Secrets of the NOTHING GRINDER

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.

100 thoughts on “Secrets of the NOTHING GRINDER

  • this do nothing machine is used in gasoline engine meany other devices of our modern world, with out it wouldn't be possible.

  • Nothing grinder ? Here on YouTube I saw a video someone use this to made a beautiful rotating expandable wood table

  • Hi mathologer,
    I recently find out a very very peculiar bug in math!!

    Here it is

    -1 = -1^(2* 1/2) = ((-1)^2)^1/2 = (1)^1/2 = 1 !!!

    I hope you can answer me, cause apparently no one else can

  • The two-axis grinder looks exactly like a floating arm trebuchet! It's apparently much more efficient than the medieval one, I wonder if there's a mathematical reason why.
    Here's a video of it:

  • I see another way
    take any point on the rod the sum of its distance from both points (those screws) remains constant its a property of an ellipse
    read this 🙂{%22issue_id%22:294160,%22view%22:%22articleBrowser%22,%22article_id%22:%222426405%22}

  • I am going to use this information to make a device that illustrates how 3 phase electricity can result in a rotating magnetic field.

  • If this looks complicated, never ever try to understand stellar motion, which is to say motion of things in space. Planets rotate and moons rotate and stars rotate, and planets orbit in ellipses of varying shapes and sizes, and all that madness moves to follow the star. Fun fact, our star moves in a cork-screw pattern along a relatively straight line, because why not.

  • now apply this Fractal mass roughness model by 3Blue1Brown to this "NOTHING GRINDER" and in turn the *Cryocoolers Ideal Stirling Cycle in real life, then make the "NOTHING GRINDER" reassessment again.
    Fractals are typically not self-similar
    ,Published on Jan 27, 2017
    An explanation of fractal dimension.


    ,Published on Nov 20, 2014
    Cryogenic Engineering by Prof. M.D. Atrey , Department of Mechanical Engineering, IIT Bombay

  • Isn't this thing popular for young children? I mean I remember playing around with this using a special ruler in elementary school.

  • Whats an elipse? Surely you mean oval? My printer is 17 years old and I just hope it doesn't fail soon. It's either eat or get a 3d printer. I choose food! Hopefully, when they come down in price to say $10, I may be able to get one, but until then they are WAY out of my price range! Man I love being dumber than 3 feet of mud! I can bypass this video with a happy heart and watch something FUN! Man this is boring! Back to the cat videos!

  • I'd say only one point, as the pattern seems to change the further/closer it get's to the center, being a circle.

  • Since ellipsis appear to be progressively becoming wider does it reach a point where it returns to a perfect circle?

  • Very cool! I never knew this mechanism existed. I bet this could be applied as a motor and pretty efficient too. Thank you for this knowledge that was unknown to me

  • the way he presents this contraption with his Germanic accent, he sounds like he could take over the world with this. I love his enthusiasm! I wish my school grade teachers showed such interest in what they taught instead of watching the countdown to their retirement.

  • Follow 'Fast-track' at – A quick observation that proves Opticks (with a 'k')(wrongly attributed to Newton) are wrong:- orient a prism to obtain 'rainbow' pattern, move prism to surface & observe 'rainbow' splits out to roy & vib patterns at apex points – thus prism does NOT split out white light, colours are NOT frequency related but arc-angle related – thus Einstein, Hubble, Higgs, CERN are wrong → e≠mc2 , etc., etc., …..
    → Understand how Optics (no 'k') and the Universe really functions ….. follow the fast-track at

    Opticks (wrongly attributed to Newton) are wrong → e≠mc2
    Refraction is NOT Refraction
    Perspective: singular proof that Einstein, Hubble, Hawking, Higgs, CERN etc. are wrong
    Quantum Theory contradicts Classical Physics because QT is wrong.
    etc., etc., etc. …….
    John Nash (Nobel Economics 1994) – models are defective -> massive socio-economic destruction
    Fermat's Last Theorem: Andrew Wiles' 'proof' is NOT a PROOF; cf Proof that 1=0; also Wiles' 'proof' too complex to PROVE a PROOF; CDADD has developed a classical PROOF.
    etc., etc., etc., ….

  • Wow. Finally something on youtube that is not BS. Worthy of being the first thing I was ever inclined to (and yes, did) subscribe to. Now, all I need to do is learn how to ignore some of the comments below that tend to make me regret it. Baby steps.

  • Well, if I didn't feel stupid before I certainly do now. It turns out that I remember absolutely nothing from high school apparently.

    If you will excuse me, I will go and find something more appropriate for my IQ, like some cat videos🤔.

  • hey, mathologer, This is not Archimedes! This is Archidamus the 3rd!!! (Spartan king) Please don't help spread this misinformation.
    For some reason, many institutions have used this bust of Archidamus, even though he is obviously wearing an armour (why would a mathematician wear an armour?)

  • The two black buttons trace ellipses, too. It's said that every line segment is more properly defined as an ellipse with a long axis equal in length to the line segment you're trying to define and a short axis of zero length. In fact, no other simple continuous function describes a line segment! Obviously, the sliders' velocity profiles follow the sinusoidal wave functions.

  • anyone else think wankle engine? could this be used in reverse so that instead of turning the crank to do nothing, a force can be used to expand the pistons and turn the crank? it would be a balanced engine too because essentially no matter how many pistols it had, they would be equally spaced and at different points of ignition based on the differences in compression of the little slidey rods. obviously this needs a cap or ring around the perimeter just beyond the apex of each rod's exterior movement in order to complete the compression chambers. anyone else getitng this or do i just see imaginary engines?

  • I might have drunk half a bottle of vodka but even I can see that that there are 4 curved points (or set of points) on the first do nothing machine, all the parts/sections can be combined to form a perfect circle and/or eliptical. Ok I'm going to bed I just took an hour to type this pray I dont have a hangover

  • Ya know, Nothing Grinders can be useful for having an array of buttons that are only supposed to have one on at a time, like a special kind of dial.

  • Danke für das Video. Hochinteressant und bereichernd. 
    An der Fachoberschule hatte ich während zweier Jahre die Fächer Darstellende Geometrie und Trigonometrie. Später an der Technischen Fachhochschule hatte ich nochmals zwei Jahre lang Darstellende Geometrie. Ich staune unaufhörlich darüber was ich immer wieder hinzulerne.
    P.S.: Danke auch dafür, dass Sie Ihren deutschen Akzent nicht krampfhaft zu bekämpfen suchen und stattdessen ein geschliffenes Englisch darbieten. Bei vielen unserer Landsleute ist es mittlerweile leider umgekehrt.

  • Actually a little more than "nothing". Woodworkers use this and variations of it as a jig for carving out out ellipses with a router.

  • I saw the MindYourDecisions video about rolling a circle around the outside of another circle n times larger a while ago. This reminded me of it as the circle rolling around the inside of another circle n times larger gets a free rotation, n – 1, instead of paying an extra rotation, n + 1, for going around.

  • I like your accent, can you please say "Of course, the whole point of a Doomsday Machine is lost, if you keep it a *secret*! Why didn't you tell the world, EH?"

  • My grandfather was a highly skilled woodworker, and he made one of these as a toy for us kids. Fifty years later, I see some of the related math!

  • I found a Trammel of Archimedes made of wood in the trash outside the arts building at my university. I seemed to recall that I had read about one before and that it was made first by an Arab or Persian mathematician….this led me to look up "Tusi Couple".   But my object was different.

    I then took it to a party of science educators. It was a hit and the host of the party decided it was a perfect ice-breaker…everybody at the party tried to guess what it was. Finally a math professor walked in and said "oh a Trammel of Archimedes!".

    Thanks for this video, since it allowed me to connect these two ancient machines.

  • 5:17 it's pronounced "ad-Din at-Tusi" – solar and lunar letters. The "al"-consonant doubles the 1st consonant for some consonants.
    That's how "Salah al-Din" became "Saladin" latinised. If you transliterate phonetically, you'll see him as "Salah ad-Din".
    "al-Gibr" stays so, "algebra".

  • Astonishing. Thank you for making and sharing. I had math up to college levels in the USA and never heard of this. 🧡 💛 💚 💙 💜 🖤

  • Looks like Archimedes might have been trying to come up with a valve to channel water or a type of water pump, beyond the one he allegedly invented. His pointless grinder reminds of the under ground aqueducts found under the Giza plateau in Egypt…where he studied.

  • Seem to remember a shaker screen mounted on a spring bed that did a great deal of sizing and separating rock and sand.., similar idea in practice. Springs replaced the sliders, and the body of the screen did the ellipse. Lots of Math required in the design and operation for Mechanical/Mining Engineering.

  • Can any two points on the arm draw the same ellipse? Intuitively I think not. The foci are moving out to infinity, and I think an ellipse can be uniquely described by it's foci. Is this right?

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