Why did we forget this simple visual solution? (Lill's method)
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- Опубликовано: 16 июн 2024
- Today's video is about Lill's method, an unexpectedly simple and highly visual way of finding solutions of polynomial equations (using turtles and lasers). After introducing the method I focus on a couple of stunning applications: pretty ways to solve quadratic equations with ruler and compass and cubic equations with origami, Horner's form, synthetic division and a newly discovered incarnation of Pascal's famous triangle.
00:00 Intro
04:14 Lill's method
07:31 Free meal
09:51 Square turtles
11:39 Origami turtles
14:16 Iterative turtles
17:32 QED
24:00 Pascal's turtle animation
Here is the page with an implementation of Lill's method for cubic polynomials that I show in the video.
www.qedcat.com/misc/lill_method/
It's an adaptation of this webpage
heim.ifi.uio.no/magho/lill/
(I have not been able to find out who put this together originally).
The article that inspired this video is this:
Thomas C. Hull, Solving Cubics With Creases: The Work of Beloch and Lill, The American Mathematical Monthly , Vol. 118, No. 4 (April 2011), pp. 307-315. Here is a link to this article on Thomas Hull's webpage: mars.wne.edu/~thull/papers/ame...
Lill's original paper:
www.numdam.org/article/NAM_186...
Other good references include:
Polynomials as polygons by Serge Tabachnikov
www.math.psu.edu/tabachni/pri...
Dan Kalman's book Uncommon Mathematical Excursions: Polynomia and Related Realms (the first chapter is about the Horner form and Lill's method)
books.google.com.au/books?id=...
Thank you very much to Marty, Karl and Danil for their help with this video.
One version of today's math t-shirt (Zombie addition): www.redbubble.com/people/mani...
The piece of music at the end is called "Fresh fallen snow" by Chris Haugen from the free RUclips music library.
Really neat 1-line Mathematica code for the generation of the Pascal turtle which appeared on Reddit after the video was posted there:
Graphics[Table[Line[ReIm[Accumulate[Table[2^(-n/2)Binomial[n,k]Exp[I(4+2k-n)Pi/4],{k,-1,n}]]]],{n,0,7}]]
and another nice implementation in Python (with a real turtle graphics turtle) by Alex Hall repl.it/repls/DeepskyblueFrac...
Enjoy :)
Mathologer Patreon: / mathologer
Mathologer PayPal: paypal.me/mathologer
(see the Patreon page for details)
“You’ve already accepted speeding turtles and weirdly bouncing lasers, so it’s definitely a bit late to start objecting now.” 🤣
if this was allowed in the proofs i wrote in my college computer science (theory) classes, I would have had a lot more fun
@@pvic6959 : You just never allowed yourself that ;-)
From 0 to 100 in a snap.
I have been the caretaker of many a turtle, most of which were not tortoises. Non-tortoise turtles are quite fast, especially amphibious river turtles.
@@ThomasKundera
He would never allowed himself anyway because teachers want you to vomit their own methods. Just like grade school math teachers require you to do things by Core mathematics, now. So much for "New Math" which is what I learned, a la Tom Lehrer. (Look up "new math." I learned it exactly that way.)
Now this Core math is so cumbersome. In fact I did learn it as jumps on a number line, but we went past that very quickly.
The "New Math" version is much quicker and streamlined.
But it would have been nice if they'd taught long square roots along with our long division. It would have been very useful.
How is it that I've been teaching math for 20 years, factoring polynomials, finding zeros, dividing them etc. and I have never known Lill's BEAUTIFUL method. The overlap with geometry makes this a fascinating topic. Time to rethink my instruction- what a great way for students to explore all the connections!
I have trouble understanding concepts that I can't visualise, so I'm sure your students will greatly appreciate that!
Bob Tivnan Yeah it’s incredible! Teaching mathematics can make you feel so powerful sometimes and then stuff like this makes you so humble again!
Well said! Your students are lucky to have you.
Luke Janicke pπp
./
;l;”o
More teachers like you for the world, please
This probably one of the coolest things I've heard/seen in math.
h.khkoyoytot
check out 3b1b's channel for many similar cool concepts and visualizations
I think I can say that to about all of Mathologer's video :-)
Wait till you try factorials
Unfortunately if it wasnt shown using a turtle it would have probably been another boring lesson
Saturday, 5 a.m. An early start for me here in Melbourne. Another long one. Hope you enjoy it :)
Mathologer Friday, 9:30 p.m. in Leipzig. An excellent start in the weekend with this long Mathologer episode. I'll enjoy it, for sure!
I did.
does this work with irreducible polynomials, looking for complex roots as well? Or they must be real?
Peter Bocan, I was wondering the same thing about complex roots. Good question.
Tbh, when I first saw the title... It wasn't interesting... The reason I watched this video is because I got bored and had nothing else to do... But man! Glad I watched it... This completely changed the way I imagine equations forever.
I can't believe that school is meant to make learning harder for people instead of checking out what high professors like Mathologer eghm, solve what we see as a big problem, with a simple and fun methods...
No disrespect, but this can be fed to 10 years old and they will still understand it.(most of it at least)
Thank you Mathologer (A.K.A the best Math teacher in the history of math) . I wish I meet you irl one day.
I have a turtle free proof for the inverted zeros result.
So we have this fast snail that we have to hit with this weird slingshot...
Weird comedy? We'll have Nunavut.
🤣🤣🤣
Imma eat yo dogs
@@denelson83 CHEERS! 🥂
Jörg Sprave just heard the word slingshot...
Not even once Mathologer, can I make it through a video of yours without learning non stop, start to finish! People like you are truly some of humanity’s most valuable gifts. To think how many new physicists, engineers, programmers, mathematicians in the making that have been added to society, spear-headed by your influence, the value is clearly demonstrated as we are all benefactors of that reality.
I graduated in Civil Engineering in the mid-sixties, and a large part of our training was spent at the drawing board, as it was at that time for architecture and other branches of engineering. A sub-set of the drafting work was learning graphical methods of deriving and then solving stress and strain equations, and their differentials and integrals, to the extent that one could determine the forces in, for example, a trestle bridge without having to do a single calculation - you just scaled off the answers. What you show in your video is a rationalisation of one application of the graphical approach, and far from being ignored for several generations, it was the everyday practice of thousands of engineers since Newton suggested the idea of gravity.
Excellent video, all the same.
Fascinating... Could you maybe point me to a video where they civil engineers are using or teaching the method?
I wish I could help, but I don’t keep up with these things these days. I doubt very much that it would be worth anyone’s while to do a RUclips lecture on civil engineering practices from over 50 years ago, particularly since no-one is taught old-fashioned drafting when computers can do most of the work automatically. I suggest a visit to the civil engineering department of a decent university and a trawl through the historical library. You would be looking for something like “Graphical Solutions to Problems in Civil Engineering”. If you can gain access to the drawings of professional civil engineering firms from anytime before the First World War you will be able to see the graphical calculations with the item being drawn. I would also suggest going through the papers of any civil engineering designer as we all kept monographs of good solutions that we came up with for use on future projects.
That's really cool, and I was thinking it would be great to make a calculating tool out of this method
@@tmm3258 there is a video by Efficient Engineer that uses this kind of techniques on trusses
I majored in math, and I enjoy it, but there's no way I'll ever have the devotion to math that Mathologer does. It takes a lot of passion to make videos this good. Very much appreciated.
Are youba maemrhematician now if I may ask? Would you have come up with something like this if you hadn't been shown it?
You hacked my brain: I can't help but click on something if it has "lasers" "turtles" & "equations" all in one title
edit: I stand by my preemptive upvote; great vid
Recalls learning Logo programming in elementary school. FORWARD 50; RIGHT 90; FORWARD 50; RIGHT 90 etc. Education with turtle graphics is about 50 years old now.
Same here.
Logo flashbacks, yeah
Same here in Italy
Someone from my class did π art with logo.
I never got the connection of Logo turtle graphics to "real" programming. I mean, yes it's algorithms, but that kind of graphical problem seems worlds away from the problems that programmers actually tackle. It's like giving a kid a remote control car, to learn how to drive a real car. They're only a tiny bit similar.
Hi Mathologer! I have been captivated by mathematics for quote some time now, and am well underway of getting my masters in mathematics. Yet each time you upload a video, I am amazed at how beautiful mathematics can be.
Having a go at those little puzzles you throw in makes me discover connections I have never seen before. While this is very humbling as I realize that, even in things I have studied in much detail such as polynomials, there is still a wealth of beauty and knowledge I have not yet seen; but this is exactly why I am studying mathematics.Thank you for always coming up with such high-quality content :)
I was hesitant to click on this video because I was like "Sounds like a high level physics or math thing I won't understand"
Boy was I wrong. Best video I've seen on math probably ever
So when are you making a Pascal's turtle shirt for purchase?
This was probably my favorite Mathologer video, thank you very much for sharing this with us! And please make a second part, too :)
For the flipped equation, the proof is as follows:
Take any polynomial p(x) and break it in to odd and even parts p(x) = o(x) + e(x). The odd part has all odd powers of x (x, x^3,x^17,...) and the even part has all the even powers (constant term, x^2, x^42,....).
If a is a solution of p, that is p(a) = o(a) + e(a) = 0, then flipping the sign of a and flipping the sign of e(x) should still work: p(-a) = o(-a) - e(-a) = - (o(a) + e(a)) = 0. Here I have used the defining properties of odd and even functions to get the answer.
we need LaTex in youtube comments
@@Qualiummusic absolutely. Latex should be integrated into JavaScript and come default with every website these days.
Shouldn't you flip o(a) instead?
@@christiandiegoalcocer we have p(x)=o(x)+e(x) and we have the degree of p (it's highest power) is odd
Since o only has odd powers
o(-x)=-o(x)
And similarly
e(-x)=e(x)
Now the flipping the sign of every second term, as it starts with an odd power, means we change the even powers and we get a new polynomial
g(x)=o(x)-e(x)
Now let a be a root if p, i.e.
p(a)=o(a)+e(a)=0
Now we check
g(-a)=o(-a)-e(-a)
Using the identities derived at the start for o and e ge get
g(-a)=-o(a)-e(a)=-(o(a)+e(a))=-p(a)=0
QED
Edit: if p is even then as we start with an even power and swap the signs of the odd ones we get
g(x)=-o(x)+e(x)
Thus fir our root a
g(-a)=-o(-a)+e(-a)
Which with the identities becomes
g(-a)=-(-o(a))+e(a)=o(a)+e(a)=p(a)=0
What?
A brilliant way to visualise Horner's method (which, incidentally, I was also taught at high school about 40 years ago).
:)
@@Mathologer Thanks for sharing but HOW could I or you have come up with this on our own?? How amd why did Lili come up with it?? I could never admit I'm not brilliant enough to come up with this...thanks for sharing..hope to hear from you..
What a jaw-breaking topic ! One of your greatest videos ! Please do keep on !
Okay, but has anyone taken this to its limit? What happens when we start throwing analytic (heck, just rational) functions at this method? What does it do to divergent Taylor series? That vertical flip to negate the even power coefficients... so close to the zeta-eta relationship...
Thanks for this, you've produced so many new questions for me to ponder.
I love all the crazy connections in math. And a really amazing presenter is just icing on the cake. Super beautiful bit at the end with Pascal's Triangle making an appearance in yet another cool way. Love the channel and been a fan for years. Keep 'em coming!
Mum: *Why are you shooting turtles with lasers son?*
Me: *I'm doing Maths homework mum!*
_Maths teacher gets arrested by the RSPCA for promoting turtle laser tag_
, son*
, mum*
It doesn't hurt the turtles. It only makes their shells sparkle different colors. (That's what you tell the kids as you get back to vaporizing turtles.)
@Jorge C. M.
Déjà vu... You Grammar Nazi! I think I have seen you before...
www. youtube. com/watch?v=7lJkRCrfW0E&lc=UgwrQy4ra_VH2rageVB4AaABAg
Keep going. 😏
The right thing to do would be to give that kid Nobel prize for right lasers, right after this happens. Right?
Beautiful. This is the first time I'm seeing this. I'm speechless! Thanks Mathologer
This has to be the best of all the mathematics videos with the most elegant property of all time wow... please make a video about complex solutions and what to do when turtle intersects path and maybe even more elegant properties this might have?
This sure was my favorite mathologer video. If you intend to make a second part with the complex form etc., I'll definitely be interested in that!
This is probably the most beautiful method to solve anything in maths i have ever seen! please make the second video about the complex roots and closed turtle paths and different angles about which you talked about i'd really like to know what happens!
Known for 150 years and faded into obscurity? Something so beautiful? Gah.
Thank you for this. And an awesome description with links. Perfect.
This feels like the sort of thing that ought to be in a video game. I think making people interact with this would be a lot more intuitive than listening to it, even with cool laser turtle graphics.
Well, for starters people should play with the online app that I show in the video :)
That Pascal's Turtle part at the end was beautiful and the music chosen to go with it was great. I almost cried (in a good way) actually. Thanks for the video and for showing us Lill's method.
I had never been so fascinated by any mathematical subject like this one. Thank you so much!!!
Zombie + Human = 2 Zombies
Human = 2 Zombies - Zombie
Human = Zombie
You are already a Zombie, Wake up!
The math checks out.
Automatic like for mentioning L.V.B.
Zombie + Human = Zumbie Human = Zombie * Human ?
Are you sure it's addition. Because I'm not sure, it may be a non-commutative function.
@@livedandletdie lol or is it zombie * + * human?
I thought I'd at least watch the first couple of minutes of this to see what it was about. I'm 6 minutes in, and I'm hooked. Damn! Now I have to watch it all!
I'm going to watch it a few times to let it sink in.
This is very interesting.I've been involved and delving into math academically, professionally, and recreationally for most of my 62 years on this planet and I still haven't seen this method until now.
I played a lot of this multiple times, and paused and rewound a lot, to make sure I got it completely. It's fantastic! This is the kind of thing kids should be taught in school. It connects geometry, trigonometry and algebra in such a cool way!
9:16
Consider the general form of a polynomial
a0 + a1 * x + a2 * x^2 +...+ an * x^n
and find its roots r1, r2, r3, ..., rn.
If you were to plug one of them in, the polynomial evaluates to 0.
Using the negative version of the roots, each term with an odd exponent would change its sign:
a0 + a1 * (-r) + a2 * (-r)^2 +...+ an * (-r)^n
= a0 - a1 * r + a2 *r^2 - ... +/- an * r^n.
As we have plugged in the negative version of the roots, every second sign has flipped.
To make this polynomial equal to the original one, every second sign needs to be flipped back.
Turning this observation on its head gives us what we are looking for:
If you have a polynomial of the general form and its roots,
by flipping the sign of every second term of the polynomial, the negative inverses of the original roots will be the roots of the new polynomial.
Ok so I just finished the video and in the beginning I had my suspicions about why the iterative way of finding roots works and how synthetic division is involved here but when I saw how magically it - and solving quadratics - works, I was invested. Never have I been as excited about math as I was watching this play out and I've been involved with it for as long as I can remember.
That geometric approach to Pascal's triangle is so elegant, it needs to go on a tee shirt. Congratulations.
Starting with a single line segment and ascending, it seems to be fractal in nature although it grows rather than shrinking like most fractal constructions (e.g Sierpinski's triangle). To ascend to the next power, the rule appears to be "Replace each line segment with an isosceles right angled triangle whose hypotenuse is the original line segment resulting in a rotation by 45% and all the new lengths being scaled down by 1/sqrt(2) = .707... Every two powers, the height appears to increase by 1/2 the length of the original line segment, so unlike many fractals, it doesn't have a finite size as n -> infinity.
Your explanations of methods are always so silly and cool! Thank you for being you and sharing this incredible method!
The ending is so beautiful and geometrical 😭
This was really cool. Would love to hear about those more complex things you hinted at for another video.
Great content, made my day! One curiosity about Horner form for polynomials: it is used in computers to actually compute the value of a polynomial, of degree say n, because it involves n multiplications and n sums, instead of computing all powers and summing up, that involves n sums but n(n+1)/2 multiplications. Horner is much more efficient, because of lower error propagation in numerical arithmetics at multiplications, and also requires less memory.
Thank you so much for making such interesting videos. I am not great at understanding all you say but if i work hard i can get it eventully at least this particular one. Your love and knowledge of Maths is amazing and I enjoy your whacky sence of humour. Your channel will probably launch a whole new batch of mathematicians so keep up the good work inspiring us to think outside the secondary school maths textbooks box.
This was so illuminating! Will wonders never cease x
Wow. Simply wow. I did not expect such an elegant way of solving an equation.
For the first homework: changing the sign of the even exponent terms actually does two things; it flips the graph horizontally and vertically. Changing the sign of the odd exponent only flips it horizontally, this is because even exponents are already horizontally symmetrical, changing the sign of every terms after that flips it vertically (this is true for any single variable equations), wich in the end leaves only the even exponents signs changed.
The turtle path demonstration at the end was amazing! And that pascal triangle correspondence is mind-blowing! Thank you for doing these videos, it's always a pleasure to watch!
Thank you Mathologer for showing us the beauty in mathematics.
Absolutely astounding video - and perfect timing for us at our school : we just started doing Polynomial Division! Your video has kept our maths faculty talking for three days :-). We really love the connections between algebraic and geometric thinking. This was a little heavy going in one sitting for our Year 11 students, but hopefully inspired at least some of them to go further. So much wonderful content in this video I'm thinking of making a guided investigation worksheet for students to explore - please keep your web page with Lill's method up. (This could also make a great GeoGebra construction activity: build the tool to generate the path and allow the user to try out laser paths). Thanks again so much for providing such high quality, inspirational videos.
That's great. Would be nice if more teachers would get to see this :)
Your videos are unlike any other math videos on RUclips in just how original the topics are - whenever I watch a Mathologer video, I feel like I've gained some secret knowledge.
Which is to say, great job!
:)
This is actually incredible!
More, please more. Also finding the complex roots using this method...... I don't have a pen and paper, so your home works, are no good for me :( But this is beautiful method. It should be taught across the world. Thank You.
The Pascal's turtles graphic at the end was really something special! Like all Mathologer videos I think this will take more than a single viewing to truly sink in.
This is mindblowing beautiful. Math keeps suprising me again and again
This is so mind-blowing, my neighbors called the fire department.
Now everyone here is truly amazed after I showed them this on my phone.
This is so mind-blowing, my father -whom 10 years ago went to grocery store to buy milk- came back to see it!
Sad thing he went back to the store because he says he forgot to buy the milk
9:23 All even-order terms of a polynomial are multiples of a power of x^2. All the odd-order terms are multiples of x times a power of x^2. Call P(x) with all the signs reversed on the even-order terms P*(x). If we take P*(-x), all of the even-order terms will be equal to those in -P(x) because x^2=(-x)^2, and those terms are the same multiple of the same power of either x^2 or (-x)^2. The odd-order terms will also be equal. Any odd-order term in P*(-x) is the opposite of the corresponding term in P*(x), since it is equal to some multiple of -x times some power of (-x)^2=x^2, while the corresponding term in P*(x) is equal to the same multiple of x times the same power of x^2. Similarly, any odd-ordered term in -P(x) is the opposite of the corresponding term in P(x). Since the odd-order terms of P*(x) and P(x) are equal, and the odd-order terms of P*(-x) and P(x) are their respective opposites, the odd-order terms of P*(-x) are the same as those of -P(x).
Since all the terms in P*(-x) are equal to the trms in -P(x), P*(-x)=-P(x) Therefore, if -P(x)=0, P*(-x)=0. Since 0=-0, this also implies that if P(x)=0, P*(-x)=0.
QED
Thank you for your wonderful, clear and fascinating methods for solving polynomials. Inspiring!
Pure coincidence: we had origami and solving the cubic at our Montreal Math Circle just two weeks ago! Adding to the origami part of the "bouncing" laser: the crease line from taking each point to the line is equivalent with constructing a tangent to a parabola with the focus and directrix being the point and line, respectively. Practically, the slope of the common tangent to two parabolas is the solution of a cubic equation. Very nicely explained in R. Geretschlager's Geometric origami book. Maybe a future video? Since this topic is so rich, it would be nice to see more videos about. Thank you for your videos and the links, they are a rich source of wonder.
There's a harmonic series hidden in the pascal's turtle: look at the little spiral of the triangles curling to the side and calculate the visible area! I'm sure there are other neat little things too.
I would love to see the complex root generalization! Please make a video about it. I can't find any working sources
eferences about that one, at least not in English.
This is simply beautiful. I plan to use this when I next come across a cubic or quadratic
لا يمكن لشخص أن يرى هذه الروعة ثم لا يشعر بالإلهام، محتوى مذهل، نوع من السحر، أشياء مختلفة تترابط مع بعضها البعض بشكل جميل ومذهل، شيء يستحق التقدير
I'm really excited now, could there ever be a video explaining Galois Theory and algebraic solvability using this turtle method? Like giving a proof we can't solve the general quintics, sextics or equations of higher degree, by radicals agebraically, using turtles?
That'd be awesome if it is possible without making it too abstract. Anyways, awesome video, loved it.
No, because this turtle method (as it stands) only locates real numbers and ignores complex solutions altogether (while some complex solutions are expressible by radicals). Also, trying to find a nice visual criterion for telling whether a slope is expressible by radicals seems hopelessly difficult.
Nice try though
@@neutralcriticism4017 Fair enough, I don't know if the generalization he talked about could help or not, but it seems like your second point is more decisive.
I actually had asked this to see if there was a nice way to tell, you never know.
@@Balequalm There was a comment on this page somewhere claiming that a generalization to account for complex numbers is possible as well as speculating about finding solutions in mod n.
It's laser-shooting turtles all the way down.
I think I have seed this reference. Where is it from?
@@ruchicharan4881 en.wikipedia.org/wiki/Turtles_all_the_way_down
@@ruchicharan4881 It's Bertrand Russell, or Oscar Wilde, or someone like that. :-)
@@RolandHutchinson - Why, I wonder, when we have the internet at our disposal, do we make guesses.
@@clearasmud376 Because nobody knows. The turtle story is apocryphal and has been atached to various names, notbly including Russell's. I threw in Wilde just for general effect, as he is one of those figures that sayings and stories get attributed to, along with Will Rogers, Benjamin Franklin, et al.
Dear Mathologer; your mind is brilliant and your videos very educational.
Another highly enjoyable video. Your animated visualizations provide a way for an inumerate person like me to appreciate the wonders of maths. The turtle triangle at the end should be on one of your T-shirts, or maybe it already is.
If the turtle starts facing left, and make right turns, the laser slope is exactly the solution without having to multiply with -1. Why is it not presented this way?
Because of the tyranny of the right-handed!
Perhaps the original focus was on factorisation rather than roots. The terms in the factorisation are the negatives of the roots, of course, so the method would give them directly.
I love the emergent golden spiral in pascal's turtle.
it's actually the spiral of theodorus en.wikipedia.org/wiki/Spiral_of_Theodorus
No, it's none of them. It's actually continuously divided for the square root of 2.
Elegant!
I love these videos because they are about my comprehension and beyond. Were I to take up mathematics in my spare time and study it beyond the necessary cramming to pass university classes, I will already have been exposed to these higher mathematical concepts. Exposure is the first step to go from clueless to mastership, so I have that first advantage.
Thank you for going beyond the Maths concepts that many instructors stop at (i.e. Algebra).
thoroughly blown my mind, the graphical nature of this technique suits me perfectly!
When I finally understood factoring polynomials, Etc… it was like the wool had been pulled from my eyes - and it was extremely frustrating. Lol. One gets the feeling that something was being deliberately miscommunicated, or not mentioned at all, as it was a little bit difficult to fail to notice all of the ways one could intuit abstractions and solve, after the fact.
Great video, but what about complex solutions? I tried to draw the turtle path for x^3+x^2+3x-5, which have a real solution of x=1 and complex solution of x=-1+-2i. The real solution worked out fine, but I was unable to draw the complex one. Should I draw it in 3D? Also, Is the same method still applies if we are solving higher order equations (with degree > 3)? From the pascal triangle animation in the end, I suppose the answer is yes, if that is the case, what happens if the turtle path intersect each other (or formed a closed path)? I hope you can address these questions in the following video.
I presume the answer would be that the laser has to hit each line in turn. You could imagine that there is a vertical dimension as well, but that we only care about the components on the horizontal plane.
Did you try having it "reflect" off the extensions of the lines like he mentioned? I doubt that helps with complex solutions but maybe with higher-order polynomials
Complex numbers are like lasers that hit a wall, go into a wormhole, then somehow end up hitting the turtle
According to wikipedia, a later paper by Lill dealt with the problem of complex roots.
Epic. Learned a lot. By the end when the distances were adding up nicely, I was already thinking 'well of course that happens!'
Best part about the solving process is that it's essentially gluing together similar triangles..
every video of yours is a gift to humanity
Today in my school, I showed this method and my friends went crazy. ''How you can solve cubic equations using turtles and lasers!?''
the future of warfare is large armored nuclear reactors bristling with arrays of lasers, kind of like turtles
The actual future of warfare is entirely digital and social. With the reliance of modern society on so much technology, you could bring a nation to its knees just by hacking and sabotaging key infrastructure. Why build weapons the international community could try to outlaw when you can just hack into a power plant and make it go haywire? Or a major water treatment facility and make it dump toxic stuff into the watertable? Oh shit, the entire communication system is down. Alternatively, hamstring their politics by messing with the information their populace and politicians get and which voices are heard. Possibly get someone working for you into high political office that way. And if you get caught, it was a group of trolling script kiddies you're making some effort to find and punish.
Or just slowly buy up their most important corporations, possibly assisted by promoting politicians that are open to loosen regulations that would hinder those acquisitions.
OR infiltrate the enemy's institutions and leak a cultural ideology of self-destruction into its mainstream.
In the end I couldn't stop the tears anymore, because it was so beautiful.
Mathloger - this is yet another example showing the mysteriousness of reality - that things are so much more than the sum of their parts, their interconnecting and varied associations creating properties that are so often not anticipated but so elegant!
I am not sure if someone has posted their solution to the proof for the problem proposed at 9:12 but I will post it regardless because I am happy that I did it all by myself.
Note: For the following, Σ means the sum from k=0 to k=n
p(x) = Σa_(n-k)*x^(n-k) with root r
p(r) = Σa_(n-k)*r^(n-k) = 0
q(x) is the new polynomial where the second, fourth, sixth, etc coefficients are multiplied by -1
q(x) = Σ(-1)^k*a_(n-k)*x^(n-k)
q(-r) = Σ(-1)^k*a_(n-k)*(-r)^(n-k)
= Σ(-1)^k*a_(n-k)*(-1)^(n-k)*r^(n-k)
= Σ(-1)^n*a_(n-k)*r^(n-k)
= (-1)^n*Σa_(n-k)*r^(n-k)
= (-1)^n*0 = 0
A lot of messy text but in any case, the result is proven.
Why do you have n minus the instead of just n or k?? Isn't that out of nowhere and needlessly complicated? And why is there n minus k in front of x.we are not taking derivatives here right? Sorry Don't get it.
@@leif1075 hello. The choice of n-k is probably just so that the polynomial is ordered from largest degree to smallest but that is ultimately meaningless. You could just do k if you want. Furthermore, to address your question about the n-k in front of the x, it is the index of a. For example, a_0 is the constant term of the polynomial and the x term has coefficient a_1. The _ symbol indicates an index. In general the coefficient of x^j is a_j.
So the laser path interpreted as a turtle path is just the result of dividing the equation by (x - laser_solution)?
That would also explain why the paths all have one segment less than the previous one
Fantastic! Surprise discoveries from the beginning to the end!! If there is the video about the complex solutions would be nice to put some reference in the description or link frame at the end of the video!
Awesome explanations as usual! Thank you Mathologer.
Thank you.
Now, I know a funny way of solving quadratic equations.
Great video. Still waiting for "2 plus 2 equals -8" you promised in "negative times negative" video :))
Mod 12 its true
I just found this channel today but my god i love it
Absolutely wonderful. Tremendous, addictive, fascinating, and mysterious. Leaves a feeling that you are about to discover a universal truth - so tantalisingly near but frustratingly out of reach. Mathologer Rules!!!!
Why haven’t I seen this before ? Why wasn’t I taught this in school ?
Pretty sure that hardly anybody knows about all this anymore. Hopefully this video will change this :)
Iterative Turtles...
Damn, so it _is_ turtles all the way down.
Please make a video extending the topic. The extensions you name at the end sound really really cool!!!
Nice! This gives an interesting way to find the golden ratio using a compass and a ruler.
Lol I couldn’t stop laughing. A turtle and lasers to solve this problem - this is mad; this is genius!
This looks like something that might be best dealt with using a square-root of -1 to represent the rotation ... makes me think a little.
Jawdropping demonstration. Animations spot on again.
Well, I'm not the one who studied maths for 40+ years, but I'm too still amazed how such an almost obvious thing like Horner's scheme is so powerful and can lead to so many beautiful results and make life so much easier. I just want to say thank you, Mathologer, for all this videos, for how much funny and enjoyable they are and for how they make you ask yourself: "What do I REALLY know about this topic in mathematics?". That's truly something magical and "pretty-netty")
P. S. Greetings from Russia))
It's been many years since I practiced origami, but now I very badly want to fold a turtle with Pascal's turtle drawn on it.
how else do you solve equations?
Mathematica
Bashing
I use the quintic formula for polynomials!
The description of pascals turtle was just amzing
I'm impressed why this was never taught in any school I went to. This makes factoring so easy and the rules are so simple; you even showed why it works. I really disliked the lengthiness of guessing square roots by hand and found the iterative turtle solution a lot faster.
Just amazing, really amazing. An absolutely different vision of a known topic. Thanks for making such an interesting video.
Dankeschön!
:)
I can imagine how you extend this method to work for complex zeros. It should be also possible to solve polynomials mod n by tracing the turtles path on a sphere divided into n equal segments?
Woah, now that's remarkable! Tell us!
When you say sphere I think you mean torus (mod n both horizontally and vertically). When you say "divided into n equal segments" I think you mean "divided into n^2 equal squares".
@@ipudisciple yes and yes
Thank you Burkard. I needed to learn something beautiful today. 💙
Great video as usual. So insightful and fun.
Liked just from the title
nobody:
youtube reccomendations: SOLVING EQUATIONS WITH SHOOTING TURTLES WITH LASERS
you have to admit, it is a damn good click bait
11:30 That made my day. How in the world, as a maths graduate, I've never heard of this beauty?!
Your explanations and animations are just beautiful poetry!!! Thanks a 10^6!