Aristotle's Wheel Paradox - To Infinity and Beyond

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*Creator*
Jade Tan-Holmes

*Script*
Simon Morrow

*Animations*
Tom Groenestyn

*Camera*
Philip Holmes

*Music*

*Sources*
Wheels, Life, and Other Mathematical Amusements - Martin Gardner
Aristotle's Wheel: Notes on the History of a Paradox - Israel E. Drabkin
The Joy of Mathematics - Theoni Pappas
The Wheel of Aristotle - David W. Ballew

*Math Posters*

[upandatom]

This video was wheely fun to make!


I'll show myself out...

[barryulrich]

I worked for a trucking company when I was younger and I remember the importance of matching tire circumference or tires when putting together duel wheels. If one of the tires is smaller it is dragged along and wears much faster as well as creating an increased load because of the drag.

[x_abyss]

Poor Cantor. He was derided and ridiculed when his proposed his diagonal argument (countable and uncountable sets) only to be heralded as a pioneer of set theory decades after his death.

[MedlifeCrisis]

This was so great, I love that you made the shapes. Animations are cool but practical demos are awesome

[stoneymcneal]

Three things I love about this video:
1. The explanations are clear and concise.
2. The tempo of the video is not too short and not too long.
3. The French husband, apparently half asleep, being such a good sport.

[culturecanvas]

I love how the husband is just glad to teach correct French even if you wake him up at 1am. That's very French.

[peterwarden]

I'm 56 years old and computer technician and thought I know a lot.
But I never ever heard anything before about number lines or this wheel.
This video was the most interesting thing I have ever seen AND I have ever seen in YouTube and I'm totally impressed to find such a pearl between all the other crap in YouTube.

[j]

For those wondering what the final answer was, here is what I thought she would say to tie it all together:

The demo at 0:47 is not a _measurement_ of length. It is a _mapping_ of points. It indeed shows a 1:1 correspondence in points, but (as explained at length later in the video) a mapping of points does not always denote equality of length (7:20).

Longer version:
At 4:20-5:10 she explains that the small wheel must be slipping—and she wonders (more or less) if this implies that, for every 2 points the big wheel moves forward, the small wheel must be rolling 1 point forward and then sliding/skipping 1 additional point forward. (At least that’s how the rolling polygons worked.) But the small circle is _not_ being dragged in a traditional sense, because there is a one-to-one correspondence between each point of the wheels’ respective journeys. Our intuition says one-to-one correspondence means they are equal, and therefore the circumferences must equal, which they clearly can’t be — so she spends the rest of the video explaining that, for continuous quantities like geometric points, one-to-one correspondence does _not_ mean they are equal. Therefore there can indeed be one-to-one correspondence in points along their rolling motion while _not_ requiring the lengths to be equal, thus permitting a rather non-traditional kind of “slipping” in which no point is ever dragged.

[Edited to add the short version at the top of this comment]

[cerealhawks]

For the line drawing of the cycloid, the opposite situation gives a more intuitive picture to me. If you roll a wheel on an inner circumference, you’d see the line traced at the outer circumference to double back on itself in a short loop

[erictaylor]

"Measuring tape doesn't exist, so what do you do?"
Get yourself a ruler and a ling thin strip of cloth. Transfer the measurements from the ruler to the cloth and invent a measuring tape.

[AzureFlash]

"I want you to imagine that you're an ancient mathematician"
>Am 32-year-old junior engineer
No need for imagination, I am within reasonable tolerances of the stated design goal

[NathanA]

"Hey French husband, can you say this dudes name in your native tongue for my youtube video?" "But I am le tired"

[Aaron-hgjo]

This video is absolutely brilliant. You're one of the best teachers I've ever seen. You make everything so interesting and clear that it keeps the viewer not only interested, but on the edge of their seat, excited to learn more all the way to the end. It made me want to binge all of your videos.
I always thought history was boring and difficult to remember. Out of all my history professors, only one ever was able to tell the story in such an interesting way that I remembered the story and did it without taking notes. He was so interesting that I didn't have to write anything down and I still remember his lectures all these years later. This is the quality you have achieved here. You made me excited to go and teach this to someone else because it was so interesting and I was able to understand it so well. Thank you so much!

[beroyaberoya]

Never seen more smile of a human talking math beside this.

[dstinnettmusic]

Galileo was so close to discovering calculus.

[JK-evuw]

I’m kind of new to your channel but the two infinites explanation has got me hooked. Well done!

[tlovehater]

I love how she woke up her husband just to make him pronounce a name lol.

[FGj-xjrd]

I guess you could say that ancient problems require ancient solutions.

[humanbass]

My explaination: the smaller circle is taking a ride on the bigger one.

[mattcole]

4:20 not slipping but floating. Slipping suggests contact intermittent grip whereas floating is what the inner circle appears to the out circle and us viewers.
Circumference = 2 Pi × radius