Wobbly Circles - Numberphile

Matt Parker on circles and centres of mass.
More links & stuff in full description below ↓↓↓
The man who loved circles: • The Man Who Loved Circ...
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Although correct, Matt does not use the most elegant method here, introducing more negatives than he would have liked. He apologises.
Correction: 45 degrees is equal to pi/4 radians
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[@Qwink27]

Dip the edges in ink, roll it. What shape does it make? Can you mathematically describe it? I want to know this.

[@mohammedyassin8064]

I feel that going to be an interesting thing

[@yingo4098]

@@mohammedyassin8064 it's gonna be a line segment

[@carlosandleon]

two parallel wavy lines with sharp crests

[@Daspletophysis]

Since the contact points occur along at most a semicircular portion of each disc, I imagine the trace of where these points contact the table would approximate 2 cycloids that are perfectly out of phase with each other.

[@Ziphoroc]

It would make lines. Duh

[@DavidRichfield]

The center of mass of a rolling square doesn't go in a zigzag; it goes in a series of circle arcs.

[@ianwubby6271]

+David Richfield I'd imagine the zig-zag was just easier to animate.

[@ARP2wefightforyou]

David Richfield Vsauce made a video about this.

[@ky-gp4sz]

David Richfield if you look closely you can tell it moves a lot more than the lines

[@kaina5467]

Looking for this

[@Pulpaldabadies]

Indeed. A rolling Parker square makes a zig-zag :)

[@TheBigBigBlues]

"Ah a new numberphile video, great!" *Watches Video* *Spends £30 on book and maths gear* "Cheers Brady, Beans again for me tonight."

[@numberphile]

TheBigBigBlues money well spent and nothing wrong with beans!!!! have steak on the weekend, but next week you can support us on Patreon and have beans again! :)

[@Number-cz1rd]

Numberphile For convenience he could perhaps offer a premium package that includes both a signed book and a supply of beans.

[@PeterBarnes2]

+Number 5 But it's the beans they used to teach you about counting at school.

[@Triantalex]

??

[@TheNefari]

Now for a little artistic experiment take these discs and put them into paint or ink and then roll them on white paper the line you will see is quite amazing :D

[@boaz2578]

Tell me, what will I see?

[@Beesman88]

Instead of Pi or Tau we should make a compromise. Pau = 1.5 Pi

[@epajarjestys9981]

It's a Ti(e).

[@munjee2]

Wau

[@frechjo]

From the perspective pi, 1.5pi is 50% away, from the perspective of tau, it's only 25% away. To be fair, it should be 1+1/3pi or 2/3tau.

[@jacobr7729]

That was an 0ld XKCD joke...

[@GeodesicBruh]

Fuci

[@jodyze5413]

''let's remove this mess of root 2 here'' *proceeds to add more root 2*

[@ob3vious]

In maths we add to subtract

[@h-Films]

no he has an abomination of root 2 not a mess =D

[@electroislove]

Matt has this subtle sense of humour, which is lovely You rock man!

[@miikeV33]

"45° is π/2..I mean you could say, 'That is τ/4', but you'd be an idiot." Oh, the irony.

[@blackmephistopheles2273]

Nah, it wasn't irony...it was Matt using his big brass ones!

[@mydemon]

Truly amazing. Only way it could've been better if he said "tau/8"

[@pedroivog.s.6870]

Actually 45° is π/4

[@kutsen39]

For those that don't get it, tau is 2π

[@dielaughing73]

I think he was talking about Parker Pi

[@classyname42]

Please feature Matt parker more often. He's the funniest man on earth.

[@numberphile]

jediknightonearth we'll try

[@AdrenalineL1fe]

Numberphile also James Grime

[@Triantalex]

false.

[@bruinflight]

Thanks Brady, Matt and everybody else on the Numberphile team for the great videos! You all rock and... as this video shows, roll!

[@hpekristiansen]

You nicely prove that the center of mass is at the same height in these two positions. -but you forget to prove, that the center of mass does not oscillate up and down in between.

[@numberphile]

hpekristiansen hi, there is a second video coming to Numberphile2 where Matt touches on this, although a proof is not show. It is proven.

[@naequs]

Numberphile ha, homework ! i'm gonna try to prove that myself before the next video comes out :D

[@salmachi9836]

I love this mathematician , so lovely and smiley guy .

[@AceTheBraveIT]

5:02 - 45° = pi/2 ??? WAT??? Great video though!

[@Fogmeister]

LOL! I thought exactly the same thing! :D

[@nonoymanoynoy]

He even dissed Tau. haha

[@MegaJORB]

He's talking in radians where pi=90 degrees

[@Fogmeister]

MegaJORB Good point... except Pi = 180degrees. 45degrees = Pi/4

[@StephandaSilva]

MegaJORB except that pi is usually 180 degrees.

[@Yldron]

1:06 That square's center of mass does not move like this!

[@Epikification]

Yeah, it moves in arcs, right?

[@anothermoth]

Epikification and the thing that makes it roll really badly is not the up and down motion of the centre of mass, but the sharp corners it turns at the bottom of it's path where it must instantaneously change direction from rotating around one corner to rotating around the next.

[@UMosNyu]

Epikification Circles around the corner that is to the ground.

[@Epikification]

Yeah, that's what I was thinking, but I wasn't certain enough to comment. A college make a square wheeled vehicle and a track composed of semi-circles to demonstrate it at some point.

[@JamesCoyle95]

anothermoth What you just described is the center of mass having to move up instantaneously when it gets to the edge. It doesn't roll well because you need to add a force to lift the center of mass up over the corner.

[@davidsweeney111]

I suspect this guy is a mathematician.

[@zxwy37]

In his attempt to diss Tau, Matt managed to illsutrate exactly why Tau is better than Pi :-P

[@TheOneTrueIgnus]

That diss was a classic Parker Square

[@austinglugla]

@@TheOneTrueIgnus Parker Circle in this case

[@shambosaha9727]

A Parker pi

[@axelrosalewski]

I love how all the phds and professors on numberphile are so happy when solving those problems! It really gives me the fun on maths back :)

[@christianchristian3467]

Did this with two circular place mats before in a pub, just stumbled upon it by accident, loved it and no no clear answer why, until now!

[@karl131058]

Nice video :) Another tiny mistake: in the animation showing the rolling square, the path aof the center of mass should be a series of circular arcs (because the center is rotating around the corner of the square which is on the ground) instead of a series of straight lines (1:03 - 1:10)

[@GoldenSax]

Did... did we just reinvent the wheel? O.o

[@MarksAwesomeness]

Golden Sax Perfect.

[@dispatch1347]

Golden Sax Going to be a lot harder to attach an axle to that centre of mass.

[@zozzy4630]

5:07 "You could also say that it's tau on four, but then you'd be an idiot." Quite right, Matt, considering 45 degrees is an eighth of a circle, so it's obviously tau over eight.

[@williamrutherford553]

Right, because that's the most important part of the identity of Pi, is whether it's simple to convert from degrees to radians.

[@robertdarcy6210]

@@williamrutherford553 indeed, since pi is defined in terms of diameter,and radians are defined in terms of the radius

[@burpie3258]

"But then you'd be an idiot."

[@Scerttle]

burpie Shots fired.

[@burpie3258]

EssThree Indeed.

[@robknightfilms]

Yup. Especially because that's tau/8 (or pi/4).

[@trickytreyperfected1482]

Which immeadiately backfires since he said "Pi on 2" when it should be "Pi on 4"

[@isavenewspapers8890]

To be fair, one would indeed be an idiot for confusing a 45-degree angle with a right angle. He was not, on technicality, wrong.

[@HYEOL]

that's why you use τ in the first place :D made my day.

[@KingLarbear]

I never really put together that that is why circles rolled I thought it was because they're round but this makes sense too

[@ZardoDhieldor]

"You could say it's tau on four, but then you'd be an idiot" Of course you would. It's obviously tau over _eight_. :D

[@uforob5601]

and pi/4

[@AlucardNoir]

Messy physical reality, what a typical mathematician thing to say.

[@mikecannon3795]

Thanks for the video, Brady and all Numberphile educators!

[@Will140f]

Hurray! New numberphile! Double hurray!! Featuring Matt Parker!!

[@hersirirminsul]

Note to animator: 1:03. The centre of mass of a rolling square would trace a scallop form not a zig zag.

[@SebBrosig]

You showed that 2 positions of the slotted-circle-assembly's rolling path have equal height of COG. That doesn't prove that it _will_ roll freely inasmuch as it's necessary not sufficient criterion. intermediat angles could be higher/lower, no? However it's enough to calculate the slot depth.. and physically playing with the toy shows it must be true! Mathematicians get a woozy stomach at this point but can work out the proof for arbitrary angles, while the rest of us just get on and play with the toy for a bit.

[@fsakdhfksajhf]

A zig zag line is used to show the change of center-of-mass of the rolling square. The CoM would actually describe circular arcs with a center at the corner touching the surface.

[@seanehle8323]

A square will roll perfectly well, without slipping - provided the surface it rolls along is curved appropriately. In the case of a square, the surface would be scalloped with repeating cycloids. How's that for a topic? "Square wheels could be real." I bet most of your viewers would find this fascinating.

[@williamrutherford553]

Didn't they cover this in the shapes of constant width video? Also, by mathematical definition, a wheel cannot be square, because a square is not a shape of constant width. If you want to say that there's a possible surface on which a square could roll, fine, but that doesn't make it a wheel. A wheel is an object that can roll on a flat surface.

[@CsharpPreza]

I understand it can happen once in a while that someone says something wrong but calling someone an idiot for using tau seems way too inappropriate to me.

[@kingsizedmidget7294]

Well than you seem pretty thin skinned, and need to learn how to take a joke. If you watch this channel at all you'd know the ongoing tau vs pie debate and find gat remark quite humorous.

[@soupisfornoobs4081]

True,

[@pad92011]

Great video, I was about to complain about the lack of an annotation but then remembered what Brady always says: "Nobody ever checks the notes." so I did.

[@mancheaseskrelpher8419]

Oh Matt, no matter how much you think pi is better than tau, the truth shall always be that tau > pi.

[@you_tube618]

You win

[@007bistromath]

τ>π Not even the most diehard pi fanatic can argue this.

[@letsgetsomeshoes1239]

x

[@007bistromath]

letsgetsomeshoes1239 ...That's adorable. Leaving aside that Prof. Parker clearly started it, WOOOOOOOOSH

[@RQLexi]

letsgetsomeshoes1239 Tau vs pi is an ongoing (harmless) discussion on this channel, partially because of the practical aspect of it, partially because of the humorous aspect of treating the matter far more seriously than necessary. In this video, Matt quite clearly referred to it for reasons of humour, knowing full well that a lot of the viewers are familiar with the debate and where he stands in the matter. I choose to believe that 007bistromath 's comment tried to reply in kind ("pi fanatic" being a clear hint). What I have more trouble justifying is your reference to toxicity, comparison to the LoL community and the use of "shitheads". Are you sure you aren't overreacting a bit? Or am I missing a joke here?

[@007bistromath]

John Smith  +letsgetsomeshoes1239 is probably swept up in the zeitgeist of current events in gaming. It's a good thing I wasn't saying anything mean to vihart. They'd probably be calling me a misogynist, too. :V ...Huh. Can't get the markup to work.

[@QuadfishTym]

letsgetsomeshoes1239 While I don't blame you for missing such a shamelessly geeky joke like that, there's no 'toxicity' here except what you're bringing with your own comment.

[@TeamDragofied]

Parker draws a lopsided circle in a video about wobbly circles that came out on 8 September, 2014. The Parker Circle video came out on 18 September, 2017. This predicted the Parker Square addition known as the Parker Circle.

[@frasernicol8868]

Love this guy. He's brilliant

[@DesmondAltairEzio]

i paused the video to solve for myself but i used sine instead even though his way was probably easier, and he was right- i got the same answer. i love how math always works itself out like that.

[@DOSTalks]

I love this guy! more matt on numberphile!

[@andvil01]

I called the cut c. The large triangle hypotenuse is 3r-2c. The small triangle hypotenuse is 2r-d. That gives (3r-2c)/r=(2r-c)/(r/sqr2). With a little algebra we have c=(1-sqr2/2)r=0,2929r. A little faster way.

[@benkao8253]

When squares roll, I think about whether a side of a square simultaneously touches the ground after each flip, or whether the edge touches from base all the way up to the next tip not instantly

[@belgaer4943]

this

[@BeautifulEntropyS]

I think it's hilarious that he disses on Tau after falling prey to the dumb Pi conversions. Congrats, this is why we people who like Tau advocate for it.

[@soupisfornoobs4081]

It's called misspeaking, everyone does that. Do you really think 360/8 is less hassle than 180/4? For a mathematician? Doing live unscripted commentary?

[@ricardojbatista]

Great video, as always. Just bought a signed copy of Matt's book.

[@GEM4sta]

Obvious mistake in the video. Many people wrong in the comments as well. Theoretically, if we only consider gravity and a force applied to the... we'll call it a "wheel", the square would roll just as efficiently as the circle. Yes, potential energy is gained when it rises on it's tip. However, potential energy is lost when it's center of mass goes back down. The kinetic energy will oscillate between two numbers, like the little graph you saw in the video. You could simplify the problem by approximating it to a straight line. Friction takes away energy, otherwise energy is usually constant.

[@freshrockpapa-e7799]

And more energy is taken when you move the center of mass.

[@xDMrGarrison]

Does anyone else find it really adorable the way this little contraption wobbles ahead? xD

[@praisethyjeebus]

I love what happens if you click on the link for the signed copy now.

[@vangildermichael1767]

so cool. Such an absolutely awesome idea, about yoking up two circles together at a right angle. And yet have them roll. And after i am so amazed with the whole idea, then you come and dissect the maths about how it is all possible. So I don't have to. thankx

[@DaveScottAggie]

I look forward to receiving my signed copy.

[@MrTurkmenistan1000]

I didn't know I needed this in my life until now!

[@lukedig11]

45 degrees is equal to pi/4 , not pi/2 as mentioned (so it is also equal to tau/8). Who's the idiot now eh Matt? :P

[@numberphile]

***** I'm not sure Matt will not reply because he's already outside sticking forks in his eyes.

[@anticorncob6]

Numberphile I hope that's because he realized he screwed up and not because people are "overreacting to his joke" or whatever.

[@kovanovsky2233]

OOF

[@iCrAz33luVmuSiC]

You should have clips of these brass circles on the next Hello Internet podcast on KZbin. They're mesmerizing! 😁

[@daddymuggle]

'Wobbly circles'. I was expecting cute Parker circles of some sort. The interlocked discs wobbling along were even cuter than that! Who ever said that maths wasn't fun?

[@JktuUekmw]

Matt: I have some circles today. Me: *shivers with anticipation*

[@krisinox888]

Parker is one of my favourite people on numberphile

[@ravenlord4]

Loved the anti-tau jab! It's icing on the cake of another sweet video :D

[@kasuha]

Center of rolling square moves along circular sections, not straight lines. Every time its corner is on the ground, the center follows a circle around that point until another corner touches the ground.

[@Ariel-ps8je]

wibbly wobbly- circle wircles... Matt Parker has always reminded me of the 11th Doctor when he's doing his stand up

[@LittleBoilover]

I suspect I'd be playing with this for HOURS

[@Formulka]

he should be called Math Parker :)

[@hssh8698]

11:02 the centre of mass may not move up and down but still it moves sideways as you can see which does take a lot of kinetic energy out of the forward movement

[@il2xbox]

once again our hero sqrt(2) saves the day

[@1GoodRiddance]

That's mesmerizing.

[@AnkhArcRod]

There was no proof offered for why the height of center of mass must stay the same in all rotations. We solved the situation for 2 specific configurations and showed that the height of center of mass remained unchanged for the two configurations shown. The really cool part was, in fact, the proof that this stays unchanged regardless of the configuration of rotation. Incidentally, for any randomly chosen flat shape, there will always be at least two configurations for which the center of mass is at the same height (proof lies in the continuity of path of center of mass and that center of mass must return to the same value after 360 degree rotation).

[@pasunurusaivineeth3739]

0:33 PARKER CIRCLE, Thanks for stopping by!

[@inothernews]

Matt hasn't let go of Pi vs Tau, great vid haha :)

[@michaellikeaboss]

I just made one of these from 11 gauge stainless steel and they work great!

[@JonJeffels]

***** elegantly describes maths in ways we can understand. Thank you! Also, thank you for my signed copy of your book :))

[@Pengochan]

(sqrt(2)-2)/(1-sqrt(2))=sqrt(2) obviously since sqrt(2)-2=sqrt(2)*(1-sqrt(2)) . The interesting part is where you explain why the two disks center of gravity stays at that height in all other positions too, why did i miss that? For now you only showed it's the same height every 45°. :)

[@mattvw9287]

At 11:03 you see LuLu Numberphile? Wow that office is cluttered, and where's the shredder? all for our leader CGP Grey

[@kinderzabawki545]

Taking only the two positions and calculating d in terms of r - all this is saying that when rolling the system, the height of the center of mass will be the same only in the positions we calculated for. We have no guarantee, that the same height of the center of the mass will be in any other arbitrary moment. In other words d=sqrt(2)*r is a sine qua non condition. How to find a sufficient condition? To find h(t) as a height of the center of the mass as a funciton of time and find this as a constant function?

[@Roman_CK]

Kind of tempted to give this one a try. 🤔

[@itsTK-421]

Could you mention the number 2520 in one of your next videos? I really like it, because it's the smallest possible number divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 :) I think that preoperty should be honored ^.^

[@No-]

Omondi Akura He multiplied both sides by sqrt(2), not 2 * sqrt(2). Sqrt(2)^2 = 2, so sqrt(2) * sqrt(2)(r + d/2) = 2(r + d/2) = 2r + d. He just shortcut through some of the repetitive steps.

[@breathless792]

at 5:32 (approx) he mentioned pandigital numbers, but had already showed one at 3:31 : 8326197504 is a pandigital number using all 10 digits and it wasn't pointed out

[@Leibowitz]

This is fabulous, might have to try this with my math class..

[@nsajko1]

At 8:30 a sqrt(2) just has to be factored out of the right parentheses, instead of all the nonsense. It's easier to notice that if you look at the numbers inside of parens as powers of 2.

[@barlowcj]

@ 1:03 - the centre of mass of a rolling square doesn't trace a triangular path, as shown. The locus comprises circular segments, of course, as it rotates around a corner.

[@fakjbf]

The irony of calling t/4 idiotic when he wasn't even right.............

[@ericcartmansh]

His sketching skills are very impressive in addition to his equation solving skills :D

[@andrijadragovic7470]

1:46 I love it when mathematicians try to describe the imperfect world of physics

[@LucidEnigma21]

I like the calm theme. :]

[@MisterFanwank]

You could make some neat D4s out of this.

[@HydroByte]

Nice video! Just a comment: the red trajectory that starts in minute 1:00 would be formed by semi-circles, not straight lines.

[@DoxalDise]

Is it just me or do I really want Numberphile videos to explain more about the maths?

[@raidedsalt7110]

"You could say that it's tau on four, but then you'd be an idiot." We got the best person to explain everything.

[@SeanStephensen]

begs the questions - is there a spherical analog? If these 2 circles were "filled out" to be spheres, it would just be 2 adjacent spheres, but what about an arrangement where the spheres oscillate in some direction like these circles do but their centre of mass, and motion of centre of mass remain constant? And also, what about a version with 3+ circular disks attached in a way that keeps the motion consistent/smooth?

[@markzero8291]

Here's a comment that's 6 years late: 1:02 those should be circular arcs centered on the lower corner of the square, with half the square's diagonal as their radii! But other than that minor detail, great video!

[@oreubens]

0:20 but... there are shapes that roll quite well and don't have the center of mass staying at the same height. like reuleaux polygons, ellipses (if they aren't too eccentric), and polygons with rounded corners and edges (they have a name, just can't recall it atm). The problem with the square isn't so much that the center of mass goes up and down, but more that it's got flat edges meaning it has no way to 'roll through', it just falls on a flat side and then recovering from that requires quite a bit of force (and friction). it does roll well on a catenary though :p

[@Kasparovwannabe]

The center of mass IS the main issue actually. It's why if you try to roll an ellipse it will stop much faster than a circle would. Could you still roll it? Sure, you could roll a stick with sufficient initial force, but the main thing here is that the center of mass is the primary factor in determining how well it rolls.

[@TimVerweij]

I'm pretty sure that at 1:03, when the square is 'rolling', the red line segments should be curved.

[@glenthemann]

Numberphile It would be cool to extend this to calculate the ideal curvature along the edge of the discs. As the discs grow thicker the rolling would become inhibited, necessitating the need for curved edges. What sort of curve would give a perfect rolling action? I wonder. I'm going to work this out ;)

[@AstroHolden]

Hey, Matt! A circle transcribes a the linear distance around the edge, which equals pi. What value do the two wobbly discs transcribe? The answer is very interesting!

[@shashankambone6920]

I just feel like the algebra can be done much more easier just in my head. The way he does it feels so much complicated than it needs to be.

[@DrDress]

5:07. He never stops his pi-tau fight :-)

[@_helium_]

4:13 he says "Two times the radius" Someone must not be a fan of defining their circle constant by the diameter🤔

[@davidhall6712]

In other words, the bottom of the notch is at the middle of an edge of the largest possible square that can be drawn inside the circle.

[@U014B]

*_ˈæk_*_ ˌtʃu ə li,_ it's about halfway between ¼ and ⅓, so it's really (¼+⅓)/2, which is 7/24.

[@Toastwig]

1:52 jaw drop....why was that so amazing??

[@Garbaz]

Going to buy the bonus edition of the book, hope it ships fast to germany ^^

[@turtledynasty]

Matt miscalculated the radian conversion because the weirdness of pi psyched him out! Tau is more fundamental to the definition of a circle. TAU ALL THE WAY!