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.

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

Great explanations but i understand nothing

Elliptical trammel… If you have read kinematics of machines

All this trig talk makes me want to watusi with a cardboard al Tusi

the emphasis in "ellipse" should be on the second syllable, not the first.

I think 3 total points for each ellipse

11:15

Why does proving alpha_red = 2*alpha_green prove that the 2 arcs are the same length?

Awesome video! I learned a ton. You got a new sub!

Fabulous! Thank you so much for sharing knowledge.

https://www.instagram.com/p/Bt_n8golYO0/ pretty cool jewelry nothing grinder piece

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: https://www.youtube.com/watch?v=ZpCWSzvy5O4

reminds me of how sound vibes appear on a sand plate also would make a great rotory engine

Wow…. My mouth was completely open for the whole vedio… Too good!!!

I see another way

take any point on the rod

the sumof its distance from both points (those screws) remains constant its a property of an ellipseread this 🙂 http://digitaleditions.walsworthprintgroup.com/publication/?i=294160&article_id=2426405&view=articleBrowser&ver=html5#{%22issue_id%22:294160,%22view%22:%22articleBrowser%22,%22article_id%22:%222426405%22}

Watching this makes me wish that I had a teacher like you when I was in school!

There have been many steam engines and at least one design of four cylinder petrol engine using this linkage.

Thanks for the inspiration: https://www.youtube.com/watch?v=LN3e6v9pYDQ

I'd love to hear you explain why, when a cone is cut by a plane we get an ellipse and not an egg shape.

The nothing grinder can be used to control flows. Very entertaining and mesmerizing video!

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

frames

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.

I'm sure that out there in the world, someone, somewhere, is interested in this……..

I use that as a weed grinder. It works great.

Smoke grinder

Halo from Dagestan, спасибо за russian language.

Love your visuals

Is that Babooshka by Kate Bush ?

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.

https://www.youtube.com/watch?v=gB9n2gHsHN4

Fractals are typically not self-similar

3Blue1Brown

,Published on Jan 27, 2017

An explanation of fractal dimension.

* https://youtu.be/_I-NTj7UaKM?t=569

nptelhrd

,Published on Nov 20, 2014

Cryogenic Engineering by Prof. M.D. Atrey , Department of Mechanical Engineering, IIT Bombay

Just saw a video that had rocks with star shaped boreholes that look just like 7:56.

The final animation you showed really reminds me of a recurring fever dream i have had as a child. 😮

each elipse would be unique, I think.

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!

thank you!

At 7:37 is a mathematical analogy of a rotary engine.

What defines the orientation of the elipsis if this is symetrical?

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

your Z offset is off.

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

you could make a fancy pizza cutter.

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

I would really like to smoke some hash oil laced with opium before I watch this video.

So would this be the opposite of an involute curve and whats the mathematical term

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 www.cdadd.com – 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 www.cdadd.com

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., ….

www.cdadd.com

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.

I have designed an engine that makes the same path as your nothing pump

Zig says he once made a tune out of spit and called it a spit-tune!

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?)

I used a homemade one as a router holder to rout ellipses in woodworking.

The nothing grinder looks like some weird combustion engine

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.

Shapes man just shapes

instructor to clear ,

head exploded, taking nap

Am I just high or is this dudes accent like an evil german

Thank you maths man

Basically an ancient fidget spinner.

Is this not the concept of single engine planes.

Really interesting video. 😀👍

make it an engine 6 cylinders 3 pistons cars would burn less gas

haha I've made a youse from a nothing grinder

Now how do I turn this into a Smash Bros. Stage?

The thing you use to spin looks like a giant Pee pee

Me barien hurtz

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?

THIS FUCKING TRIPPED ME OUT

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

I just learned about sinusoids today and he happens to have a sine joke on his T-shirt. 👌

The statue smile

Trying to defeat a goblin army on terraria while listening to this isn't funny

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.

My high school's cool so I'll try the 3d printer there

Pausing the video at 5:11 to tell you that yes, it's beautiful!

Nice music to.

isnt this the technic that rotary engines uses?

Only a couple of seconds into this vid, and I can already say, I hate your shirt.

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

outsideof another circle n times larger a while ago. This reminded me of it as the circle rolling around theinsideof 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

keepit a *secret*! Why didn't you tell the world, EH?"7:00 Why are the animations laggy? Reminds me of my own computers…

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.

great video.

😂 👍 subscribed!

Hello y'all from Suwannee County Florida 1st amendment auditors always watching an filming the Police 🚔

Do a video on topology

Your using it backwards. Spin the base, not the arm.

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?

THE BEAUTY OF "MATHEMATICS" ♥

Fabulous

why is the the ellipse squashed on one axis and not the other?