Aristotle's wheel paradox

[merjet]

1 hour ago, Jon Letendre said:

Can you see the disparity between the two black dots?

The big dot looks like if you let it fall down, it would reach its start at its white dot.

But the small wheel's black dot will never reach, because it is not making honest, adhered travel, but rather sliding along.

I will try to translate. If you drew the radius through the two black dots, then it would intersect the lower white dot, but it would not intersect the higher white dot. Is that correct? By the way, I had never heard of a dishonest black dot. 

Regarding the video I linked, can you see the cycloid path and the curtate cycloid path the black dots traveled? Did you see that one deviated farther from its horizontal line than did the other? I did. Isn't that amazing for a guy with "deficient visual/spatial/mechanical reasoning"? 

[Jon Letendre]

No, Merlin, I don't think your translation is what I'm talking about.

I have marked up and zoomed in a little to the still of your video.

Imagine the wheels turning counter-clockwise, bringing everything back to their starting points.

I have drawn in pink the course the big wheel's black dot will take back home with its white dot.

Now, focus on the small wheel's black dot.

Do you see that it will not make it all the way back to the white dot? It will fall short.

Yet, when we run the video, it DOES make it all the way back to its home white dot.

But look at the picture again - it cannot make it there, except by skidding across its black line.

[Jonathan]

46 minutes ago, merjet said:

I will try to translate. If you drew the radius through the two black dots, then it would intersect the lower white dot, but it would not intersect the higher white dot. Is that correct? By the way, I had never heard of a dishonest black dot. 

Regarding the video I linked, can you see the cycloid path and the curtate cycloid path the black dots traveled? Did you see that one deviated farther from its horizontal line than did the other? I did. Isn't that amazing for a guy with "deficient visual/spatial/mechanical reasoning"? 

Merlin,

Please watch the video below, which is a reposting of one that I posted earlier. I would find it hard to believe that anyone of your general intelligence could be so deficient in visual/spatial/mechanical reasoning that you couldn't grasp the "paradox" after viewing this clip.

As I wrote in that previous post:

Now, here’s exactly the same thing, but with the area of the small circle isolated. The same disc and the large circle are still there, but they’ve been colored black so as not to be a visual distraction:

C'mon, you get it now, don't you?

[merjet]

9 minutes ago, Jon Letendre said:

But look at the picture again - it cannot make it there, except by skidding across its black line.

You call it "skidding," which is a weak metaphor.  I call it translational-rotational motion, which is literal. Is there some physical force that causes the "skid"? No. In the setup the black dot is not "skidding across its black line." It moves behind its corresponding straight line, traveling its curtate cycloid path, and there are physical forces guiding its behavior. Did you fail to observe that curtate cycloid path?

[Jon Letendre]

Hi Jonathan,

I think this is a case of people having very different cognitive strengths. You and I it seems can run dynamic physical interactions in our mind and see what happens. Most people cannot.

Passengers in my car often report that I made this or that "close call."  I sincerely ask what they are talking about and they describe this or that that we "almost collided" with. Their sense of what almost happened is so off I have to hold back chuckles.

I have concluded that most people can visualize where the moving parts will be in the near future only so long as there are only one or two parts (autos in both directions, pedestrians, bicycles.)

I honestly can see, and I see it right now, where all six or seven moving parts are going to be in 1 and in 2 and in 3....and in 6 seconds. 

[Jon Letendre]

3 minutes ago, merjet said:

You call it "skidding," which is a weak metaphor.  I call it translational-rotational motion, which is literal. Is there some physical force that causes the "skid"? No. In the setup the black dot is not "skidding across its black line." It moves behind its corresponding straight line, traveling its curtate cycloid path, and there are physical forces guiding its behavior. Did you fail to observe that curtate cycloid path?

It is possible neither of us I s understanding the other's language, so I'm not sure yet what you are thinking.

Let me ask you this: do you agree that if the wheels and their lines had gear teeth, then the wheel could not spin at all, but would be locked permanently in place?

[Jonathan]

4 minutes ago, Jon Letendre said:

Hi Jonathan,

I think this is a case of people having very different cognitive strengths. You and I it seems can run dynamic physical interactions in our mind and see what happens. Most people cannot.

Passengers in my car often report that I made this or that "close call."  I sincerely ask what they are talking about and they describe this or that that we "almost collided" with. Their sense of what almost happened is so off I have to hold back chuckles.

I have concluded that most people can visualize where the moving parts will be in the near future only so long as there are only one or two parts (autos in both directions, pedestrians, bicycles.)

I honestly can see, and I see it right now, where all six or seven moving parts are going to be in 1 and in 2 and in 3....and in 6 seconds. 

Yup, I understand. I'm just sometimes amazed at the differences in cognitive strengths, and perplexed by others' inability or willful refusal to recognize and accept their limitations. That's something that I just don't identify with. The defensiveness. Personally, I suck at math. But I don't get upset in the least about that fact. I don't understand the attitude of someone wanting/needing to believe that they're good at, or even average at, something that they are showing themselves to really, really suck at.

J

[Jon Letendre]

Merlin,

Do you see that only one wheel and it's line can be geared to each other with gear teeth?

Do you see that the wheels cannot turn at all if both of them are geared to their line?

[Jon Letendre]

2 minutes ago, Jonathan said:

Yup, I understand. I'm just sometimes amazed at the differences in cognitive strengths, and perplexed by others' inability or willful refusal to recognize and accept their limitations. That's something that I just don't identify with. The defensiveness. Personally, I suck at math. But I don't get upset in the least about that fact. I don't understand the attitude of someone wanting/needing to believe that they're good at, or even average at, something that they are showing themselves to really, really suck at.

J

Agreed.

I enjoy it, though. It's fun trying to figure out what on earth is going on in their head and how it might be cleared up. I take it as a challenge - to find out how good I might be at detecting what's gone wrong and how to present the right correction.

So far with Merlin, it seems the answer is that I'm not good enough at it. ?

[Jonathan]

58 minutes ago, Jon Letendre said:

Agreed.

I enjoy it, though. It's fun trying to figure out what on earth is going on in their head and how it might be cleared up. I take it as a challenge - to find out how good I might be at detecting what's gone wrong and how to present the right correction.

So far with Merlin, it seems the answer is that I'm not good enough at it. ?

Ditto! I began by creating a video which I thought would make it abundantly clear. It was the first one that I posted. To me, the motion -- the slip/skid -- was obvious as hell. Apparently it wasn't enough for Merlin, so I reduce the size of the smaller circle, to make the differences even more visible, and added color sections and markings so as to ease people's efforts at visual tracking. Still no success. So then I took it to the stage of visually isolating the small circle while maintaining the physics of the initial setup, and also offering zoomed in views. Perhaps even all of that is not enough!

J

[Jon Letendre]

So Merlin,

I can't tell if you understand and accept the resolution Jonathan and I have provided but are looking for some other form of resolution, or if you don't accept that we have presented a resolution.

[BaalChatzaf]

5 hours ago, Jonathan said:

Merlin,

Please watch the video below, which is a reposting of one that I posted earlier. I would find it hard to believe that anyone of your general intelligence could be so deficient in visual/spatial/mechanical reasoning that you couldn't grasp the "paradox" after viewing this clip.

As I wrote in that previous post:

Now, here’s exactly the same thing, but with the area of the small circle isolated. The same disc and the large circle are still there, but they’ve been colored black so as not to be a visual distraction:

C'mon, you get it now, don't you?

That is an excellent display. 

[BaalChatzaf]

I would like to bring the matter of wheels and infinitesimals together.  A wheel rolls  when there is an infinitesimal rotation of the line connecting the point of tangency to the center of the wheel combined with an infinitesimal translation of the center in the direction of motion.  This distinguishes rolling from rotation in place (the center is not translated in the direction that the wheel turns) and from sliding where there is translation of the center and no rotation.  

This may seem a bit complicated but if one practices  "seeing" infinitesimal motions  one can begin to grasp some complicated  rotational dynamics.  It is a very handy skill for someone wishing to do physics or any kind of mechanical engineering. 

Newton and Leibniz  both invented infinitesimal motions and this constituted the basis for their version of calculus  which is more accurately called calculus 1.0.  Bishop Berkeley who was no mean mathematician  pointed out the logical uncertainty of the concept and he referred to infinitesimals as "the ghost of departed quantities".   His criticism was quite cogent and eventually the mathematicians had to invent the limit concept to avoid  some the the logical paradoxes that arose from infinitesimals.   This is the way calculus was taught from the early 19 th century onward and it is still the way calculus is taught in most schools.  However in 1964 Abraham Robinson invented  non-standard analysis and re-introduced infinitesimals in a rigorous non contradictory manner.   Robinson based his non-standard analysis on formal set theory and ultra filters which, to say the least is abstract and not intuitively easy to grasp even for a profession mathematician.  But Ed Nelson came to the rescue with a less formal version of set theory and a mathematician who knows standard set theory and Zermelo Frankel axiomatics can pick it up rather swiftly.   It remains to be seen whether bring brack the infinitesimals will make proofs in analysis of real and complex variables  any simpler. 

[anthony]

This article's factual accuracy is disputed. Relevant discussion may be found on the talk page. Please help to ensure that disputed statements are reliably sourced. (February 2017) (Learn how and when to remove this template message)

This article needs attention from an expert in Philosophy. The specific problem is: Aristotle's wheel paradox is misstated. See the talk page for details. WikiProject Philosophy may be able to help recruit an expert. (February 2017)

Aristotle's wheel paradox is a paradox appearing in the Greek work Mechanica traditionally attributed to Aristotle.^[1] There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution. The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels'circumferences. But the two lines have the same length, so the wheels must have the same circumference, contradicting the assumption that they have different sizes: a paradox.

[anthony]

Settled? Not so fast. I had a look at wiki to find what "the paradox" there could possibly be, since "slipping" clearly can't be ruled out, as is insisted in this version, and see a "disputed/mis-stated" error has been noted (above).  Check the "talk page" link. The problem has been ineptly put. As presently posed - i.e. *two* wheels, their relative movements and "without slipping"- this is nonsensical. But there is one wheel in the original problem, and concentric circles - imagine a car tyre and within it the metal wheel rim (different diameters). Must be a maths solution.

[merjet]

23 hours ago, Jon Letendre said:

It is possible neither of us I s understanding the other's language, so I'm not sure yet what you are thinking.

Let me ask you this: do you agree that if the wheels and their lines had gear teeth, then the wheel could not spin at all, but would be locked permanently in place?

 

23 hours ago, Jon Letendre said:

Merlin,

Do you see that only one wheel and it's line can be geared to each other with gear teeth?

Do you see that the wheels cannot turn at all if both of them are geared to their line?

I'm not going to waste time trying to figure out what you mean with such mumbo-jumbo. If you wish, post an image or a video for me to see. "A picture is worth a 1000 words."

Also, the paradox can be depicted with one wheel. See Anthony's posts.  To assume there are two wheels mostly just adds confusion.

  

Either of these could be used to illustrate "Aristotle's wheel." For the duct tape the hole can serve as the inner circle. For the other there is more than one inner circle to choose from. For either of them only the biggest, outer circle rolls on a surface. Any inner circle does not roll on another surface. Any horizontal line drawn for the inner circle is a contrivance. It's imaginary. There is nothing for the inner circle to slip/skid on, despite Jonathan's pretensions that there is. His asking me if I "get it" is built on a fabrication. Do you get that?

That the inner circle slips/skids is "as obvious as hell" to Jonathan. Heh. Centuries ago it was "as obvious as hell" to many people that the sun revolved around the earth.  Grab a roll of duct tape and watch it roll. Do you see the inner circle slip or skid? [Deleted. A little after posting what I deleted, I decided it wasn't a good comparison. But I got too busy to delete it immediately. My apology.] 

[anthony]

I've just tuned in but I gather you've all been arguing apples against pineapples, Merjet. Nobody was ultimately wrong, since the proposition was cocked up by Wikipedia's writer. I kinda suspected a paradox by Aristotle would not be so simple.

[merjet]

11 hours ago, BaalChatzaf said:

 A wheel rolls  when there is an infinitesimal rotation of the line connecting the point of tangency to the center of the wheel combined with an infinitesimal translation of the center in the direction of motion.  This distinguishes rolling from rotation in place (the center is not translated in the direction that the wheel turns) and from sliding where there is translation of the center and no rotation.  

I will address these sentences in reverse order.

Good. I think these different things are very relevant to thinking about Aristotle’s wheel or a model for it, e.g. a tire or roll of duct tape. It is only in very recent days that I observed “translate” used with that meaning. I had the concept, but the word for me in connection with Aristotle’s wheel was pure horizontal or lateral movement (no rotation). I think it could be confusing to use translate and slide as synonyms, since slide is a synonym of slip and skid. So I will try to avoid using slide to mean translate.

That way of thinking about rolling is very different from the ordinary idea of rolling. The straight surface rolled on rotates rather than the straight surface is stationary. Am I correct to assume that the different approach makes the math more tractable because polar coordinates can be used?

 

[merjet]

5 hours ago, anthony said:

I've just tuned in but I gather you've all been arguing apples against pineapples, Merjet. 

I think that fits pretty well. Observe how my vocal opponents (not Baal) have latched onto cycloids and so carefully distinguished between rolling, rotation in place, and translate. Not! 

[BaalChatzaf]

2 hours ago, merjet said:

I will address these sentences in reverse order.

Good. I think these different things are very relevant to thinking about Aristotle’s wheel or a model for it, e.g. a tire or roll of duct tape. It is only in very recent days that I observed “translate” used with that meaning. I had the concept, but the word for me in connection with Aristotle’s wheel was pure horizontal or lateral movement (no rotation). I think it could be confusing to use translate and slide as synonyms, since slide is a synonym of slip and skid. So I will try to avoid using slide to mean translate.

That way of thinking about rolling is very different from the ordinary idea of rolling. The straight surface rolled on rotates rather than the straight surface is stationary. Am I correct to assume that the different approach makes the math more tractable because polar coordinates can be used?

 

Any problem involving  rolling, rotating or phase shifting  better understood in terms of polar co-ordinates.   By the way complex variable are best expressed in polar form

Z = R*exp(i theta). where  Z has the magnitude R  and the direction theta.  Complex numbers  are really vectors that also have algebraic properties. 

[Jon Letendre]

7 hours ago, merjet said:

 

I'm not going to waste time trying to figure out what you mean with such mumbo-jumbo. If you wish, post an image or a video for me to see. "A picture is worth a 1000 words."

Also, the paradox can be depicted with one wheel. See Anthony's posts.  To assume there are two wheels mostly just adds confusion.

  

Either of these could be used to illustrate "Aristotle's wheel." For the duct tape the hole can serve as the inner circle. For the other there is more than one inner circle to choose from. For either of them only the biggest, outer circle rolls on a surface. Any inner circle does not roll on another surface. Any horizontal line drawn for the inner circle is a contrivance. It's imaginary. There is nothing for the inner circle to slip/skid on, despite Jonathan's pretensions that there is. His asking me if I "get it" is built on a fabrication. Do you get that?

That the inner circle slips/skids is "as obvious as hell" to Jonathan. Heh. Centuries ago it was "as obvious as hell" to many people that the sun revolved around the earth.  Grab a roll of duct tape and watch it roll. Do you see the inner circle slip or skid? If you respond 'yes', then draw a dot midway down a pendulum, and then swing the pendulum. Does the dot midway down slip/skid? Draw another dot even higher up. Does it slip/skid even more?

Meaning, no, you cannot see the things I asked about.

Its not jumbo-jumbo, it's you incapable of getting out of your erroneous mental constructs.

Its not built on a fabrication, but on an abstraction you are not capable of holding.

Your pendulum idea is horrible, not at all a good idea for the problem at hand, further reinforcing the observation that you are still miles from grasping the "paradox."

You comment that I am your "opponent"? What the hell is that? I've been trying to help you understand.

Merlin, if I am a waste of your time, just fuck off. It's simple. Fuck off and stop responding to me.

 

[Jonathan]

Here's another "paradox" for Merlin. There is a wheel whose circumference we wish to discover. A guy named Dave suggests measuring it by wrapping a string around it. He holds one end of a string against the wheel with one hand, and then wraps the string around the wheel with the other. He marks the string, removes it from the wheel, and straightens it next to a long ruler. The circumference turns out to be eight feet and five inches.

That length doesn't seem right to Dave's friend, Harold, who insists on measuring the wheel's circumference himself. Harold is missing his left arm, so he can't hold the end of the string against the wheel, but just starts wrapping the string with his one arm. Well, using this method, Harold can't end up back at the beginning of the string, so it turns out that the circumference is infinity!!! How is this paradox possible? How can a wheel's circumference be both 8' 5" and infinity? The same string was used both times. It's unresolvable! 

[BaalChatzaf]

8 hours ago, merjet said:

 

I'm not going to waste time trying to figure out what you mean with such mumbo-jumbo. If you wish, post an image or a video for me to see. "A picture is worth a 1000 words."

Also, the paradox can be depicted with one wheel. See Anthony's posts.  To assume there are two wheels mostly just adds confusion.

  

Either of these could be used to illustrate "Aristotle's wheel." For the duct tape the hole can serve as the inner circle. For the other there is more than one inner circle to choose from. For either of them only the biggest, outer circle rolls on a surface. Any inner circle does not roll on another surface. Any horizontal line drawn for the inner circle is a contrivance. It's imaginary. There is nothing for the inner circle to slip/skid on, despite Jonathan's pretensions that there is. His asking me if I "get it" is built on a fabrication. Do you get that?

That the inner circle slips/skids is "as obvious as hell" to Jonathan. Heh. Centuries ago it was "as obvious as hell" to many people that the sun revolved around the earth.  Grab a roll of duct tape and watch it roll. Do you see the inner circle slip or skid? [Deleted. A little after posting what I deleted, I decided it wasn't a good comparison. But I got too busy to delete it immediately. My apology.] 

The inner circle rotates at the same rate as the outer circle (rigidity is assumed).  But it does not -roll- the same.  Just consider a simple wheel for the moment  on a surface with friction between the wheel and the surface.  The wheel rolls around the point point of contact between the wheel and the surface.  Infinitesimally the center of the wheel gets a little ahead of the point of tangency.  Draw a line connected the point of tangency to the center.  When the wheel  rolls (not rotates)  the line leans infinitesimally forward. Of course the point of tangency catches up (no skid).  If one can manage infinitesimal visualization the difference between rolling, skidding, sliding and rotation becomes apparent. 

[merjet]

26 minutes ago, Jon Letendre said:

Meaning, no, you cannot see the things I asked about.

........

Its not jumbo-jumbo, it's you incapable of getting out of your erroneous mental constructs.

Its not built on a fabrication, but on an abstraction you are not capable of holding.

You comment that I am your "opponent"? What the hell is that? I've been trying to help you understand.

.......

Merlin, if I am a waste of your time, just fuck off. It's simple. Fuck off and stop responding to me.

Evader.

.....

LOL.

.....

Then don't try to confuse the matter with gears. No gears are needed to model the paradox. 

[Jonathan]

Thanks, Merlin, for a wonderful thread! It's a collector's item. I'lll be referring and linking to it many times in the future. It presents absolutely perfect examples of many different issues. It's a gem.

Thanks again!

J