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59 minutes ago, Jon Letendre said:

Let’s take the Hurricane twice around IMI, a 1.1 mile go-kart track.

The guys who pass me are old men, in their sixties! But riding modern setups. I am on my stock original 1987 HONDA CBR600F HURRICANE. 2nd gear only.

Wicked.

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Like any other, it is something that appears to be so, but cannot be so.

What something?

A wheel rolls a certain distance and it appears the smaller diameters within the wheel roll the same distance, but they can’t.

Who says it appears they roll the same distance?

Well, draw a line under any smaller diameter “wheel” and it rolls the same distance, on that “line/road.”  **

If you hold that there is a paradox at this point, where is it?

It is in the belief that something impossible can happen, namely, a diameter, considered as a wheel of its own, with its own draw “road” can traverse distances outsized to its circumference. It’s a paradox. Something has to give, something is wrong.

What has to give? Not the notions of the smaller diameters and considering them as wheels themselves there’s nothing wrong with that, I t’s what we did with the lid on the table. Line AB was outsized to the actual circumference of your lid, just like with any inner circle.

What actually happens when we play with the lid / the inner circle?

You found that it CAN  get to B, just like we know it does and must, but only by skidding.

Insert “sure, but only by skidding” above, at the ** and keep reading from there.

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1 hour ago, jts said:

Draw a horizontal straight line with end points A and B.

Draw a circle (or wheel) sitting on point A. Put an X on the circle where it coincides with point A.

Make line AB longer than the circumference of the circle.

Problem:  Rotate the circle so X again comes down on line AB at point B. Remember line AB is longer than the circumference of the circle. Skidding, slipping, sliding, etc are not allowed.

Maybe Merlin the magician can do it. I doubt anyone else can.

Missed the point completely. You're depicting an inner circle, out of context.

Known: The outer one travels its circumference exactly (and no slippage). Known: The inner circle has a lesser circumference than distance traveled.

Therefore: Established beyond debate, is that the inner circle's circumference has absolutely *ZERO* bearing on distance traveled.

Only the outer one determines this. The inner one follows the exact distance, while revolving at less rotational speed.

The 'skippers' and 'sliders' believe that "slippage" is how you 'fix' a perfectly valid revolving wheel.

Incredible, doesn't anyone see properly and think in reality?

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3 hours ago, anthony said:

1 Missed the point completely. You're depicting an inner circle, out of context.

2 Known: The outer one travels its circumference exactly (and no slippage). Known: The inner circle has a lesser circumference than distance traveled.

3. Therefore: Established beyond debate, is that the inner circle's circumference has absolutely *ZERO* bearing on distance traveled.

4. A. Only the outer one determines this. B. The inner one follows the exact distance, C. while revolving at less speed.

The 'skippers' and 'sliders' believe that "slippage" is how you 'fix' a perfectly valid revolving wheel.

Incredible, doesn't anyone see properly and think in reality?

1.  No. Isolating, focusing on what happens to the inner circle, in relation to its drawn road.

2. Correct.

3. Correct. However, the inner circle’s circumference WILL determine how much skidding it performs.

4. A. Yes, only the outer one determines distance traversed. B  Correct. C  while revolving ONCE. The wheel and the inner circle revolve at the same rate and they traverse the road at the SAME speed.

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4. C. I do know what you mean by less speed.

The tips of the hands of a clock are “going faster” than their middles, which means: given a particular amount of rotation, the tips will “cover more ground” than the middle of the hands do.

The wheel and the inner circle are similar. They want to cover different amounts of ground, given the same amount of rotation. But they can’t do that, they have to “stay together.” So what gives? The inner circle skids to make up the difference.

The straw cannot roll without slip while the lid rolls without slip, that’s impossible. So what gives? The straw skids.

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4 hours ago, Jon Letendre said:

Tony wrote:

”Bob: Nothing changes if there is to be a 'track' the inner wheel is on, or if the line is an imaginary 'path'. You get the same result with two surfaces, or one. Set up or picture an experiment with any bottle (best, because it has a protruding "inner circle", the neck and cap seen from the side). Construct two parallel tracks for the bottle to rest on. One higher than the other to finely adjust for the differing diameters. When the bottle is evenly balanced and leveled on both tracks, roll it along them and observe that both bottle and neck will roll evenly, with no skipping, slipping or jamming. (I'm sure if it's not﻿ balanced ﻿well, it will veer﻿ off course)..﻿﻿”

Tony thinks the straw is NOT skidding it’s track...

Full marks for your effort, Jon. Of course, this is not rigorous enough for science. Too much flexibility and too little weight in a plastic straw; and any slight imbalance of levels or in the manually applied downward force - is naturally going to cause drag.  A controlled scientific experiment will do the utmost to obviate these variables.

And as I keep repeating, friction, drag, rolling resistance or gravity are not explicitly or implicitly imparted by the paradox diagram.

It is not a mechanical puzzle. It's a thinking exercise about circles/wheels.

Even so. You've got what, 20-30 times differences of circumferences of straw to that of the lid? That's approx 25: 1 ratio. For them to travel the same distance, by your theory, the straw must drag almost continuously. I don't see that in the video, I see it revolve at least half way, and I had no 'drag and skid' in my practical tests.

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1 hour ago, Jon Letendre said:

That’s a great teaching suggestion, jts.

The lid is the “small wheel” or “inner circle.”

It doesn’t reach B. It can’t, because it lacks circumference to reach B, just like the “inner circle” lacks circumference to roll the whole length of the dotted lines. But the inner circle DOES get there, so the lid HAS to get there. How? By skidding. You still have the lid in your hand, it didn’t reach B, get a good grip on it, hold it tight, now push and slide that motherfucker across the table to point B. THAT’s how the inner circle makes it all the way. It doesn’t skid only at the end, like I just told you to do, but it is skidding a little bit all the way along since starting at A.

jts's suggestion is a great teaching suggestion.

M-a-y-b-e this will help a bit with the visualization:

The outer circle rolls thus:

A................B - point-by-point contact with its horizontal tangent.

The inner circle rolls thus:

A.#.#.#.#.#.B - not point-by-point contact with its horizontal tangent.

Ellen

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2 hours ago, anthony said:

It is not a mechanical puzzle. It's a thinking exercise about circles/wheels.

I am glad to hear that.

So think of the inner circle as a wheel and the drawn line as it’s road.

Is said wheel rolling without over-spin and without skid? Is it performing true point-to-point rolling?

If you say it is, then the paradox is still unresolved for you, because for you, the inner circle can and does roll distances in excess of its circumference, which is impossible.

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If you would rather throw out the line that is the inner circle’s road, like Merlin did to the Wiki page, then what paradox remains?

Two differently colored dotted lines are the same length. So what? Is that a paradox? If that is a paradox, kindly state what it is that appears to be so and yet cannot be so.

Maybe the response is: Well, both the wheel and all the smaller circles within the wheel travel an identical distance.

Well, of course they do, they are all just parts of one wheel. That’s not a paradox. The center point of the wheel travels that distance, too, and yet, as a geometric point, it cannot even rotate, let alone have a circumference - the poor shit - and yet it goes the same distance as all those inner circles and the wheel do. Well, so what?

After you throw out the line that is the inner circle’s road, there is no longer the appearance of the inner circles rolling road lengths in excess of their circumferences. THAT was the something that appeared to be so and cannot be so. Throwing out elements that give rise to a paradox is no resolution of a paradox. That is just coming up with something else, carefully devised to be unparadoxical.

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I haven't chimed in yet, so why not?

Now that I've looked at the paradox through the prism of a jar lid, a book and a straw, I only have one question.

For the straw (or equivalent), will ball bearings help?

I mean, isn't that one of the main reasons for them to exist in the first place?

Michael

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8 minutes ago, Michael Stuart Kelly said:

I haven't chimed in yet, so why not?

Now that I've looked at the paradox through the prism of a jar lid, book and a straw, I only have one question.

For the straw (or equivalent), will ball bearings help?

I mean, isn't that one of the main reasons for them to exist in the first place?

Michael

Put KY on the straw. And then...

No, wait, a lid. A BIG lid. Put KY on it, and then...

Seriously, this is a serious thread. Don’t make me come down on you like Peter arranged for himself.

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Jon,

If I understand the argument correctly, people (and I don't mean you) are pretending that the smaller circle is in contact with the same surface the bigger circle is riding on. It's a blank-out of the reality of where the surface is.

With the straw in the example, just disconnect it from the jar lid and see if it runs across the book with the same low number of rotations the jar makes it do when connected. It can't.

To use the language of this entire discussion, there has to be slippage by the smaller circle if it's in contact with an equivalent surface and connected to a larger circle.

I don't understand the argument to the contrary.

Did I understand that correctly or am I missing something?

Michael

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44 minutes ago, Michael Stuart Kelly said:

Jon,

1  If I understand the argument correctly, people (and I don't mean you) are pretending that the smaller circle is in contact with the same surface the bigger circle is riding on. It's a blank-out of the reality of where the surface is.

2  With the straw in the example, just disconnect it from the jar lid and see if it runs across the book with the same low number of rotations the jar makes it do when connected. It can't.

3  To use the language of this entire discussion, there has to be slippage by the smaller circle if it's in contact with an equivalent surface and connected to a larger circle.

4  I don't understand the argument to the contrary.

Did I understand that correctly or am I missing something?

Michael

1  Yes, exactly. Pretending that the inner circle is a wheel in contact with its drawn road. Me, Bob, jts, Jonathan, Ellen, Max, we are all doing that. We are, because the setup of the paradox, named after Aristotle and passed to us through history, asks one to so pretend. Indeed, said pretending is what gives rise to the paradox in the first place. The paradox is: *The inner circle rolls its road in excess of its circumference, but that is impossible*

2  We mustn't disconnect the straw and lid, that would disturb their comparability to the inner circle and wheel, which do not come apart.

We CAN roll the straw (the inner wheel) without slip over the book (the drawn road.) That looks like this:

3  Almost correct. The last words should be ... connected to a larger wheel rolling without slip on a road.

4  Lets finish dealing with 1-3, first.

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An aside, a visuospatial﻿ and mechanical reasoning test.

I have the item and will reveal the manifest truth in a video later.

You have seen the straw in two videos.

How many blue stripes does it have?

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18 minutes ago, Jon Letendre said:

We CAN roll the straw (the inner wheel) without slip over the book (the drawn road.) That looks like this:

3  Almost correct. The last words should be ... connected to a larger wheel rolling without slip on a road.

Jon,

Of course. I didn't think about reversing the two.

If there is no slippage with the smaller circle, then there has to be slippage with the outer if they are connected and running on same-distance surfaces. But then there will be a lot more rotations by the larger circle to get to the end of the surface than it would have taken without slipping.

Is that correct? It sure looks like it.

What's in my mind is that you have to set the circle that is not slipping as the baseline for determining (measuring) the slippage of the other circle. And, given that, in practical terms (i.e., out here in reality), it's probably true that some things are made using this principle with both circles slipping a little (bouncing the baseline back and forth for calculations). Without the quip this time, the friction is often eliminated with ball bearings if both circles are in contact with same-distance surfaces. My father worked with this stuff in the garage at home (and in the factory where he worked). I remember seeing it as I was growing up.

I'm mechanically-challenged, but this doesn't look that hard to understand.

Michael

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1 minute ago, Michael Stuart Kelly said:

Jon,

Of course. I didn't think about reversing the two.

If there is no slippage with the smaller circle, then there has to be slippage with the outer if they are connected and running on same-distance surfaces. But then there will be a lot more rotations by the larger circle to get to the end of the surface than it would have taken without slipping.

Is that correct? It sure looks like it.

What's in my mind is that you have to set the circle that is not slipping as the baseline for determining (measuring) the slippage of the other circle. And, given that, in practical terms (i.e., out here in reality), it's probably true that some things are made using this principle with both circles slipping a little (bouncing the baseline back and forth for calculations). Without the quip this time, the friction is often eliminated with ball bearings if both circles are in contact with same-distance surfaces. My father worked with this stuff in the garage at home (and in the factory where he worked). I remember seeing it as I was growing up.

I'm mechanically-challenged, but this doesn't look that hard to understand.

Michael

It appears to me that you are comprehending it correctly.

I rebuild motors and see nothing wrong with what you've expressed so far, except for what I corrected before.

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33 minutes ago, Jon Letendre said:

You have seen the straw in two videos.

How many blue stripes does it have?

Jon,

Hell, I'll play. I don't mind being the stooge when I'm learning something.

So...

drum roll...

3?

Michael

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50 minutes ago, Jon Letendre said:

It appears to me that you are comprehending it correctly.

Jon,

Then the true nature of the paradox that Aristotle came up with is essentially an optical illusion.

Right?

(Note: My tone below might sound like I am teaching you or whatever. That's not my intent. You know this shit way better than me. I merely wrote as I worked it out, thinking out loud, so to speak. Now it's late and I don't feel like fixing it for niceties. Somehow, I don't think you are bothered by tone, anyway.  )

I know the animation below has already been given in this thread, but I want to have it here for my comments.

In fact, on reflection, the term optical illusion does not quite do it for me. A better way of saying it, at least for me right now, is that there is an optical inducement to do a switcharoonie between the center of the circles and their respective circumferences. The reason this switcharoonie doesn't work in reality is that the center moves in one dimension only--it moves from left to right in a straight line. It's movement has no top or bottom.

The circumferences move in two dimensions, not just left to right, but also up and and down.

That means more stuff because of more space.

In other words, this is not a two-dimensional drawing. It is a combination of a one-dimensional drawing of movement (a straight line), and a two-dimensional drawing of movement (the circles). Notice that the illusion or inducement is created by the straight line being displaced from the center and repeated. Also, there is a line from the center of the circles to the outer circumference as if to imply that center and circumference are the same. They are not.

The paradox appears when the straight line is treated as if it represents two dimensional movement. It doesn't. Moving from left to right only happens in one dimension only.

I haven't straightened out the circumferences to compare them against the straight line, and God knows I couldn't anyway. At least not right now. My current pathetic state of math skills are not up to measuring the circumference of a circle.   But I bet if you stopped the animation and straightened out the circumferences of the circles so they became straight lines, they would both be longer than the original straight line.

Michael

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8 hours ago, anthony said:

Full marks for your effort, Jon. Of course, this is not rigorous enough for science. Too much flexibility and too little weight in a plastic straw; and any slight imbalance of levels or in the manually applied downward force - is naturally going to cause drag.  A controlled scientific experiment will do the utmost to obviate these variables.

And as I keep repeating, friction, drag, rolling resistance or gravity are not explicitly or implicitly imparted by the paradox diagram.

It is not a mechanical puzzle. It's a thinking exercise about circles/wheels.

Even so. You've got what, 20-30 times differences of circumferences of straw to that of the lid? That's approx 25: 1 ratio. For them to travel the same distance, by your theory, the straw must drag almost continuously. I don't see that in the video, I see it revolve at least half way, and I had no 'drag and skid' in my practical tests.

The above looks like a rather extremely weak substitute for trying to understand, Tony.

It looks like someone who is trying something, anything, to avoid the message.

I would number it and patiently address each point like I have been doing elsewhere, but I’m hoping that you will read and experiment and read again the other stuff I have posted recently and wait and see if you want to withdraw the above and say something else entirely.

You continue to resist abstracting the inner circle as a wheel itself, in contact with its drawn road. You hint that modeling that would be impossible, but it is not impossible to do, it is easy and natural to do. The inner circle goes along with the wheel, it rotates, it is translated, it traverses the drawn line. All the relevant motions can be modeled, and have been.

May I suggest, if you want to continue talking it over with me, that we dial way down how much we deal with in each post? Pick one crazy thing that Jonathan, Bob, jts, Ellen, Max and I all seem to agree about, and it’s totally crazy, and we will deal with just that one thing. Or something that you are sure has to be correct and we don’t seem to see it is correct.

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1 hour ago, Michael Stuart Kelly said:

Jon,

Then the true nature of the paradox that Aristotle came up with is essentially an optical illusion.

Right?

(Note: My tone below might sound like I am teaching you or whatever. That's not my intent. You know this shit way better than me. I merely wrote as I worked it out, thinking out loud, so to speak. Now it's late and I don't feel like fixing it for niceties. Somehow, I don't think you are bothered by tone, anyway.  )

I know the animation below has already been given in this thread, but I want to have it here for my comments.

In fact, on reflection, the term optical illusion does not quite do it for me. A better way of saying it, at least for me right now, is that there is an optical inducement to do a switcharoonie between the center of the circles and their respective circumferences. The reason this switcharoonie doesn't work in reality is that the center moves in one dimension only--it moves from left to right in a straight line. It's movement has no top or bottom.

The circumferences move in two dimensions, not just left to right, but also up and and down.

That means more stuff because of more space.

In other words, this is not a two-dimensional drawing. It is a combination of a one-dimensional drawing of movement (a straight line), and a two-dimensional drawing of movement (the circles). Notice that the illusion or inducement is created by the straight line being displaced from the center and repeated. Also, there is a line from the center of the circles to the outer circumference as if to imply that center and circumference are the same. They are not.

The paradox appears when the straight line is treated as if it represents two dimensional movement. It doesn't. Moving from left to right only happens in one dimension only.

I haven't straightened out the circumferences to compare them against the straight line, and God knows I couldn't anyway. At least not right now. My current pathetic state of math skills are not up to measuring the circumference of a circle.   But I bet if you stopped the animation and straightened out the circumferences of the circles so they became straight lines, they would both be longer than the original straight line.

Michael

I think I follow you in the last sentence and I think what you say there is correct. I think.

Please review my contributions on page 35, (and all my posts since page 35.) I cover the video you site above extensively.

There are several perfectly valid ways to depict a rolling wheel. With  pictures, drawings, videos, video animations.

The above video is not a valid or accurate depiction of a rolling wheel. See page 35, near the top.

I have to confess - I am not following the vast majority of your post, I just don’t know what you mean to convey. I will take more time and read it more tomorrow.

The Paradox is not so much an illusion as it is an erroneous suggestion - that the inner circle is rolling it’s road like the wheel rolls the road.

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12 hours ago, Jon Letendre said:

If you would rather throw out the line that is the inner circle’s road, like Merlin did to the Wiki page, then what paradox remains?

Two differently colored dotted lines are the same length. So what? Is that a paradox? If that is a paradox, kindly state what it is that appears to be so and yet cannot be so.

Maybe the response is: Well, both the wheel and all the smaller circles within the wheel travel an identical distance.

Well, of course they do, they are all just parts of one wheel. That’s not a paradox. The center point of the wheel travels that distance, too, and yet, as a geometric point, it cannot even rotate, let alone have a circumference - the poor shit - and yet it goes the same distance as all those inner circles and the wheel do. Well, so what?

After you throw out the line that is the inner circle’s road, there is no longer the appearance of the inner circles rolling road lengths in excess of their circumferences. THAT was the something that appeared to be so and cannot be so. Throwing out elements that give rise to a paradox is no resolution of a paradox. That is just coming up with something else, carefully devised to be unparadoxical.

Well said. The idea of there being a "paradox" was based on ancient thinkers making the false assumption that the smaller wheel was rolling freely on the line or surface at its base. In his attempt to pretend to be right, Merlin has eliminated that condition from the setup, including at Wikipedia, and has therefore rendered it nonsensical.

I had already covered this issue, back on page 30 of this thread, by placing the small wheel's line on top of the wheel rather than beneath it. Merlin didn't grasp it. Here's what I had written and illustrated:

Quote

Here's a variation on the "paradox" which is as equally retarded as the original, but which helps to illustrate the retardation of the falsely assumed premises:

It starts out the same as the original "paradox". 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.

Here's where it changes: The path traced by the bottom of the large wheel, and the path traced by the top of the small wheel are straight lines, which are apparently (to visuospatial retards) 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.

Solve the "paradox" without referring to reality. It's a "thought experiment," so you have to accept the premise that you're visually retarded enough to believe that the smaller wheel rolls freely on the orange line and that the orange line is therefore its circumference. Don't argue that there is slippage/skidding/friction from the small wheel's rotating in the opposite direction of the line that it contacts.

So, how is it possible that the orange line, which traces the motion of the inner wheel's rim, is equal to the circumference of the outer wheel?

Merlin's retarded response was a non-response. He didn't grasp what I had written, and instead of commenting on my positioning of the orange line on top of the smaller wheel, he commented on the yellow line and the center points of both wheels, which had nothing to do with anything that I had written:

Quote

I've already answered that multiple times, freaking retard. The center of the outer wheel and the center of the inner "wheel" are the same. Do you know what concentric means? If one center moves a given distance, then the "other" center necessarily moves the same distance. Ditto for each wheel/"wheel"/circle as a whole, necessarily. Are you so geometrically deficient that you can't grasp that and translation?

I informed him that he had missed the point, and challenged him to focus harder and to try to figure it out:

Quote

Dopey, you missed the point of my post. See, the idea was to notice that I moved the smaller wheel's line to the top of the wheel rather than leaving it at the bottom. You don't seem to understand why I did that. Try to figure out my purpose in moving the line to the top of the smaller wheel. Think about it. Think hard, gramps.

Here's the image again:

Personally, I doubt that you can figure it out. In fact, I'll go so far as to say that I know that you can't. You don't have the cognitive capacity to understand why I changed the setup by putting the smaller wheel's line on top of the smaller wheel.

He was not able to grasp it.

J

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Tony and Merlin,

Please volitionally choose to exercise mental focus, honesty and integrity, and apply your full attention and effort to grasping and addressing the following. Don't evade it as you did last time that I posted it.

Quote

The large wheel is 15 inches in circumference. The small one is 5 inches in circumference. Each of the orange and yellow marks on the wheels' edges are a quarter of an inch (0.25"). The black horizontal lines contact the bottoms of the wheels. The yellow and orange segments of the lines are the lengths that the yellow and orange marks on the wheels are in contact with the lines as the large wheel rolls freely and without slippage on the lower line.

Identify those lengths. What is the length of each orange segment on the bottom line, and what is the length of each yellow segment on the top line.

What do the lengths reveal about what is happening?

Should we call this another "paradox," since the yellow marks are covering a greater distance than the length of their sides which contact the line? Or should we stop being retarded, and instead reject the false premise that the smaller wheel doesn't slip or skid in comparison to its line?

J

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MSK,

Welcome to discussion!

You may have missed it earlier, but here's a video of what happens when a large wheel rolls freely on a surface without slipping, and it carries along the smaller wheel which is firmly attached to it. The small wheel necessarily slips/skids while it rolls on the surface at its base (as described in the actual "Aristotle's Wheel Paradox," rather than the one that Merlin has recently dishonestly edited at Wikipedia in order to fake reality).

The issue here is that Merlin and Tony can't visually track and grasp the small wheel's slippage/skidding on its surface under conditions which don't include as much visual information as in the above. In the above video example, I've included all sorts of textures and markings so that anyone should be able to track the motion and see what's happening. The problem has been that other visual representations haven't included any such markings, and, without them, people like Merlin and Tony very easily get lost and confused.

But they don't want to accept the fact that they've been fooled, so they choose to believe their mistaken interpretations of the visual representations which don't have textures and markings, and they therefore conclude that above representation, in which the slippage/skidding is clearly visually obvious, is, as Merlin has claimed, a "con job," a "scam," and an "optical illusion."

Jon had also posted videos of marked wheels and surfaces in which the slippage/skidding is undeniable, and none of it has gotten through to Merlin or Tony.

They are not cognitively suited to grasping it. They are visuospatially/mechanically inept (that's not a moral judgment, but a simple, objective evaluation of their cognitive abilities in this area). And they are also stubborn, and refusing to consider others' arguments and evidence.

J

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I'd merely say take one wheel at a time as real wheels, no abstractions, covering the same distance. The big wheel turns x number of times to cover y distance and the little wheel turns x plus whatever to cover y. If "whatever" isn't anything then the paradox is real. If it's something then it can only be nothing by applying slippage.

--Brant

no visualization is necessary, just counting plus a smidgen of logic, not if you use real wheels

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Slippage