the world still seems to keep spinning. But its a great shame this awful perception of Mathematics is perpetuated generation to generation. Children hear and see their parents throw their hands up in mock disgust when Maths is mentioned. Children are sponges and pick up on the slightest thing, good and bad. How can this bad chain be broken? You Mathologer, I see you try so hard, but I think your p****sing into the wind. Sorry to have to say so.
    Any books you might mention for a child just starting GCSE Maths? (Not sure what the equivalent is down under)

  55. 0:14 "I know it's hard but ignore the toys ."
    spends the next 30 seconds playing with a UFO hologram machine and a Star Wars coin.

    Also, speaking of dorky amusements: this is quite literally the least important thing in the video, but for anyone else who was wondering about his T-shirt at 6:44 and couldn't find it on the regular Mathologer page: It's pi written in ternary (base 3), and you can buy it on the mathologe page (which exists separately for some reason):

  56. 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.

  57. 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!

  58. 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.

  59. 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!

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