Why don't they teach this simple visual solution? (Lill's method)

Welcome to another Mathologer video. Let me show you something really amazing: how to solve an equation playing laser tag with a turtle. That sounds very strange, but wait and see. Okay, here's an equation, here's our pet turtle, and here's my laser gun. Here is the turtle at its starting point, looking to the right. The leading coefficient of our polynomial is 1. This tells the turtle to walk one unit to the right. Now make a quarter turn counterclockwise. Now you are ready to upload. The second coefficient is 5, which tells the turtle to move forward 5 units. Yes, yes, I know he is a very fast turtle.

You probably don't like the look of that laser. Anyway, another quarter turn counterclockwise. The next coefficient is 7. Advance 7. One last turn. The last coefficient is three and the turtle ends its journey by moving another three units. Now we place our laser at the starting point and aim to mark the turtle. But of course a direct shot wouldn't be very sporting. So, like the pool players of old, we'll try to kill the turtle with a bank shot like

this

. BUT, our bounce rules are special. We have a strange laser that always bounces at a right angle, like

this

one.

And again. Omitted! But by tilting our laser shot slightly differently we can make an impact. There! Very funny, except maybe for the turtle. But what's the point? Well, believe it or not, we also found a

solution

to our polynomial equation. It turns out that to make the killer shot our laser beam started with a slope exactly equal to 1. And for the game we're playing here it turns out that this promises us that minus 1 is a

solution

to our equation. You do not believe me? Let's check. We'll see. Alright, five plus three, that's eight minus one minus seven.

That's zero. Ok, let's allow the turtle to return to the game. That's a cubic equation, so we hope we can find another solution and it turns out we can. Tilt the laser a little more. Okay, continue. There, a second blow. Here the initial slope is 3, which means -3 is a second solution and yes, there is a third solution too, but we'll get to that later. Surprised? I hope so, I definitely was. And with the right settings, our turtle laser

method

works for any polynomial equation. What are the settings? Well, obviously a higher degree polynomial will require more segments in the turtle's path, but as in our example, the turtle always makes a quarter turn counterclockwise after each segment.

Additionally, the turtle handles a negative coefficient in our polynomial by walking backwards instead of forwards. For a coefficient of 0, the turtle does not move but still makes a quarter turn for this zero segment. Now, one thing that can go wrong is that the next turtle segment may not be long enough for the laser beam to miss. We take care of this by allowing the beam to bounce off the full line extents of the turtle segments. Take a look at this. Then it misses but bounces anyway, bounces and ends up in one of those extensions too. It may seem a little strange to allow this.

On the other hand, you've already accepted fast turtles and strangely bouncing lasers, so it's definitely a little late to start objecting now. Anyway, trust me for now and keep going and you'll see that everything ends up being quite natural. So why does this crazy system work? Now let me take you on a tour of the strange mathematical world of turtle markings. This turtle shooting game is called the Lill

method

and is named after Austrian engineer Edward Lill, who discovered the method about 150 years ago. It seems that Lill's method was quite well known for a time, but is now largely forgotten.

I learned it thanks to a very nice article by Thomas Hull, a mathematician specialized in origami mathematics. We'll get to the origami angle soon. As for the turtles, I imagine some of you have already guessed how

they

got involved. If not, just Google turtle charts and everything will be clear. Anyway, as far as I can tell, it was also Thomas Hull who was the first to employ Turtles to explain Lill's method. I've set up this web page there so you can play with Lill's method. The link is shown at the top and also in the video description.

By entering the coefficients of a cubic polynomial, it draws the corresponding turtle path. You can then aim your laser by dragging your mouse. It really is quite impressive, isn't it? Now let's take a closer look at the changing numbers at the top. At the top is the polynomial p(x) whose zeros we are looking for. We have fired the laser with a positive slope of approximately 0.28. Then, evaluating p(x) minus that slope, we get approximately 1.4. Close but without banana. Now, and this is great, 1,4, etc. is exactly the distance between the blue and red points. I'll give the super cool test near the end of this video.

So in effect what we are doing when we shoot the turtle on some slope is evaluating the polynomial at some point x and then graphically adjusting things until p(x) becomes zero, of course p(x) can also take in a negative value that would look like this in the image. Well, I think we can all agree that solving equations alone by shooting turtles is really cool. But there is much more. What I want to do now is show you some of the super cute features and applications of this method. That includes a clever way to reinterpret turtle shooting to get free solutions to closely related equations and a clever way to solve quadratic equations by simply drawing circles, and a clever way to solve cubic equations using origami and some super efficient and fast iteratives. turtle hunt and finally I will also show you a very surprising, very beautiful and apparently new incarnation of the super famous Pascal's triangle.

And of course, this is Mathologer and along the way I'll also show you some evidence of how this all works. Quite a program but we will go at a snail's pace, at that pace. Okay, well, anyway I'll start with the

simple

st ideas and then move on to the increasingly challenging ones. Feel free to quit or start skimming when things get too scary. Okay, we're leaving. Here's the cubic again and the two ways we found to shoot our turtle to get the solutions minus 1 and minus 3. Let's flip the image vertically. Alright, this inverted diagram translates to solving another closely related equation.

What is the equation and what are its solutions? Well our turtle knows everything. So let's chase him down the road. Well, the turtle is on its way and again begins to walk one unit to the right. Then the usual counterclockwise turn. But that means the turtle walks backwards in the next stage. Therefore, there has to be a minus sign in front of the five. Spin again. Well, the turtle is facing forward again, so we can leave the plus sign in front of the seven in place. Spin once more. Back again and that's why the sign has to change again.

So we know the inverted path equation, but what about its two solutions? Well, because of the change, the initial slopes of our laser beam are just the negatives of the slopes we had previously and that means that the solutions to this new equation are plus one and plus three, the negatives of the original solutions. And of course this works for any polynomial equation. By changing every second sign in the polynomial, the solutions of the new equation are the negatives of the original solutions. Very cute, huh? :) Anyway, here's an easy first challenge: try to find proof of this fact without turtles.

As always you can respond in the comments. Now, by inverting and rotating this diagram further, we get more free solutions and information about other related equations. Here's a rotation that leads to something quite surprising. Then your second challenge: chase the turtle along its path and discover the equation it represents. How are this equation and its solutions related to the following original equation and what is the general principle? Share your opinion in the comments. So in my last video I tried to convince you that parabolas and quadratics are much more interesting than the tedious and aimless school exercises that we all seem to have suffered through.

Now here is more evidence. A really cool way to solve quadratic equations. Well, here is a quadratic equation that has solutions minus 1 and minus 2. A quadratic has three coefficients, so the corresponding turtle path has three segments and a laser beam solution consists of two segments that form a right angle. But there is a very old and very beautiful theorem about right angles and this theorem allows us to replace the trial and error method of turning the laser with drawing a

simple

circle. Do you know the theorem? No? Well, maybe he ever did? We'll find out in a second.

Start with a circle, draw one of its diameters and choose a point on the circumference. So this triangle will always be right angled. Remember that? This beauty is called Thales' theorem. Now look at this. Magic, magic, magic. Very nice, isn't it? And you can see what it does, right? It tells us that to solve the quadratic equation using the turtle's path we can simply connect the end points, find the midpoint, and draw the blue circle. Then the two points of intersection tell us where to point our laser and also the solutions of the quadratic. How super nice that is.

It definitely makes my day when I hear about something like this. And you? If you draw the turtle's path on a physical sheet of paper, it is also possible to use paper folding based on Lill's method to solve quadratic equations, but there is more. Lill's method shows how cubic equations can also be solved by folding paper. This surprising discovery was made by mathematician Margarita Piazzola Beloch in 1936. Let me demonstrate how this paper folding trick works using our cubic example from earlier. Here we go. Here is the distance from the last red dot to the top horizontal line.

Draw or fold another horizontal line the same distance above. Here we go. Now do the same for this distance and draw a new vertical line. Okay, copy the final segment of the laser beam and notice that this copy can now slide comfortably between the two horizontal lines, it just fits. And we can do the same thing with the first segment of the laser beam that is there. It's time to start folding. Take the paper and fold it along the middle segment of the laser beam. So since we have a right angle here, this segment of the laser beam will fall just above its blue copy and therefore the red dot will end up on the green horizontal.

Likewise, the black dot will end up on the green vertical. Let's do this. There, magic :) There the red and black dots end on the green lines. This means that starting with the turtle's path we can find solutions to our equation by adding the red and black points on the green lines. Okay, let's do it. And deploy. So the folding of the paper fixes the middle segment of a successful laser path, right? And the rest is autopilot. Also a super nice construction, don't you think? And in case you're wondering, here's our second solution. Excellent! And maybe you've heard people say that origami is more powerful than ruler and compass.

Have you heard that? I don't have time to go into details here, but it is exactly the fact that paper folding can solve cubic equations that proves that it is more powerful than the ruler and compass, which can only handle quadratic equations. Well, one more super nice property before we get some testing details in order. Here's our cubic again. Recall that we found the solutions minus 1 and minus 3 corresponding to slopes 1 and 3 of the beam. Of course, we will find all possible real solutions by running the laser over all possible slopes, but there is also another iterative way to find new solutions.

To start, search for a solution as usual. Now forget about the turtle path for a minute and pretend that the laser beam path is a new turtle path. How strange, huh? Turn your laser to find a solution for our new path. Now the surprising thing is that this solution to our new equation is also a solution to our original equation. In this case it is the minus 3 corresponding to slope 3. But why stop now? Let's do this one more time. Forget the turtle path again and turn the laser beam into a new turtle path. And here is a solution.

That's another minus 1, corresponding to slope 1 which, as we know, is also a solution to our original equation. Let's combine all our solutions into one diagram. Then we get the same solutions as before. However, the minus one solution appears twice, corresponding to two green angles. Why is that? Good whenIf we look closely at the polynomial, it is evident that minus one is a zero of multiplicity two. In fact, we can factor our cubic polynomial like this. There, the green solution minus 1 has multiplicity 2. This works in general. The iterative method collects the multiplicities of the zeros while Lill's basic method does not.

Very neat and also very, very mysterious. Why on earth should converting laser trajectories into turtle trajectories do what it does? I'll give some details on the test at the end of the video. But quickly, here's a sketch of what's happening. After finding the first green root minus 1, we get rid of one of the green factors found there. This leaves us with a quadratic and the second path of the turtle corresponds to this quadratic. After finding the blue zero of this quadratic, we get rid of the blue factor, leaving us with the only green factor. The turtle's final path corresponds to this linear equation and calculates the remaining green zero.

Very nice things. It's time to move on to testing. Now I'll show you why Lill's method and its iterated form work. As an incentive to hold on, at the end I will show you that beautiful incarnation of Pascal's triangle that I discovered or perhaps rediscovered while preparing this video. Math seat belts on? Here we go. I'll start with a sketch of a proof of Lill's basic method, again focusing on our cubic. So what I want to show is that our cubic function evaluated at minus thisThe slope is equal to the signed distance between the red and blue points.

To do this, here we are going to calculate these four distances successively. 1, 2, 3 and 4. Okay, so the first distance down is just the first coefficient which is 1, right? To calculate the other distances, note that the same green angle appears three times in the diagram and that the slope of the initial laser beam is just the tangent of the green angle. And then minus that slope is the input x for our cubic polynomial. Now do you remember your Sohcahtoa? The length of this yellow segment is so the green angle multiplied by water 1, which is minus x. Next, this blue side of the turtle passage has a length of 5.

Therefore, this water segment has a length of 5 minus minus X. Now Sohcahtoa again. The next yellow segment is as long as the green angle times 5 plus x, which is this. The next blue segment of the turtle passage has a length of 7, so the next water segment has this length. Do you feel like it, mmm? Repeating this calculation one more time gives us the distance between the blue and red points. There. And this turns out to be our newly written cubic polynomial in a very special way. To check that this is really our initial cubic, we simply expand.

So once, twice, yeah, that's our cubic. And so if we find an x ​​to make this final water distance 0, then we also find a solution to our cubic equation. And that's why Lill's method works. Tada, magic, right? But it gets even more magical. This special form of our cubic is called Horner's form. I still remember learning about Horner's form and a mathematical miracle related to Herrn Schwenkert, the incredible high school

teach

er who sparked what became my lifelong obsession with mathematics. That was more than 40 years ago in Germany. Yes, I'm that old :) So it was a wonderful surprise to discover that, in essence, our turtle labeling game is also simply "synthetic division", the miracle of Horner's form that Herrn Schwenkert showed me all those years ago.

Let me finish by telling you about this miracle which will then also explain why turtle iteration works. In the first instance, Horner's form offers a very simple and efficient way of evaluating a polynomial. Let's say we are seeing that the specific value x is equal to negative 2 corresponding to an initial slope of +2. There. Then we evaluate the polynomial from the inside out like this: 1 times negative 2 plus 5, that's 3. Minus 2 times 3 plus 7, that's 1. And finally, negative 2 times 1 plus 3, which is equal to 1. So , the lengths of our water segments are just the intermediate results of this super efficient way of evaluating the polynomial.

However, and this is the miraculous thing, when calculating these numbers we also perform what is called synthetic division, which means that we also divide our polynomial by x plus 2 in a very sneaky way. How, what, why, where? Well, the result of dividing a cubic like this is a quadratic plus a remainder, right? Now it turns out, and you should really check this, the three coefficients of this quadratic polynomial are the first three intermediate results of our evaluation, those three over there, and the rest is the final water number, just like that. So again, we can divide a polynomial by a linear factor simply by evaluating its previous form.

Pretty miraculous, right? And in itself it is a trick worth learning and memorizing for the rest of our lives, don't you agree? Now, for iterating turtles, the important case is when the linear factor corresponds to one of the solutions to our equation. In this case the rest fades like this... We're almost there. The remaining solutions of our original equation are then the solutions of this quadratic equation,..., whose total path is this. But, as you can see, and as a simple trigonometry exercise demonstrates, this quadratic turtle path has exactly the same shape as our cubic laser path.

And that means we can use the laser path to solve the quadratic equation. And that's why turtles iterate. Let it sink in. Understood? Very, very cool, right? And that's all for today. There were some more fascinating aspects that I was tempted to cover, such as the generalization of Lill's method to also find the complex zeros of our equations, the beautiful characterization of the equations that correspond to closed turtle paths, and what happens when we we fold. turtle and laser trajectories at angles other than 90 degrees. Maybe another video. But for now, if you're interested, check out some of the references in the description.

In closing, let me keep my promise and show you an animation of that beautiful incarnation of Pascal's triangle in the path of the tortoise. I haven't seen it mentioned anywhere, so this might be a nice little original discovery. However, if any of you have seen it before, please let me know. Anyway enjoy and goodbye for now.