Aristotle's wheel paradox

[Jon Letendre]

The pink wheel is rolling in contact with the bottom of the groove in the table.

The pink arc is one-eighth of the pink wheel circumference and has length EQUAL to the pink line, demonstrating contact with the table and true constant-contact rolling.

The yello arc is one-eighth of the yellow "wheel's" circumference and is a mere FRACTION the length of the yellow line, demonstrating that the yellow "wheel" is NOT in contact with its imaginary "table" represented by the yellow line. The yellow "wheel" must be skipping, sliding across the yellow line "table."

[Jon Letendre]

17 hours ago, Brant Gaede said:

The Earth rotates at, what, 24,000 mph at the equator.

How come at the true North Pole you seem to be as standing still as anywhere else on the planet if the North Pole doesn't slip or slide?

--Brant

Still not sure what your question is, but I have guesses.

Are you wondering why we can't tell, why we don't feel any different standing on the equator going 1,000 mph versus standing on a pole and not?

Bob can correct me on the magnitudes of the matter, but I think we CAN tell the difference. We weigh less on a scale at the equator than on a pole, due to that 1,000 mph trying to fling us off the surface of earth. But only a tiny, tiny, tiny, bit less.

If it spun much, much faster, like once per couple minutes instead of several dozen hours, then we would weigh much less at the equator than we would at the poles (which would be unchanged from what we weigh at the pole with once per 24 hour rotation.)

[merjet]

1 hour ago, Jonathan said:

Heh. It doesn't matter what the larger circle was "designed to represent." It is not the outer circumference, despite what anyone wishes to label it. The wheel's actual outer circumference is its actual outer circumference, and any point on a concentric circle on the wheel which is smaller than that circumference will trace a curtate cycloid. Both circles in the video that you posted are smaller than the circumference of the wheel, and both of their points trace curtate cycloids, despite your misperceptions due to your massive cognitive deficiencies.

See, the way that reality works is that wishing doesn't make things so. If you arbitrarily wish a random line to "represent" the line on which a wheel rolls, any points on the wheel don't magically obey your wish. They move in retaliation to the actual rolling circumference, not to the one that you've falsely labeled as the circumference. You've hilariously misidentified reality once again.

But you don't have to take my word for it. You can test it yourself. The method would be for you to try to wish that a different line represents the surface on which the wheel rolls. Let's go with the smaller circle in the video. Now that you wish it to represent the line on which the wheel rolls, is the path traced by the larger circle's dot suddenly a prolate cycloid? Do you now imagine that you can see a prolate cycloid just as you earlier imagined that you could see a common cycloid when it was actually a curtate cycloid the you were looking at?

False. Both are curtate cycloids. Neither is a common cycloid. Just do a motion-track with a multiplier filter on the dots and sustain them from frame to frame of video. Then you'll see reality rather than the fantasy that you're believing in.

The absence of a dot on the wheel's circumference doesn't change the fact of reality that it is the circumference which is rolling on the table. The large circle inside of the circumference doesn't magically become the circumference upon which the wheel rolls just because you've labeled it that way!

 

Yay!!! Finally!!! And what does that skidding tell you? What does it mean? Think it through! You can do it!

J

Whoop-de-doo! Reality is what Jonathan says it is, regardless of any facts he doesn’t know about. We are not in a position to look at the wheel and the rest of the setup in real life. I came to the conclusion I did for three reasons.

1. I watched the path of the dot on the lower circle in the neighborhood of its starting and ending points. The path looked much more cuspy than the path the inner circle took in the neighborhood of its starting and ending points. The latter path is shaped much more like that of a child’s playground slide near its low/ending point.

2. It appears to me that there is a board or surface behind the wheel that another part of the wheel that we can’t see rolls on. It could be as high as the larger circle and a big part of the weight of the wheel is on it. The circumference of the white disk appears to roll in the tan groove. The major purpose of the groove could be to guide the disk’s roll, direction, and help keep it vertical, parallel to the horizontal wires. We have no evidence at all that the part of the disk that rides in the groove bears much of the weight. The wheel obviously has an axle that we don’t see and the guy turns. He manually turns it and if I were doing it, I would want the design of the apparatus to assist me as much as feasible to keep it rolling straight and staying vertical. Have a look at the wheel in the video you posted. The circumference of it extends below the horizontal bar that the wheel rolls on. The larger circle on the wheel is the “road surface”, not the bottom of the wheel.

3. The designer’s intended purpose, of course – to have a wheel that gives a great demonstration of Aristotle’s wheel paradox.

All the above is apparently something you could not imagine. You saw a gotcha opportunity to latch onto an infinitesimal difference between a perfect cycloid and one that might be a tiny bit curtate so that you could start slinging insults. Why is such an iota of difference so important to you? There is no strong need to use the term cycloid and say what particular kind anyway. “Curved path” works as well most of the time for the topic at hand.

I assure you that my using “skidding” was not a change in my position. It is the same as two days ago: “You call it "skidding," which is a weak metaphor. I call it translational-rotational motion, which is literal.” It is also a term that Baal uses. I’m also confident that he knows many times more physics and math than you and Jon combined. You and Jon could also note that he has not pounced on me, played gotcha with me, and insulted me like you have. He is a gentleman, something you obviously are not.

I haven’t quite yet figured out what the metaphor might mean. Regardless, I am confident it will not match what you believe it means. It has brought me to thinking about a very complicated phenomena.

[Jon Letendre]

5 minutes ago, merjet said:

Whoop-de-doo! Reality is what Jonathan says it is, regardless of any facts he doesn’t know about. We are not in a position to look at the wheel and the rest of the setup in real life. I came to the conclusion I did for three reasons.

1. I watched the path of the dot on the lower circle in the neighborhood of its starting and ending points. The path looked much more cuspy than the path the inner circle took in the neighborhood of its starting and ending points. The latter path is shaped much more like that of a child’s playground slide near its low/ending point.

2. It appears to me that there is a board or surface behind the wheel that another part of the wheel that we can’t see rolls on. It could be as high as the larger circle and a big part of the weight of the wheel is on it. The circumference of the white disk appears to roll in the tan groove. The major purpose of the groove could be to guide the disk’s roll, direction, and help keep it vertical, parallel to the horizontal wires. We have no evidence at all that the part of the disk that rides in the groove bears much of the weight. The wheel obviously has an axle that we don’t see and the guy turns. He manually turns it and if I were doing it, I would want the design of the apparatus to assist me as much as feasible to keep it rolling straight and staying vertical. Have a look at the wheel in the video you posted. The circumference of it extends below the horizontal bar that the wheel rolls on. The larger circle on the wheel is the “road surface”, not the bottom of the wheel.

3. The designer’s intended purpose, of course – to have a wheel that gives a great demonstration of Aristotle’s wheel paradox.

All the above is apparently something you could not imagine. You saw a gotcha opportunity to latch onto an infinitesimal difference between a perfect cycloid and one that might be a tiny bit curtate so that you could start slinging insults. Why is such an iota of difference so important to you? There is no strong need to use the term cycloid and say what particular kind anyway. “Curved path” works as well most of the time for the topic at hand.

I assure you that my using “skidding” was not a change in my position. It is the same as two days ago: “You call it "skidding," which is a weak metaphor. I call it translational-rotational motion, which is literal.” It is also a term that Baal uses. I’m also confident that he knows many times more physics and math than you and Jon combined. You and Jon could also note that he has not pounced on me, played gotcha with me, and insulted me like you have. He is a gentleman, something you obviously are not.

I haven’t quite yet figured out what the metaphor might mean. Regardless, I am confident it will not match what you believe it means. It has brought me to thinking about a very complicated phenomena.

I'm just memorializing this before you can it delete later, like you did your deeply unfortunate pendulum analogy.

[Jonathan]

38 minutes ago, Jon Letendre said:

The outside circumference of the wheel is rolling on the table.

Only dots such as the pink one, at the outside edge of the wheel, will trace a cycloid.

The black dots are not at the edge of the wheel where rolling contact with the table is made, so they trace curtate cycloids.

But Merlin has arbitrarily declared that the large black circle was "designed to represent" the line on which the wheel rolls! And therefore, to him, it IS the line on which the wheel rolls, even though it's clearly not! And therefore any point on the actual circumference on which the wheel is actually rolling MUST trace a prolate cycloid according to Merlin's stupid theory, which it clearly doesn't.

Um, what is it with Objectivish-types? Over the years I've experienced so many of them who reach a point where they can't resist intellectual self-immolation. They're suddenly willing to shit all over their own reputations, dump gasoline on themselves, and strike the match by arguing in favor of the most irrational nonsense. WTF? It's just this weird, obstinate, petulant, self-destructive idiocy.

J

[Jonathan]

19 minutes ago, merjet said:

Whoop-de-doo! Reality is what Jonathan says it is, regardless of any facts he doesn’t know about. We are not in a position to look at the wheel and the rest of the setup in real life. I came to the conclusion I did for three reasons.

1. I watched the path of the dot on the lower circle in the neighborhood of its starting and ending points. The path looked much more cuspy than the path the inner circle took in the neighborhood of its starting and ending points. The latter path is shaped much more like that of a child’s playground slide near its low/ending point.

2. It appears to me that there is a board or surface behind the wheel that another part of the wheel that we can’t see rolls on. It could be as high as the larger circle and a big part of the weight of the wheel is on it. The circumference of the white disk appears to roll in the tan groove. The major purpose of the groove could be to guide the disk’s roll, direction, and help keep it vertical, parallel to the horizontal wires. We have no evidence at all that the part of the disk that rides in the groove bears much of the weight. The wheel obviously has an axle that we don’t see and the guy turns. He manually turns it and if I were doing it, I would want the design of the apparatus to assist me as much as feasible to keep it rolling straight and staying vertical. Have a look at the wheel in the video you posted. The circumference of it extends below the horizontal bar that the wheel rolls on. The larger circle on the wheel is the “road surface”, not the bottom of the wheel.

3. The designer’s intended purpose, of course – to have a wheel that gives a great demonstration of Aristotle’s wheel paradox.

All the above is apparently something you could not imagine. You saw a gotcha opportunity to latch onto an infinitesimal difference between a perfect cycloid and one that might be a tiny bit curtate so that you could start slinging insults. Why is such an iota of difference so important to you? There is no strong need to use the term cycloid and say what particular kind anyway. “Curved path” works as well most of the time for the topic at hand.

I assure you that my using “skidding” was not a change in my position. It is the same as two days ago: “You call it "skidding," which is a weak metaphor. I call it translational-rotational motion, which is literal.” It is also a term that Baal uses. I’m also confident that he knows many times more physics and math than you and Jon combined. You and Jon could also note that he has not pounced on me, played gotcha with me, and insulted me like you have. He is a gentleman, something you obviously are not.

I haven’t quite yet figured out what the metaphor might mean. Regardless, I am confident it will not match what you believe it means. It has brought me to thinking about a very complicated phenomena.

Hahahahahahaha!!!!!!!

 

[Jonathan]

Is it even possible for this thread to get any more entertaining?

Perhaps!

What if another pompous twat O-job, like, say, His Royal Published Highness, the Majestic Roger Bissell, happens to share Merlin's degree of visual/spatial/mechanical ineptitude, agrees with his nonsensical position on this thread, and would be willing to step up and help to argue Merlin's case?

That would be just freaking amazingly awesome!

J

[Jon Letendre]

3 hours ago, Jon Letendre said:

The pink wheel is rolling in contact with the bottom of the groove in the table.

The pink arc is one-eighth of the pink wheel circumference and has length EQUAL to the pink line, demonstrating contact with the table and true constant-contact rolling.

The yello arc is one-eighth of the yellow "wheel's" circumference and is a mere FRACTION the length of the yellow line, demonstrating that the yellow "wheel" is NOT in contact with its imaginary "table" represented by the yellow line. The yellow "wheel" must be skipping, sliding across the yellow line "table."

To add to the precision of my language use, by skidding, slipping and sliding, I mean the "wheel" is turning at a rate that is lower than the keeping-traction-rate..

When a vehicle applies heavy braking, the tires screech because, while the auto is actually moving at 60 mph, the braked wheels are rotating at a rate lower than consistent with 60 mph per hour travel. The braked wheels are forced to rotate at a slower rate than the road moving underneath them, and so, they slide, slip past, the road underneath them. They still rotate, but slower than the road moves underneath them.

In the picture above, the green, blue and yellow "wheels" are slipping over their imaginary horizontal "roads." Like a braked, screeching wheel on a car, they are rotating, but not fast enough to keep up with their imaginary "roads."

Any and every "wheel" inside of the circumference of the wheel pictured above will skip, slip, slide over its imaginary "road." They all will, because they are not rolling wheels, but circles smaller than and drawn on, a rolling wheel. 

Only the pink wheel is staying in contact with its road, and we can see the equality of lengths of the pink ARC (which depicts the portion of the wheel that has rolled, so far) and the pink LINE (which depicts the portion of the road that has been rolled over.)

Their equal lengths prove traction was kept, there was constant contact of the wheel with the road.

The inequal lengths of the yellow ARC and yellow LINE prove that only a small length of yellow "wheel" circumference (the yellow arc) was rotated through, while many times more that length of "road" was passed over (the yellow line.) The yellow "wheel" spent a half-inch of rotational circumference and got two inches down its road. Also called skidding, sliding or slipping.

[Brant Gaede]

10 hours ago, Jon Letendre said:

Still not sure what your question is, but I have guesses.

Are you wondering why we can't tell, why we don't feel any different standing on the equator going 1,000 mph versus standing on a pole and not?

Bob can correct me on the magnitudes of the matter, but I think we CAN tell the difference. We weigh less on a scale at the equator than on a pole, due to that 1,000 mph trying to fling us off the surface of earth. But only a tiny, tiny, tiny, bit less.

If it spun much, much faster, like once per couple minutes instead of several dozen hours, then we would weigh much less at the equator than we would at the poles (which would be unchanged from what we weigh at the pole with once per 24 hour rotation.)

It's just me trying to illustrate wheels within wheels another way.

--Brant

[BaalChatzaf]

2 hours ago, Brant Gaede said:

It's just me trying to illustrate wheels within wheels another way.

--Brant

The weight difference at the equator is barely detectable, but it is detectable.  Its main effect is to make uncompensated pendulum clocks unsuitable for engineering use.  Back in the days before quartz oscillators were developed this was an annoying problem..

[Jonathan]

20 hours ago, merjet said:

2. It appears to me that there is a board or surface behind the wheel that another part of the wheel that we can’t see rolls on. It could be as high as the larger circle and a big part of the weight of the wheel is on it. The circumference of the white disk appears to roll in the tan groove. The major purpose of the groove could be to guide the disk’s roll, direction, and help keep it vertical, parallel to the horizontal wires. We have no evidence at all that the part of the disk that rides in the groove bears much of the weight. The wheel obviously has an axle that we don’t see and the guy turns. He manually turns it and if I were doing it, I would want the design of the apparatus to assist me as much as feasible to keep it rolling straight and staying vertical.

I’m trying make sense of the above kookiness and to figure out what in the hell Merlin imagines that he’s seeing, and I came up with this:

Merlin, are the above images representative of what you’re saying? Do you think that the setup is not a flat board with a groove in it, as in the lefthand images, but that there is a piece sticking up behind the wheel, as in the righthand images?

J

[merjet]

1 hour ago, Jonathan said:

Merlin, are the above images representative of what you’re saying? Do you think that the setup is not a flat board with a groove in it, as in the lefthand images, but that there is a piece sticking up behind the wheel, as in the righthand images?

An answer to that is going to cost you. It could be money, more respectful wording and apologies, or telling me how you did those drawings. 

[Jon Letendre]

1 hour ago, Jonathan said:

I’m trying make sense of the above kookiness and to figure out what in the hell Merlin imagines that he’s seeing, and I came up with this:

Merlin, are the above images representative of what you’re saying? Do you think that the setup is not a flat board with a groove in it, as in the lefthand images, but that there is a piece sticking up behind the wheel, as in the righthand images?

J

Yes, Jonathan, the above drawings are a correct representation of what Merlin proposed is the truth about the wheel in his video. He's so incredibly lost.

[Jonathan]

27 minutes ago, merjet said:

An answer to that is going to cost you. It could be money, more respectful wording and apologies, or telling me how you did those drawings. 

I'll agree to be "more respectful" if you do as well. Sound fair?

As to how I did the drawings, I began by creating the basic shapes as vector art in Adobe Illustrator, imported them into Carrara (https://www.daz3d.com/carrara-8-5-pro), "extruded" them in the "spline" modeler, added lighting and shadow specifications, and then rendered them in "photorealistic" mode.

Now, if the drawings do represent something close to how you're envisioning the setup in the video that you posted near the beginning of this thread, I'll just inform you now that there are various pieces of evidence in the video which invalidate the notion that there is a ledge sticking up, including brief glimpses of the ends of the setup which conform to the perspective of a flat board rather than a vertical one, and various shadows and reflections which necessitate that the board is flat with a groove in it versus the mistaken opinion that it has a vertical section. I would advise you to look more closely at such details. Pay particularly close attention to the initial still image, and how the section that you're mistaking as a ledge is so short as to be almost nonexistent. Notice in the video that the "height" of the alleged ledge changes depending on the altitude of the camera. (The lower a camera, the more foreshortening occurs of the ground plane).

J

[merjet]

Apologies? You first. And you have time to edit your post with the images in it.

[Jonathan]

No thanks.

[merjet]

25 minutes ago, Jon Letendre said:

He's so incredibly lost.

It took incredible stupidity to say that.

[Jonathan]

4 minutes ago, merjet said:

It took incredible stupidity to say that.

No, I think he's right. Sorry, no disrespect intended, but you do seem to be lost, and not only that, but also grasping at straws. You seem to be trying your hardest to salvage a mistaken point of view, indulging in confirmation bias, and not giving due consideration to the arguments that have been presented which refute your position.

[Jon Letendre]

6 minutes ago, merjet said:

It took incredible stupidity to say that.

Says the only idiot in the room still perplexed by the "paradox."

[merjet]

2 hours ago, Jonathan said:

Merlin, are the above images representative of what you’re saying? Do you think that the setup is not a flat board with a groove in it, as in the lefthand images, but that there is a piece sticking up behind the wheel, as in the righthand images?

The images on the right are roughly what I imagined.  Regarding the third on the right, I was thinking the groove narrower, a "shoulder" on the right of the piece sticking up supporting the wheel (the "head"), and the disc in the groove being much closer to the bottom of the groove.

The effect of all that would hugely undercut J's assertions that the larger circle on the wheel traces a curtate cycloid. Ideally it would trace an ordinary cycloid. Also, that's by engineering standards, not abstract math standards.

[Jonathan]

8 minutes ago, merjet said:

The images on the right are roughly what I imagined...

That's what I had suspected.

So, as I said in a recent post:

"I'll just inform you now that there are various pieces of evidence in the video which invalidate the notion that there is a ledge sticking up, including brief glimpses of the ends of the setup which conform to the perspective of a flat board rather than a vertical one, and various shadows and reflections which necessitate that the board is flat with a groove in it versus the mistaken opinion that it has a vertical section. I would advise you to look more closely at such details. Pay particularly close attention to the initial still image, and how the section that you're mistaking as a ledge is so short as to be almost nonexistent. Notice in the video that the 'height' of the alleged ledge changes depending on the altitude of the camera. (The lower a camera, the more foreshortening occurs of the ground plane)."

[Jon Letendre]

 

 

[merjet]

40 minutes ago, merjet said:

The images on the right are roughly what I imagined ...

The effect of all that would hugely undercut J's assertions that the larger circle on the wheel traces a curtate cycloid. Ideally it would trace an ordinary cycloid. Also, that's by engineering standards, not abstract math standards.

Note how J tried to sweep my second part under the rug by lopping it, saying "That's what I had suspected." after including only the first part in his post.

 

 

[Jon Letendre]

6 minutes ago, merjet said:

Note how J tried to sweep my second part under the rug by lopping it, saying "That's what I had suspected." after including only the first part in his post.

 

 

If the wheel actually was constructed the way you imagine, then yes, the black dot on the circumference of that rolling wheel would trace a cycloid.

But the wheel is not constructed the way you imagine it to be. It is not rolling on an imaginary road we cannot see.

The wheel in your video is rolling on the bottom of the groove. We know because the pink arc is the same length as the pink line.

[merjet]

I begin with defining some symbols to use rather than words for ease of reference and economizing keystrokes.

VJ = video posted by Jonathan Link

VM = video posted by me Link

Pb = point on the larger circle in VM , 6:00 o'clock at the starting position 

Ps = point on the smaller circle in VM, 6:00 o'clock at the starting position

The wheel in VJ and the wheel in VM are isometric. For both it is the larger circle that represents a real world wheel that rides on a road. The road is represented by the lower bar in VJ and by the lower wire in VM.

J posted that only the inner, smaller circle in VJ slips/skids while the outer, larger circle does not. “Can you see how the large black circle properly "rides" the bottom line, but the small black circle skids along the top line?" Link.

Suppose a third bar were placed at the very bottom of the wheel in VJ. You can easily do the experiment. Simply hold a ruler at the bottom edge of the wheel and watch it roll. The video plays exactly as before. It thus appears that the perimeter of the wheel “rides” the third bar or ruler and the large circle skids a bit along the middle line. In other words, simply assuming adding another bar or holding a ruler changes the large black circle from “riding” to skidding. That seems problematic to me.

Anyway, now that we have two basically alike models to observe, there are still some key differences regarding what they tell us. The purpose of VM is apparently to highlight what happens to a given point on the wheel as the wheel rolls normally. It doesn’t have the different pie-shaped colors VJ has. Without the pie-shaped colors it doesn’t convey the “slipping” like VJ does. On the other hand, the purpose of VJ is apparently to highlight what happens to a given circle, or a pie-shaped part of it, on the wheel as it rolls normally. Also, lacking VM’s points, it does not trace the simultaneous paths of Pb and Ps. One could focus on how a point moves by looking at a pie piece corner, but it’s impossible to trace one on the large circle and one on the small circle simultaneously the way VM does.

I think the different perspectives the two models show has added a lot of the friction on this thread. Of course, that's not all bad. It affords a learning opportunity.