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Welcome to another Mathologer video. Today's mission is to do nothing. Well
sort of. Today we'll reveal the secrets of the
mysterious trammel of Archimedes also known as the nothing grinder. This gadget
here is the basic model but there are many
more complicated incarnations. Lots of really satisfying visual aha moments and
beautiful maths coming your way. Enjoy :) Ok, let's have a look at what this
thing does. And, yes, at first glance it really does
seem to do nothing. It just spins and spins like a particularly pointless
fidget spinner. Hence the colloquial name nothing
grinder or do nothing machine. A lot of people even call it the bullshit grinder.
I did not make this up, promise. But first impressions can be misleading. Let's zoom
in to have a closer look. I've highlighted the point on the arm exactly
in the middle between the two screws. What curve do you think it draws? Well of
course any time someone asked you that it's a good bet that the answer is "a
circle". And it sure looks like a circle. And looks are not deceiving, yep it's a
circle. Neat! Here I've marked a couple more points along the arm. The blue
button traces a perfect ellipse and so do all the other buttons. Now of course
ellipses are some of the most fundamental curves in mathematics and
nature with planets zooming around the Sun on elliptical orbits and so on.
Turns out the do-nothing machine produces ellipses of all possible shapes.
Super neat don't you think? Mathematically probably the easiest way
to construct all ellipses is to simply squish a circle in one direction. For
example, here are the ellipses that we just saw produced by the nothing grinder.
Alright, neat huh. Here's a puzzle for you:
Given one ellipse of a particular shape, say the blue ellipse, how many points on the
arm of the nothing grinder trace an ellipse of the same overall shape. Here
I'm assuming, in typical mathematical denial of reality,
that the arm is in fact an infinitely long ray that continues beyond where
the physical arm stops. Share your thoughts in the comments.
Now since ellipses are super important and since nothing grinders are super
good at drawing them is there maybe a practical use for our nothing grinder. Well
not so much now but in the good old pre-computer days the ellipseograph was
indeed a standard and important mechanical drawing tool. So there's a
picture of a really beautiful antique ellipseograph. You can adjust the
positions of these bits over there to draw ellipses of many shapes and sizes.
Here is a different nothing grinder featuring three sliders instead of two.
Mesmerizing isn't it. Also pretty amazing when you think about it. Two linear
sliders giving two degrees of freedom to allow the arm to spin in a fixed way
makes sense. But how come it is possible to insert a third linear slider into
this setup without the whole thing seizing up? Oh, and by the way, I 3d
printed the model over there and I'll link to 3d printable STL files of this
and other nothing grinders in the description. Some early Christmas
presents for all of you. These models print out perfectly without adding any
supports on my monster Zortrex 3d printer but mileage will almost
certainly vary depending on what sort of printer you have. Let me know in the
comments if you succeeded in printing a copy. Okay so what sort of curves does
this more complicated do-nothing machine trace, what do you think? Maybe it's a
little surprising but nothing new happens. This thing also traces ellipses
and nothing else. So the three screws you see here are the corners of an
equilateral triangle and the midpoint of this triangle again traces a circle.
Unfortunately my aim was slightly off when I pushed the pink pin in and so we
don't see a perfect circle here but a slightly squished one. All very pretty
but where do these circles in the middle come from? Why can you have
more than two linear sliders? And why all those ellipses? I know you won't be able to
sleep tonight unless you know the answers to these questions
so let me inflict some really beautiful and surprising explanations on you. What
do you see? A little circle of points rolling inside a large circle? Sure, but
do you also see a bunch of lines? No? Let's make it clearer. Whoa, I bet you did
not see that one coming. Really amazing don't you think? I still
remember being very taken by this the first time I saw it. So what's going on
here? This phenomenon is known as the Tusi couple named after its discoverer the
13th century mathematician and astronomer Nasir al-Deen al-Tusi
Regular mathologerers will remember the Tusi couple from our recent video on
epicycles and Fourier series: if a circle rolls inside a circle of twice the size
then any point on the circumference of the small circle traces out a diameter
of the larger circle. Super duper pretty :) That's exactly what you see in this
animation: eight points on the circumference of the small circle
tracing diameters of the large circle. And when we focus on just these two
diameters here and the points moving on them we're looking at an exact replica
of our original nothing grinder. The Tusi couple also makes it clear at a glance
why nothing grinders can have as many linear sliders as we wish. So another way
of looking at this animation is to interpret it as a nothing grinder with
eight sliders and with pivot points evenly placed around an invisible rolling
circle. Here is a six point grinder I printed, complete with the stationary
large circle and the small rolling circle.
It's also now really easy to see that the midpoint of the pivot points is
tracing a circle. Why, well this midpoint is the center of the rolling circle,
which of course traces another circle. At the end of this video I'll also explain
where all those ellipses come from and why the Tusi
couple does what it does but before I do this here is a quick show-and-tell of some
other pretty stuff. Here again is the basic setup with the rolling circle
highlighted. Let's first play with the position of the pivot points on the
rolling circle and move them inside the circle.
Alright here we go. Then, as shown, instead of line segments these pivots will now
trace ellipses this means that we could have the sliders run in elliptical
grooves instead of straight grooves and still have a smoothly working nothing
grinder. So let's have a look at this. That's what it would look like. Next, if
we modify the size of the rolling circle other interesting things start happening.
Here we go. Let's roll! Yep it's spirograph time. If we have
both sliders move along the red trefoil groove, then other points on the arm
trace rounded triangles. And we can get rounded squares... and pentagon's and a
lot of other spriography curves that I talked about in the epicycle video. The
3d printing part of all this is still work in progress but you can see I'm
having a lot of fun again. Now to mathematically round of things, let me
show you where all those ellipses come from. We begin with the familiar unit
circle in the familiar xy-plane and head out from the origin at an angle theta. Then
the point on the circle has x-coordinate cos theta and y coordinate sine theta.
Now let's stomp on the circle squishing it into an ellipse. This amounts to
multiplying the y-coordinate by some small scaling factor a. As theta varies the point sweeps out our
ellipse and so this gives the parameterization of the ellipse. The
theta is the theta of the original circle. We can still clearly see the
x-coordinate cos theta of the original triangle in the ellipse. So there we go.
We can also visualize the y-coordinate in a scaled down triangle,
with hypotenus a, like this. Ponder this for a moment. All under control?
Great! Now just bring these two triangles into alignment and the do-nothing
machine materializes right there in front of our eyes:) Now as we change the
theta the arm traces our ellipse. Super neat and very natural, isn't it? And what
this also shows is that our picture that goes with the standard parameterization
of an ellipse is a natural generalization of the picture that goes
with the standard parameterization of the circle that most of you will have
done to death in school, right? Let's go back and forth a couple of times, really
pretty, isn't it? So unbeknownst to you, every time you drew the circle diagram
you were just a mini step away from understanding the fabulous do-nothing
machine. Recently 3blue1brown did two nice videos in which he talked
ellipses. What I just showed you also makes a nice addition to these videos, so
definitely also check out the 3blue1brown videos if you haven't seen them
yet. And that finishes the official part for today. Hope you enjoyed this video.
BUT for those of you who like their maths to be even more mathsy stick around a
little longer and I'll show you a pretty visual proof that the Tusi couple
draws straight lines. Okay, here's the starting position for the little rolling
circle. I want to convince you that the red point will really run along the orange
diameter. Let's roll it a little bit. So if al-Tusi is correct, where in this picture
should the red point now be? Well, obviously, here on the orange diameter.
How can we prove that it's really there? Well what we have to show is that these
two arcs along which the two circles have touched during the rolling action
have the same length. Remember that the larger circle has
twice the radius of the smaller circle with proportionally larger arcs. So to
prove that the green and red arcs are the same length, we simply have to show
that this green angle here is half this red angle. But showing that the red is
twice the green is easy. Here's the first green angle inside the red one, there we
go. Now here is an isosceles triangle with pink sides equal and that means we also
have a green angle over there. But then this zigzag here shows that we've got
yet another green angle here and so two green angles make a red. Tada
the magic of maths :) and that's really it for today.