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.
Loved the 3d printing aspect of this video. Why is it you think that there seems to be so little interest in 3d printing on this subreddit? At least I cannot remember any post about anything 3d printing related and there is so much great stuff being done in this space, for example, Henry Segerman's topological prints.
Interesting, two posts of the same Mathologer video. Which one will do better?
Here is an interactive simulation of Nothing Grinder.
Anyone knows where can I buy something like this? I'd really want to gift this as a Xmas present, but it's tricky when you don't own or have access to a 3D printer.