Aristotle's wheel paradox

[Jon Letendre]

Here are two roads

 

[Michael Stuart Kelly]

40 minutes ago, Max said:

The diagram is in so far incorrect, that it doesn't represent a wheel that is rolling without slipping...

Max,

That is an even better way of saying it than I did.

We assume the diagram is showing a wheel that is rolling and not slipping. But there's no reason to assume that.

As to your second point, I agree. I was addressing where the misunderstanding mostly arises. (Apropos, in cases like this, I use posts and discussion to think through the issue, not teach others what I know.) Thus, since most people are arguing as if at least one of the wheels is not slipping, I took that as the default. I should have qualified my thoughts rather than presumed this was clear.

In fact, presumption in lieu of qualification where something seems off is the same epistemological error that the diagram induces people to do.

 

In further fact:

40 minutes ago, Max said:

Those two intersection points rotate completely synchronously. However, a different thing is that at least one of those points is also slipping along its tangent line.

This is exactly what I was saying. Except I was presuming (to use just one example) that the larger wheel's circumference, if unrolled on the ground like a roll of toilet paper, would be the same as the length of the road. In that case, the point on the smaller circle represents that distance of rotation in relation to the actual road length (i.e., the rotation circumference of the larger circle), not the rotation length of its own circumference.

This is because it is not a separate circle, but part of an assembly of circles where the larger circle rules, so to speak. (Remember, this only applies to the case where the large circle does not slip.)

I know that sounds a bit convoluted, but conceptually, I know it's correct. As you say, the point represents rotation. I get that. But rotation can be represented by a straight line length after rolling. If the road line measured is not the same length as the circumference of the rotation, that means the wheel slipped. And there's nothing in the diagram that says the rotation represents a non-slipping wheel. Either wheel. And there's nothing that says it represents a slipping wheel, either. Those are merely presumptions.

I make no apology. I can get simple later. Convoluted is exactly what working through a thorny idea looks like. I mean, why be simple when complicated also works?

 

Michael

[Jon Letendre]

Here is what honest, point-to-point rolling without skid or slip looks like.

Point-to-point is easily confirmed; each successive tooth is falling into the next pocket in the chain.

 

[Jon Letendre]

Here is what happens when the wheel rolls without any slip on the road.

Why isn’t the small wheel staying in point-to-point contact?

Is Jon performing a trick, or is there something real, something about all of this that is, in reality, keeping the small wheel from performing honest roll?

 

[Jon Letendre]

Here the small wheel rolls it’s road without slip. Watch it once to confirm this fact. Then watch it again, this time focusing on the large wheel and the lower road - does it look like the large wheel is rolling that road without slip?

 

[Max]

11 minutes ago, Jon Letendre said:

Here is what happens when the wheel rolls without any slip on the road.

Why isn’t the small wheel staying in point-to-point contact?

Is Jon performing a trick, or is there something real, something about all of this that is, in reality, keeping the small wheel from performing honest roll?

 

When I saw your previous video, I was sure that this would be the next one, all the attributes were already there...

[Jon Letendre]

16 minutes ago, Jon Letendre said:

Here the small wheel rolls it’s road without slip. Watch it once to confirm this fact. Then watch it again, this time focusing on the large wheel and the lower road - does it look like the large wheel is rolling that road without slip?

 

Maybe you are not sure about this one.

Let’s do another one.

The small wheel rolls it’s road without slip, and what is the large wheel doing?

 

[Jonathan]

13 minutes ago, Jon Letendre said:

Here the small wheel rolls it’s road without slip. Watch it once to confirm this fact. Then watch it again, this time focusing on the large wheel and the lower road - does it look like the large wheel is rolling that road without slip?

 

This will not help them to grasp it. Sorry. Just sayin'. Be prepared to be further astounded by levels of stupidity that you haven't imagined possible, despite having witnessed all that has been expressed on this thread.

Brace yourselves. It's coming.

J

[Jon Letendre]

1 hour ago, Jonathan said:

This will not help them to grasp it. Sorry. Just sayin'. Be prepared to be further astounded by levels of stupidity that you haven't imagined possible, despite having witnessed all that has been expressed on this thread.

Brace yourselves. It's coming.

J

I think you will be proved correct.

Tony has a chance, if he would go back to the beginning and state what is the paradox.

Up to now he does not understand what is the paradox.

His attempt, “the assumption that the small wheel dictates distance” is absurd and incoherent. Many small wheels of various diameters can be drawn onto the wheel - which one does he imagine someone believes dictates distance? No one would think that and that’s not Aristotle’s Wheel Paradox.

He can’t state the paradox, because up to now, he doesn’t grasp that there is a paradox.

Once he sees there is a paradox and he truly understands the paradox, then he will soon thereafter understand the resolution easily. He has well more than the cognitive capacity required.

But the horse has to want to drink.

[Jon Letendre]

On 11/20/2018 at 1:57 PM, merjet said:

Hogwash.

More hogwash.

As molested by you, here is the illustration that goes with the Wiki article https://en.wikipedia.org/wiki/Aristotle's_wheel_paradox

But wait, the small drawn circle is a red-herring, right Merlin? It could be any size and it is just a drawing on a real, physical, rolling wheel, it doesn’t get to DO anything, it doesn’t MATTER, right Merlin? Right Tony?

Lets fix the illustration

Looking good.

A brown line discloses road length traversed by the wheel in one rotation.

Two dotted lines also disclose road length traversed by the wheel in one rotation.

The blue dotted line is the highest point on the wheel before it starts rolling. Naturally, it is in the same position at the start as it is at the end. Indeed, we can choose any point on the wheel, penciled-on or imagined, and be assured that it will still be there at the end, and be assured further that a line connecting it from start to end will also reliably disclose road length traversed in one rotation of the wheel.

Merlin thinks there is something paradoxical here. Some “problem” to resolve.

[Jon Letendre]

In the Wiki, Merlin (I assume Merlin) says “The paradox is that the smaller inner circle moves 2πR, the circumference of the larger outer circle with radius R, rather than its own circumference.”

 

Also “The distances moved by both circles are the same length, as depicted by the blue and red dashed lines. The distance for the larger circle equals its circumference, but the distance for the smaller circle is longer than its circumference: a paradox or problem.”

 

Wlhich illustration goes best with the above descriptions?

 

[Jon Letendre]

On 11/21/2018 at 11:02 AM, anthony said:

The usual barrage and smoke screens. Nope, I clearly recognize everyone wants a 'track' to 'slip on' to validate their theory. I've mentioned that word, ad nauseam.

BUT everyone evades replying to what a free-rolling bottle 'achieves' without 'track' intervention. It don't need it.

 

Tony, read the post directly above this one.

The small wheel and the appearance that it goes farther than it should when it rolls IS the paradox.

Do you see now the need to look into this small wheel and it’s alleged traversing of road lengths equal to other, larger wheels?

If you refuse to do it, then the paradox remains unresolved.

[merjet]

On 11/21/2018 at 12:42 PM, Ellen Stuttle said:

Seriously??!!  That's what they think you et al mean by "slippage"?

No, Ellen, speaking for myself. I have not said such garbage or even remotely suggested it. That garbage is Jon's infantile fabrication.

[merjet]

On 11/19/2018 at 11:05 AM, Max said:

That story of the cycloids is completely unnecessary, as the only thing you conclude from that story is that the center(s) of both wheels travel over the same distance. You don’t need any cycloid to “prove” such a trivial thing.

Wrong. It's obvious that the smaller circle's cycloid is shorter than the outer circle's cycloid, and longer than the center's path. I conclude that the length of the cycloid path of every point on the smaller circle is identical. Since the center's path is 2*pi*R, every cycloid of the smaller circle is greater than 2*pi*r, i.e. its circumference. Therefore, the smaller circle rolls farther than it would by pure rolling. 

That's made with no mention of any slipping or skidding on some fantasized horizontal line tangent to the smaller circle.  

Max: "There is nothing to understand" (link).

You missed the boat.

 

[merjet]

On 11/21/2018 at 8:49 AM, Max said:

Yes, it will slip, as I've proved mathematically (see my post of February 4) and as Jon and Jonathan have very clearly visualized in their videos and animations (do you insinuate that these are optical illusions?). Show me were I made an error in my proof, if you can, and with real hard arguments, not with some confused metaphysical nonsense like "self-contradiction to the identity of the wheel".

Yes, it will "slip", that term being a confused metaphor. Do you know what scare quotes are? Why have you never used them around slips? You even get the metaphor wrong. The correct metaphor is "skid", meaning that the smaller circle rotates less than it would by pure rolling as an independent entity for the same horizontal distances.  But even that is not literal skidding. Literal slipping or skidding requires a point of contact. See here. Do you see a point of contact tangent to the bottom of the hub in the picture there? It is reasonable to assume a surface tangent to the bottom of the wheel that the wheel rolls on, but the authors omitted it. In reality a wheel on a car or truck rolls on a surface. It is NOT reasonable to assume a surface tangent to the bottom of the hub/rim that the hub/rim rolls on. In reality no car or truck wheel does such a thing. In reality, is there a horizontal surface tangent to the inner round surface of a roll of duct tape? In reality, is there a horizontal surface tangent to the neck or mouth of a rolling round wine bottle? Like Jonathan, you arbitrarily add a tangent to the bottom of an inner circle that is totally unnecessary. Like Jonathan, it is your "crutch" to fake reality, and you are helpless without it. 

To be consistent you would need to fake reality 5 times for the wheel shown on the page I linked above, one for every circle between the center and the largest circle.

I have proven why the inner circle behaves as it does mathematically in two ways and with no "crutch." I have no emotional, psychological, or epistemological need to fake reality like you, Jonathan, Jon, and Ellen. Show me where I made an error in my proofs, if you can, and with arguments firmly tied to reality, not faking reality with a confused metaphor and a "crutch." 

All four of you epitomize closed-minded dogmatism. You FEEL you have the only possibly correct answer, and any other answer must be wrong. Are you agents from the Ministry of Truth? Jonathan and Jon even behave like hysteric jackasses to defend their closed-minded dogma. It matters little what I actually say; they distort it. For example: "But wait, the small drawn circle is a red-herring, right Merlin?" (Jon, followed by his eliminating the red circles entirely and changing the length of the brown line).

[merjet]

On 11/22/2018 at 3:37 AM, Michael Stuart Kelly said:

The line going through the small circle and ending on the rim of the large circle has nothing to do with both circumferences. Nothing at all. It exists only to represent the circumference of only one of the circles. And, it doesn't matter which.

Scherk also copied the GIF from Wolfram MathWorld and posted it here several days ago. As I said then, it is a defective representation of Aristotle's Wheel Paradox, which the page's title implies it is. The distance from start to stop should be much longer and equal to the circumference of the larger circle.

By the way and referring to your earlier post, you don't need to measure a circumference to know its length. If you have a measurement of the diameter or radius, then the circumference is pi*diameter = 2*pi*radius. Still, it's easy to measure the circumference of some things. Grab (#1) a flexible tape measure, such as the one shown here, and (#2) a roll of duct tape or masking tape. Wrap #1 around #2. Bingo. 

What you say about the line is not true. Circumference = 2*pi*radius. The radius in the GIF exists to show a 360 degree revolution. If you watch the point where the radius meets either circle as it moves left to right, the point follows a cycloid path. Or see here and here. If the GIF were the correct length, the cycloids would have the correct shape and length for the paradox. The second image on Wikipedia shows these cycloids, except that the points start and end at "12:00 o'clock" rather than "6:00 o'clock".

Being that the GIF is defective, the larger circle represents true slipping. If it were not defective, there would be no slipping, period. The inner circle would "skid", the scare quotes indicating a metaphor. 

All the above assumes that there is one road that the larger circle rolls on. The hub or rim of a real wheel, such as on a car or truck, does not have another road tangent to the hub or rim. If the smaller circle were assumed to have its own "road", which is the case for a flanged train wheel, then the narrative would differ. The cycloid shapes and lengths would differ. However, this other narrative is not needed to understand the paradox.   

[merjet]

Everything that Jonathan and Jon have said about my "molesting" the Wikipedia article or similar is hogwash. I vastly improved it. I have received thanks and compliments for doing so. J and Jon insist the paradox is about two wheels each on their own road. That contradicts both history and the 'Wrong problem entirely' section on the Wikipedia Talk page. Note how Jonathan conveniently omits mentioning that section when he accuses me of "molesting" Wikipedia and similar such rot.

[merjet]

11 hours ago, Jon Letendre said:

Wlhich illustration goes best with the above descriptions?

The top one, nitwit.

[Brant Gaede]

Neither "slip" nor "skid" is a metaphor for they are not reducible. They only work that way as part of a total floating  abstraction which may or may not be coherent unto itself. In that context calling them metaphors is redundant and confusing for the implication of actual physicality.

--Brant

[Jonathan]

3 hours ago, merjet said:

Everything that Jonathan and Jon have said about my "molesting" the Wikipedia article or similar is hogwash. I vastly improved it. I have received thanks and compliments for doing so. J and Jon insist the paradox is about two wheels each on their own road. That contradicts both history and the 'Wrong problem entirely' section on the Wikipedia Talk page. Note how Jonathan conveniently omits mentioning that section when he accuses me of "molesting" Wikipedia and similar such rot.

Hawrgwarsh!!!

[Jonathan]

3 hours ago, merjet said:

Jonathan and Jon even behave like hysteric jackasses to defend their closed-minded dogma.

Gramps, you've said, several times, that my diagrams and animated diagram sequences are illusions, scams and con art.

But you haven't identified anything that is actually false about them.

Show any aspect of any of my diagrams to be geomtrically inaccurate. Show that any measurement, angle or relationship is geometrically false.

How is it that any point on the rim of the rolling large circle in any of my animations takes the path of a proper cycloid, and any point inside of the rim takes the path of a curtate cycloid in correct proportion to the point's distance from the center point, yet you claim that my animated diagrams are scams? Wouldn't my diagrams have to fail to comply with your own method of attempting to solve the "paradox," rather than precisely complying with it?

You claim that you can track or trace cycloids. Well, do so. It's elementary geometry.

Identify specifically what you think is a "con" or a "scam" in my diagrams/sequences.

J

[Max]

1 hour ago, merjet said:

Wrong. It's obvious that the smaller circle's cycloid is shorter than the outer circle's cycloid, and longer than the center's path. I conclude that the length of the cycloid path of every point on the smaller circle is identical. Since the center's path is 2*pi*R, every cycloid of the smaller circle is greater than 2*pi*r, i.e. its circumference. Therefore, the smaller circle rolls farther than it would by pure rolling. 

Applying the same argument to the large circle: every cycloid of the large circle (cycloid length = 8*R) is greater than 2*pi*R . Therefore, the large circle rolls farther than it would by pure rolling? You've created a new paradox!

1 hour ago, merjet said:

You completely missed the boat.

At least I'm not the one who is drowning.

[Jon Letendre]

5 hours ago, merjet said:

No, Ellen, speaking for myself. I have not said such garbage or even remotely suggested it. That garbage is Jon's infantile fabrication.

You had the wildest misinterpretations of our use of slippage, Merlin. 

And you still do.

You call it metaphorical. It is real.

You call it metaphorical because there is no surface there to slip on.

But I put a surface there, now there is one there, and the small wheel slips over it.

Real.

[Jon Letendre]

5 hours ago, merjet said:

Wrong. It's obvious that the smaller circle's cycloid is shorter than the outer circle's cycloid, and longer than the center's path. I conclude that the length of the cycloid path of every point on the smaller circle is identical. Since the center's path is 2*pi*R, every cycloid of the smaller circle is greater than 2*pi*r, i.e. its circumference. Therefore, the smaller circle rolls farther than it would by pure rolling. 

That's made with no mention of any slipping or skidding on some fantasized horizontal line tangent to the smaller circle.  

Max: "There is nothing to understand" (link).

You missed the boat.

 

“Therefore, the smaller circle rolls farther than it would by pure rolling.”

Yes, Merlin. Well done. Your cycloids prove that, they make that big breakthrough.

Problem is, one can see that the small wheel rolls farther than it should the moment one perceives the paradox. That IS the paradox.

It doesn’t need to proved, but the impossibility of it needs to be resolved.

Your cycloids don’t resolve how can the small wheel do that. They just re-establish that it does.

[Jon Letendre]

5 hours ago, merjet said:

I have proven why the inner circle behaves as it does mathematically in two ways and with no "crutch." I have no emotional, psychological, or epistemological need to fake reality like you, Jonathan, Jon, and Ellen. Show me where I made an error in my proofs, if you can, and with arguments firmly tied to reality, not faking reality with a confused metaphor and a "crutch." 

All four of you epitomize closed-minded dogmatism. You FEEL you have the only possibly correct answer, and any other answer must be wrong. Are you agents from the Ministry of Truth? Jonathan and Jon even behave like hysteric jackasses to defend their closed-minded dogma. It matters little what I actually say; they distort it. For example: "But wait, the small drawn circle is a red-herring, right Merlin?" (Jon, followed by his eliminating the red circles entirely and changing the length of the brown line)

Oh, we don’t understand there can be multiple resolutions? Cute.

No, that’s you, Merlin.

Go to page one. Bob says “in that physical instantiation, the small wheel slips”

You then go on and on and on rejecting.

Do we have to reprint it? Do you need reminding?

We all would love to hear another resolution, do you have one?

So far, you have a cycloid observation that helps solidify that the small wheel really does get farther than its circumference should allow it to... any ideas for a resolution of that paradox?