Wobbly Circles - Numberphile
ฝัง
- เผยแพร่เมื่อ 7 ก.ย. 2014
- 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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Dip the edges in ink, roll it. What shape does it make? Can you mathematically describe it? I want to know this.
I feel that going to be an interesting thing
@@mohammedyassin8064 it's gonna be a line segment
two parallel wavy lines with sharp crests
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.
It would make lines. Duh
The center of mass of a rolling square doesn't go in a zigzag; it goes in a series of circle arcs.
+David Richfield I'd imagine the zig-zag was just easier to animate.
David Richfield Vsauce made a video about this.
David Richfield if you look closely you can tell it moves a lot more than the lines
Looking for this
Indeed. A rolling Parker square makes a zig-zag :)
Instead of Pi or Tau we should make a compromise.
Pau = 1.5 Pi
It's a Ti(e).
Wau
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.
That was an 0ld XKCD joke...
Fuci
"45° is π/2..I mean you could say, 'That is τ/4', but you'd be an idiot."
Oh, the irony.
Nah, it wasn't irony...it was Matt using his big brass ones!
Truly amazing. Only way it could've been better if he said "tau/8"
Actually 45° is π/4
For those that don't get it, tau is 2π
I think he was talking about Parker Pi
''let's remove this mess of root 2 here'' *proceeds to add more root 2*
In maths we add to subtract
no he has an abomination of root 2 not a mess =D
"Ah a new numberphile video, great!"
*Watches Video*
*Spends £30 on book and maths gear*
"Cheers Brady, Beans again for me tonight."
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! :)
Numberphile For convenience he could perhaps offer a premium package that includes both a signed book and a supply of beans.
+Number 5 But it's the beans they used to teach you about counting at school.
??
Please feature Matt parker more often. He's the funniest man on earth.
jediknightonearth we'll try
Numberphile also James Grime
jediknightonearth pARkeR squARe
false.
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
Tell me, what will I see?
Matt has this subtle sense of humour, which is lovely
You rock man!
The tau burn is cancelled out by his mistake using pi so I am content
Exactly what I was thinking.
I suspect this guy is a mathematician.
Thanks Brady, Matt and everybody else on the Numberphile team for the great videos! You all rock and... as this video shows, roll!
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.
hpekristiansen hi, there is a second video coming to Numberphile2 where Matt touches on this, although a proof is not show. It is proven.
Numberphile ha, homework ! i'm gonna try to prove that myself before the next video comes out :D
I love this mathematician , so lovely and smiley guy .
5:02 - 45° = pi/2 ??? WAT???
Great video though!
LOL! I thought exactly the same thing! :D
He even dissed Tau. haha
He's talking in radians where pi=90 degrees
MegaJORB Good point... except Pi = 180degrees. 45degrees = Pi/4
MegaJORB except that pi is usually 180 degrees.
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!
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 :)
Love this guy. He's brilliant
Great video, as always. Just bought a signed copy of Matt's book.
I love this guy! more matt on numberphile!
1:06 That square's center of mass does not move like this!
Yeah, it moves in arcs, right?
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.
Epikification
Circles around the corner that is to the ground.
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.
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.
In his attempt to diss Tau, Matt managed to illsutrate exactly why Tau is better than Pi :-P
That diss was a classic Parker Square
@@theonetrueignus Parker Circle in this case
A Parker pi
Thanks for the video, Brady and all Numberphile educators!
Hurray! New numberphile! Double hurray!! Featuring Matt Parker!!
I didn't know I needed this in my life until now!
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
Did... did we just reinvent the wheel? O.o
Golden Sax Perfect.
Golden Sax Going to be a lot harder to attach an axle to that centre of mass.
Messy physical reality, what a typical mathematician thing to say.
Loved the anti-tau jab! It's icing on the cake of another sweet video :D
"But then you'd be an idiot."
burpie Shots fired.
EssThree Indeed.
Yup. Especially because that's tau/8 (or pi/4).
Which immeadiately backfires since he said "Pi on 2" when it should be "Pi on 4"
That's mesmerizing.
Awesome work! I like it every time more, please keep going :-).
I look forward to receiving my signed copy.
***** elegantly describes maths in ways we can understand. Thank you! Also, thank you for my signed copy of your book :))
that's why you use τ in the first place :D made my day.
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.
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)
Going to buy the bonus edition of the book, hope it ships fast to germany ^^
I just made one of these from 11 gauge stainless steel and they work great!
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.
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
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.
This is fabulous, might have to try this with my math class..
'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?
Matt hasn't let go of Pi vs Tau, great vid haha :)
Very nice Thank you
Parker is one of my favourite people on numberphile
"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
and pi/4
Great video guys 😁
how can you say it was a great video if the video was uploaded 4 minutes ago and it has 11 minutes.You didn't even watched the video omg!!!WHAT A LIAR!!SHAME
Razvan Gabor I agree I've placed a large amount of faith in my Australian brothers, but I doubt I'll be disappointed
Very cool!
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
this
Oh Matt, no matter how much you think pi is better than tau, the truth shall always be that tau > pi.
You win
Nice video :)
Note to animator: 1:03. The centre of mass of a rolling square would trace a scallop form not a zig zag.
AMAZING
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.
Right, because that's the most important part of the identity of Pi, is whether it's simple to convert from degrees to radians.
@@williamrutherford553 indeed, since pi is defined in terms of diameter,and radians are defined in terms of the radius
You should have clips of these brass circles on the next Hello Internet podcast on TH-cam. They're mesmerizing! 😁
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.
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.
True,
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!
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?
As it rolls forward would the center of mass also move in a complete linear direction forward? Though the you said the c.o.m. stays at the same height as it rolls forward, does the path it takes also maintain a linear one? Say if from a top view would the wobble from side to side in it's progression?
Also curious as to how different thicknesses would affect the physics, if you have any thoughts on the subject.
So, here's an interesting question in regard to the wobbling circles above. What if you placed the inner joined wobbling circles inside a container (think large bowl) that, by it's own nature, naturally pushes the wobbling circles back away from one side of the container towards the other side and then back again towards the other side once it hits the curvature of that side. Back and forth forever?
I never really put together that that is why circles rolled I thought it was because they're round but this makes sense too
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.
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.
I like the calm theme. :]
I just thought of an interesting generalization. What if the two discs have different radii ? Of course we can work out d. But will they roll?
I am curious as to how the thickness of the disc plays into a perfect calculation. You can imagine that if the discs shown were very thick (ex. t = r), that it may not roll properly... or would it? I have no idea, can you help?
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.
Matt: I have some circles today.
Me: *shivers with anticipation*
Is this really true for a CD, as it is is not of uniform density, it lacks a piece in the center? Does that shift the center of gravity of the connected circles when they roll compared to the solid disks, or is there a theorem explaining what happens when two objects gets connected which are not solid?
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 ;)
If you make a circle with uneven mass distribution will it roll less? I had no idea, but come to think of it, it does make sense
Does anyone else find it really adorable the way this little contraption wobbles ahead? xD
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.
Looking through older numberphile videos I wonder why this isn't a Tadashi Tokieda video. On the other side it reminds me of the shapes of constant width. Wonder how you could stick together shapes of constant width to have the same property.
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?
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
***** I'm not sure Matt will not reply because he's already outside sticking forks in his eyes.
Numberphile I hope that's because he realized he screwed up and not because people are "overreacting to his joke" or whatever.
OOF
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.
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?
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.
Hmmm ... this video remembered me about involute gear profiles ... might be a nice idea for a video :]
Would it be possible to do this with any other shapes of contant width? I know the center would move up and down but would that affect it enough for this not to work?
Why are the covers different in different countries? Do they run on different voltages?
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
once again our hero sqrt(2) saves the day
The prove only includes three points on the circe : those touching ground when being rotated multiples of 90° to the ground (Because two of the points are simmetrical to each other, the prove actually only had to deal with two). But what about the points in between? Does each disk have to be a circle? Could i connect those points by straight lines or other patterns?
I wonder if this configuration would be good for wheels of some kind - like the mars rover. Any thoughts on that?
So, call the height of the center of mass "Z", the table in the direction of travel, is "X",and the table perpendicular to the direction of travel is "Y". Does the center of mass "wobble" along the "X" axis in the direction of +/- "Y"?
Very interesting
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 ^.^
I tryed to buy the signed book from the link and pay with paypal but it dosent seem to work.
His sketching skills are very impressive in addition to his equation solving skills :D
what a nice vid.
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.
You rock
I realise it's a bit late but I'm interested, can you put ink from a marker on those wheels and roll them across paper, I'd be interested to see what pattern they form.
Nice video!
Just a comment: the red trajectory that starts in minute 1:00 would be formed by semi-circles, not straight lines.
My turbulence professor has a great phrase for the algebra (or calculus) parts of a problem, he calls it "turning the algebra crank". I've also heard my fluid stability professor call it "plug and chug". Any one else have clever names for the tedious parts of a problem?
Does the center of mass wobble side to side as it rolls forward as well? Would this cause energy loss more so than a rolling sphere?
it would be great if we could buy these wobbly circles somewhere. i cant find them and I don't have the materials to make them.
EPIC!!!