Welcome to another Mathologer video. Let
me show you something really amazing, how to solve an equation by playing laser
tag with a turtle. That sounds very strange but just wait and see. Okay
here's an equation, here's our pet turtle and here's my laser tag gun. Here's the
turtle at his starting point, facing to the right. The leading coefficient of our
polynomial is 1. This tells the turtle to walk one unit
to the right. Now he makes a quarter turn in the counterclockwise direction.
He's now ready to go up. The second coefficient is 5 which tells the turtle
to advance 5 units. Yes, yes I know it's a very speedy turtle. Probably doesn'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 finishes his trip by advancing another three units. Now we place our laser
at the starting point and aim to tag the turtle. But of course a straight shot
wouldn't be very sporting. So like billiard players of old we will try
to zap the turtle with a bank sho,t like this. BUT, our bouncing rules are special.
We have a weird laser that always bounces off making a right angle, like
this. And again. Missed! But by angling our laser shot slightly differently we can
arrange a hit. There! Great fun, except maybe for the turtle. But what's the
point? Well, believe it or not, we've also found a solution to our polynomial
equation. It turns out that to make the killer shot our laser beam began with a
slope exactly equal to 1. And for the game we are playing here it turns out
this promises us that minus 1 is a solution of our equation. Don't believe
me? Let's check. Let's see. All right five plus three that's eight minus one
minus seven. That's zero. Ok let's allow the turtle back into the game.
That's a cubic equation down there so hopefully we can find another solution
and it turns out we can. Tilt the laser a little more. Okay keep going. There, a
second hit. Here the initial slope is 3 which means that -3 is a second solution
and yes there's also a third solution but we'll worry about that later ok.
Surprised? I sure hope so I definitely was. And with suitable adjustments our
turtle laser method works for any polynomial equation. What are the
adjustments. Well, obviously, a higher degree polynomial will require more
segments in the turtles path but as in our example the turtle always always
makes a counterclockwise quarter turn after each segment. Also the turtle deals
with a negative coefficient in our polynomial by walking backwards instead
of forwards. For a 0 coefficient the turtle doesn't move but he still makes a
quarter turn for this zero segment. Now one thing that may go wrong is that the
next turtle segment may not be long enough so that the laser beam will miss.
We take care of this by allowing the beam to bounce off the full line
extensions of the turtle segments. Have a look at this. So it misses but
bounces anyway, it bounces and also ends on one of those extensions. It may seem a
little strange to allow this. On the other hand, you've already accepted speedy
turtles and weirdly bouncing lasers so it's definitely a little late to start
objecting now. Anyway trust me for now and just run with it you'll see it all
ends up being quite natural. So why does this crazy system work? Let me now take
you on a tour of the bizarre mathematical world of turtle tag. This
turtle shooting game is called Lill's method named after the Austrian engineer
Edward Lill who discovered the method about 150 years ago. It seems that Lill's
method was quite well known for a while but it is now largely forgotten. I
learned about it from a very nice article by Thomas Hull,
a mathematician who specializes in origami mathematics. We'll get to the
origami angle soon. As for the turtles I imagine a number of you have already
guessed how they got involved. If not just Google turtle graphics and all will
be clear. Anyway, as far as I can tell it was also Thomas Hull who was the first
employed Turtles to explain Lill's method. I've set up this web page over
there so you can play with Lill's method. The link is shown at the top and also in
the description of the video. On input of the coefficients of a cubic polynomial
it draws the corresponding turtle path. You can then aim your laser by dragging
the mouse. There, really quite stunning isn't it? Let's now have a closer look at
the changing numbers at the top. At the top is the polynomial p(x) whose zeros
we are hunting. We've shot the laser at a positive slope of about 0.28. Then
evaluating p(x) at minus that slope we get about 1.4. Close but no banana. Now,
and this is super cool, 1.4 etc. is exactly the distance between the blue
and red points. I'll give the super super cool proof near the end of this video. So,
in effect, what we're doing when we shoot at the turtle at some slope is to
evaluate the polynomial at some point x and then we graphically adjust things
until p(x) becomes zero of course p(x) can also take on a negative value
which in the picture would look like this. Okay I think we can all agree that
just on its own solving equations by shooting turtles is really really cool.
But there's lots and lots 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 the turtle shooting to get free solutions to closely related
equations and a slick way to solve quadratic equations by simply drawing
circles, and an ingenious way to solve cubic
equations using origami, and some super efficient rapid-fire iterative turtle
shooting, and, finally I'll also show you a very surprising very beautiful and
apparently new incarnation of the super famous Pascal triangle. And, of course, this is Mathologer and so
along the way I'll also show you some proofs of how all this works. Quite a
program but we'll go at a turtle's pace, that pace. Okay, well anyway I'll begin
with the simpler ideas and then go on to the ever more challenging.
Feel free to bail or begin to skim when things get too scary. Okay off we go.
Here's the cubic again and the two ways we found of shooting our turtle to give
the solutions minus 1 and minus 3. Let's flip the picture vertically. Alright this
flipped diagram translates into solving another closely related equation. What's
the equation and what are its solutions? Well, our turtle knows it all. So, let's
just chase it around the path. Okay, the turtle is on its way and again begins by
walking one unit to the right. Then the usual counterclockwise turn. But that
means the turtle walks backwards on the next stage. Therefore there's got to be a
minus sign in front of the five. Turn again.
Okay the turtle is facing forward again and so we can leave the plus sign in
front of the seven in place. Turn once more. Backwards again and so the
sign has to change again. So we know the equation for the flipped path but what
about its two solutions. Well, because of the flip the starting 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.
Changing every second sign in the polynomial the solutions to the new
equation are the negatives of the original solutions. Very cute, hmm? :) Anyway here's a first easy challenge for you: Try to find a turtle-free proof of this
fact. As always you can answer in the comments. Now by
further flipping and rotating this diagram we get more free solutions and
insights into other related equations. Here's a rotation that gets us to
something pretty surprising. So your second challenge: chase the turtle around
his path and figure out the equation it represents. How are this equation and its
solutions related to the original equation below and what's the general
principle? Share your thoughts in the comments. So in my last video I tried to convince
you that parabolas and quadratics are much more interesting than the aimless
and tedious school exercises we all seem to have suffered through. Now here's some
more evidence. A really, really cool way to solve quadratic equations. Okay here's
a quadratic equation which happens to have solutions minus 1 and minus 2. A
quadratic has three coefficients and so the corresponding turtle path has three
segments and a laser beam solution consists of two segments making 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 approach of swivelling
the laser with drawing a simple circle. Do you know the theorem? No? Well maybe once did? We'll find out in a second. Start with a circle, draw one of its
diameters and pick a point on the circumference. Then this triangle there
will always be right-angled. Remember that? This beauty is called Thales theorem.
Now watch this. Magic, magic, magic. Very pretty, isn't it and you can see what it
does, right? It tells us that to solve the quadratic equation using the turtle path
we can just connect the end points find the midpoint and draw the blue circle.
Then the two intersection points tell us where to aim our laser and so also the
solutions to the quadratic. How super pretty is that. Definitely
makes my day when I learn about something like this. How about you? If you draw the turtle path on a
physical piece of paper it is also possible to use paper folding based on
Lill's method to solve quadratic equations but there's more. Lill's method
shows how you can also solve cubic equations with paper folding. This amazing
discovery was made by the mathematician Margarita Piazzola Beloch in 1936.
Let me demonstrate how this paper folding trick works using our cubic
example from before. Here we go. Here's the distance from the final red point to the
top horizontal line. Draw or fold another horizontal line the same distance above.
There 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 this copy can now slide snugly between the two horizontal lines, there
just fits. And we can do the same with the first segment of the laser beam over
there. Time to begin folding. Take the paper and
fold along the middle segment of the laser beam. Then because we've got a
right angle here this laser beam segment will fall smack on top of its blue copy
and therefore the red point will end up on the green horizontal. Similarly, the
black point will end up on the green vertical. Let's do this.There, magic :) There
the red and black points end up on the green lines. This means that starting with the path of the turtle we can find solutions to
our equation by folding the red and black points onto the green lines. Okay
let's do it. And unfold. Then the paper crease pins down the middle segment of a
successful laser path, right? And the rest is auto pilot. Also a super nice
construction, don't you think? And just in case you're wondering, here's our second
solution. Great! 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 shows it's more powerful than ruler and compass
which can only handle quadratic equations. Okay, one more super nice property before
I tidy up with some proofy details. Here's our cubic again. Recall that we
found the solutions minus 1 and minus 3 corresponding to beam slopes 1 and 3. Of
course, we'll come across all possible real solutions by sweeping the laser
through all possible slopes but there's also another iterative way of finding
new solutions. To begin find one solution as usual. Now forget about the turtle
path for a minute and pretend that the laser beam path is a new turtle path.
Weird, hmm? Swivel your laser to find a solution for our new path. Now what's
amazing is that this solution to our new equation is also a solution of our
original equation. In this case it's the minus 3 corresponding to the slope 3. But
why stop now. Let's do this one more time. Forget about the turtle path again and
make the laser beam into a new turtle path. And here's a solution. That's
another minus 1, corresponding to the slope 1 which, as we know, is also a
solution of our original equation. Let's combine all our solutions into one
diagram. So we get the same solutions as before.
However, the solution minus one appears twice, corresponding to two green angles.
Why is that? Well when we have a close look at the polynomial it becomes
apparent 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 turns out to work in general. The iterative method picks up the
multiplicities of the zeros whereas the basic Lill's method does not. Very neat
and also very very mysterious. Why on Earth should turning laser paths into
turtle paths do what it does? I'll give some details in the proof at the end of
the video. But just 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 down there.
This leaves us with a quadratic and the second turtle path corresponds to this
quadratic. After we find the blue zero of this quadratic, we get rid of the blue
factor, leaving us with the single green factor. The final turtle path corresponds
to this linear equation and it calculates the remaining green zero. Very
pretty stuff. Time to get down to the proofs. I'll now show you why Lill's
method and it's iterated form work. As an incentive to stick it out, at the very
end I'll show you that very beautiful incarnation of Pascal's triangle that I
discovered or perhaps rediscovered while preparing this video.
Mathematical seatbelts on? Here we go. I'll begin 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 this slope is equal to the signed distance
between the red and blue points. In order to do this, we are going to successively
figure out these four distances here. 1, 2 3 and 4. Okay, so that first distance down there is just the first coefficient which is 1,
right? To calculate the other distances notice that the same green angle pops up
three times in the diagram and 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 remember your Sohcahtoa? The length of this yellow
segment is tan of the green angle times the aqua 1 which is minus x. Next, this
blue side of the turtle pass has length 5. Therefore this aqua segment has length
5 minus minus X. Now Sohcahtoa again. The next yellow segment has length tan of green
angle times 5 plus x which is this. The next blue segment of the turtle pass is
length 7 and so the next aqua segment is this long. Fancy, hmm? Repeating this
calculation one more time gives us the distance between the blue and red points.
There. And this turns out to be our cubic polynomial just written in a very
special form. To check that this really is our beginning cubic we just expand. So
once, twice yep that's our cubic. And so if we found an x to make this final
aqua distance 0, then we've also found a solution to our cubic equation. And
that's why Lill's method works. Tada, magic, isn't it. But it gets even more magical.
this special form of our cubic is called its Horner form. I still remember
learning about the Horner form and a related mathematical miracle from Herrn
Schwenkert the amazing high school teacher who sparked what became my
lifelong obsession with mathematics. That was over 40 years ago back in Germany.
Yep, I'm that old :) And so it was a wonderful surprise to discover that at
its core our turtle tag game is also just "synthetic division", the Horner form
miracle I was shown by Herrn Schwenkert all those years ago. Let me finish by
telling you about this miracle which then will also explain why iterating
turtles works. In the first instance the Horner form gives a very simple and
efficient way to evaluate a polynomial. Let's say we are looking at the specific
value x is equal to minus 2 corresponding to an initial slope of +2.
There. Then we evaluate the polynomial from the inside out like this: 1 times
minus 2 plus 5 that's 3. Minus 2 times 3 plus 7 that's 1. And, finally, minus 2
times 1 plus 3 which equals 1. So the lengths of our aqua segments are just
the intermediate results of this super efficient way of evaluating the
polynomial. However, and this is the miraculous bit, by calculating these
numbers we've also perform what's called synthetic division, which means that
we've also divided 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 that, and you should really check this, the
three coefficients of this quadratic polynomial are the first three
intermediate results of our evaluation, those three there, and that the remainder
is the final aqua number, so like that. So, again, we can divide a polynomial by a
linear factor by simply evaluating its horner form. Pretty damn miraculous, isn't
it? And just in itself a trick worth learning about and committing to memory
for the rest of your lives, don't you agree? Now for iterating turtles the
important case is where the linear factor corresponds to one of the
solutions of our equation. In this case the remainder vanishes like this... 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 one easy
bit of trigonometry proves, this quadratic turtle path has exactly the
same shape as our cubic laser path. And that means that we can use the laser
path to solve the quadratic equation. And that is why turtles iterate. Let it
sink in. Got it? Very, very cool, right? And that's just
about it for today. There were a few more fascinating aspects that I was tempted
to cover like, for example, the generalization of Lill's method to also
find the complex zeros of our equations, the beautiful characterization of
equations that correspond to closed turtle paths and what happens when we
bend turtle and laser paths at angles different from 90 degrees. Maybe another
video. But for now, if you're interested then check out some of the references in
the description. To end, let me fulfill my promise and show you an animation of
that beautiful turtle path incarnation of Pascal's triangle. I haven't seen it
mentioned anywhere so this may very well be a cute little original discovery.
However, if any of you have seen it before, please let me know. Anyway enjoy
and bye for now.
One of the coolest things I've seen in the last decade
There's a Python package called "turtle" that works exactly like the turtle shown in video.
Initially thought this was going to involve complex analysis, since the turtle's final position is basically the polynomial evaluated at i (up to some isometry)
Why does walker + runner = walker walker?
You'd think it was basic digit math, aka 1 + 10 = 11. But this doesn't work out unless runner = 10 (in whatever base you want), which isn't consistent with digit math.