Aristotle's wheel paradox

[merjet]

5 hours ago, Jonathan said:

With . . . [more hogwash and another attempted con game]

Written by the idiot who imagines he sees a catenary and a non-existent hanging chain or cable in this video.

[Jon Letendre]

5 hours ago, merjet said:

LOL, stupid moron and 100% duped.

"It notes interesting things about cycloid-drawing points on a rolling wheel, but does not directly address the alleged equality that the paradox is about" (link).

LOL, stupid.

 

Ellen has made the same comment, idiot.

Your resolution makes a giant leap, as I recall you wrote, "Most people don't consider the curved paths, considering them resolves the paradox." That's impressive hand-waving, but doesn't work as a proof or resolution.

You know that, and you are a lying piece of shit.

[Jonathan]

2 hours ago, merjet said:

Written by the idiot who imagines he sees a catenary and a non-existent hanging chain or cable in this video.

I said nothing about seeing a hanging chain or cable. You made that up. In other words, you lied.

A catenary is a type of curve. Do you not know that? The video that I posted contains catenary curves on which the squares roll.

You, on the other hand, DID claim to see a non-existent ledge. You wanted it to be there because you had misidentified the path that the point on the larger circle makes, and you don't have the integrity to admit to your error.

Jon's right. You're a lying piece of shit. 

[Max]

4 hours ago, merjet said:

Written by the idiot who imagines he sees a catenary and a non-existent hanging chain or cable in this video.

Perhaps you should read this first: https://tinyurl.com/yas5draz

 

 

[Ellen Stuttle]

On October 10, 2017 at 1:01 PM, Max said:

The cycloids are not relevant for the solution of the paradox, as they are a description of the movement of one point of the wheel in the z-x plane (z = up, x = direction of rail/ledge/road). The paradox is about the interface wheel-rail/ledge/road, however. That is: the points of the wheel and of the rail/ledge/road where they touch each other. The position of these points form a straight line along the rail/ledge/road. When the wheel rotates without slipping on its support, the length of that line is 2πR after one revolution of the wheel with radius R.

 

With the two concentric wheels (radius R and r, r < R) in the paradox, the length of those lines would after one revolution without slipping be equal to 2πR and 2πr respectively. However, the actual length can only have one value, as those wheels are part of a rigid body, so at most only one wheel can rotate without slipping, for example the larger, outer wheel. The smaller inner wheel then has to travel the same distance 2πR over its support. In the same time interval its proper slip-free rotation distance is only 2πr, which is not enough, so it has also to slip over a distance 2π(R – r) to keep up with the outer wheel. Jonathan’s animation shows this clearly.

 

Further, I’m reminded of this joke: https://tinyurl.com/y7hly2al

 

Max,

I think that's an excellently clear and precise exposition.

Just one eensy verbal quibble.  Instead of referring to "the paradox," I recommend saying "the supposed paradox" or the "so-called paradox" or using scarequotes or in some other way of your choosing indicating that the problem presented isn't properly a paradox since it includes in its very formation a contradiction with reality.  The problem set-up ignores that - quoting you - "the actual length can only have one value, as those wheels are part of a rigid body, so at most only one wheel can rotate without slipping."

Ellen

[Ellen Stuttle]

On October 10, 2017 at 1:01 PM, Max said:

The cycloids are not relevant for the solution of the paradox, as they are a description of the movement of one point of the wheel in the z-x plane (z = up, x = direction of rail/ledge/road). The paradox is about the interface wheel-rail/ledge/road, however. That is: the points of the wheel and of the rail/ledge/road where they touch each other. The position of these points form a straight line along the rail/ledge/road. When the wheel rotates without slipping on its support, the length of that line is 2πR after one revolution of the wheel with radius R.

Elaborating on the irrelevance of the cycloids:

The distance a point travels in getting from start to end of a revolution doesn't matter to the "paradox" problem, since that's about the linear distance between the start and end 6:00 o'clock positions.  The cycloid distance traveled by respective points on the inner and outer circumferences is different.  The linear distance between the start and end of the respective 6:00 o'clock positions is the same and must be the same since the configuration is a rigid body.  As Max indicates in his second paragraph (see my post above for the full quote), the set-up could be done so that both wheels slip respective to their "roads" or roads, but the linear distances between the start and end of the respective 6:00 o'clock positions would still be the same - and would be equal to the distance the center moved.

Ellen

[Max]

9 hours ago, Ellen Stuttle said:

Max,

I think that's an excellently clear and precise exposition.

Just one eensy verbal quibble.  Instead of referring to "the paradox," I recommend saying "the supposed paradox" or the "so-called paradox" or using scarequotes or in some other way of your choosing indicating that the problem presented isn't properly a paradox since it includes in its very formation a contradiction with reality.  The problem set-up ignores that - quoting you - "the actual length can only have one value, as those wheels are part of a rigid body, so at most only one wheel can rotate without slipping."

Ellen

I think that's just a matter of definition. As I said before: a paradox is an argument that leads to an apparent contradiction. The contradiction doesn't exist in reality, so there must be some error in the argument. One can solve the paradox by showing were the error in the argument lies. The bad argument, i.e. the paradox, doesn't disappear in my opinion, it has only been shown what was wrong in the argument Well-known paradoxes are for example those special-relativity paradoxes, such as the barn-pole paradox, which seem to imply contradictions in reality, by incorrectly supposing that simultaneity is an invariant. There is no contradiction, but the paradox does exist and is well-known. I think the quibble is about the distinction between a paradox and a contradiction.

[Brant Gaede]

Solve the paradox of the existence of this thread beyond the first page of posts.

--Brant

[Jonathan]

3 hours ago, Brant Gaede said:

Solve the paradox of the existence of this thread beyond the first page of posts.

--Brant

 

Stupidity + stubbornness x enjoyment of ridiculing stubborn stupidity = this thread.

...You're welcome.

[Ellen Stuttle]

On October 10, 2017 at 11:43 AM, merjet said:

[below]

Me to Merlin - here:

"You do seem to be missing what the purported problem is about.  The question asked is why the 6:00 o'clock position on the small circle or wheel hasn't moved, after one revolution, only the distance of the small circle or wheel's circumference along a horizontal line or imagined road."

Merlin - here:

"I haven't missed it any. My resolution said it clearly."

Your "resolution" post describes "the paradox, or oddity" thus:

[MJ]  "The paradox, or oddity, is that the straight lengths are equal but the circular lengths are not, despite Pb's straight and circular lengths being equal."

But then, as I quoted in a post you didn't respond to, you go on to say:

[MJ]  "The circular paths are not part of the rolling experiment and make a 'red herring.'"

The question raised by the supposed paradox is about the respective relationships of the circular paths to the straight paths, so, yes, the circular paths are part of, and not a "red herring" to, the rolling experiment.  Cycloids, which you think resolve the "paradox," don't address the circular path/linear distance issue.

Plus, you somehow have gotten the idea, displayed in several of your posts, that people have been talking about slippage of the smaller circle/wheel in relationship to the larger figure, but that isn't what anyone was talking about or what the supposed paradox is about.  Instead, the fixity of the smaller circle/wheel in relationship to the larger figure is why there isn't any real paradox.

Ellen

[Ellen Stuttle]

6 hours ago, Jonathan said:

 

Stupidity + stubbornness x enjoyment of ridiculing stubborn stupidity = this thread

Not totally.  The thread does contain material of mathematics and physics interest, some of which material you've contributed along with ridicule.

Ellen

[Ellen Stuttle]

11 hours ago, Max said:

I think that's just a matter of definition. [....] Well-known paradoxes are for example those special-relativity paradoxes, such as the barn-pole paradox, which seem to imply contradictions in reality, by incorrectly supposing that simultaneity is an invariant. There is no contradiction, but the paradox does exist and is well-known. I think the quibble is about the distinction between a paradox and a contradiction.

I won't continue with the quibble, but you raise a question I've been wondering about.  Seems to me that this wheel problem differs from the special-relativity paradoxes in that those come from reasonable assumptions whereas the wheel problem doesn't.

It's an issue I'd like to discuss with my husband, who's a physicist and is very knowledgeable about relativity, but I haven't yet mentioned this thread to him.  He's superbusy with a special course he's teaching this fall, plus with preparations for a university-wide seminar series talk he's scheduled to give in March, and I've hesitated to distract him with an issue which pertains to symmetry relationships.  Symmetry is a particular love of his, and he's easily distracted by symmetry problems.

Ellen

[merjet]

On 10/11/2017 at 11:47 AM, Jonathan said:

I said nothing about seeing a hanging chain or cable. You made that up. In other words, you lied.

You said you saw a catenary. If you are too stupid to know that a catenary is formed by a hanging wire chain or cable, so be it.

You are the liar. I didn't make it up. It came from WolframMathWorld: "The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force" (link).   Like the rabidly dishonest scumbag you are, you tried to sweep that reference under the rug by misquoting. 

[Jonathan]

5 hours ago, Ellen Stuttle said:

Not totally.  The thread does contain material of mathematics and physics interest, some of which material you've contributed along with ridicule.

Ellen

Good point.

[Jonathan]

1 hour ago, merjet said:

You said you saw a catenary. If you are too stupid to know that a catenary is formed by a hanging wire chain or cable, so be it.

You are the liar. I didn't make it up. It came from WolframMathWorld: "The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force" (link).   Like the rabidly dishonest scumbag you are, you tried to sweep that reference under the rug by misquoting. 

>yawn<

[merjet]

18 hours ago, Ellen Stuttle said:

But then, as I quoted in a post you didn't respond to, you go on to say:

[MJ]  "The circular paths are not part of the rolling experiment and make a 'red herring.'"

The question raised by the supposed paradox is about the respective relationships of the circular paths to the straight paths, so, yes, the circular paths are part of, and not a "red herring" to, the rolling experiment.  

Your have it backward to what I said. On Sep 23 I wrote, "The circular path's length is obviously the circumference and ignores the fact that the circle is moving. The curved paths do not ignore the circles moving."

The circular path is only rotation with no horizontal movement. Since the circles do move horizontally in the experiment, the circular path is not part of the experiment. The circular path is part of the paradox.

[Brant Gaede]

1 hour ago, merjet said:

Your have it totally backward to what I said. On Sep 23 I wrote, "The circular path's length is obviously the circumference and ignores the fact that the circle is moving. The curved paths do not ignore the circles moving."

The circular path is only rotation with no horizontal movement. Since the circles do move horizontally in the experiment, the circular path is not part of the experiment. The circular path is part of the paradox.

What is "the paradox" part of (if there were no humans on earth)?

--Brant

[Jonathan]

2 hours ago, merjet said:

Your have it totally backward to what I said. On Sep 23 I wrote, "The circular path's length is obviously the circumference and ignores the fact that the circle is moving. The curved paths do not ignore the circles moving."

The circular path is only rotation with no horizontal movement. Since the circles do move horizontally in the experiment, the circular path is not part of the experiment. The circular path is part of the paradox.

Actually, curved paths do ignore the circles moving. You eliminate the circles from the setup, and replace them with three points on a line segment which undergoes rotation and translation. The "paradox" setup is about circles/wheels traveling on lines/surfaces, not about a line/stick with points/dots on it flipping through the air

Also, you haven't demonstrated -- proven -- that the points make the paths that you claim that they do. You've only trusted online sources that you've borrowed from. You reject our (original, unborrowed) presentations with the claim that they are, or could be, optical illusions. So, your sources could be as well. You would have to eliminate that double standard by actually showing that your borrowed math isn't merely describing an illusion. Get out your compass and straightedge and do the work. Show it, or else your claims about curved paths are just hearsay based on optical illusions.

[Jonathan]

1 hour ago, Brant Gaede said:

What is "the paradox" part of (if there were no humans on earth)?

--Brant

Judging by the comments at the various sites which address the "paradox," it's only a "paradox" to a very small portion of the population. I'm glad Merlin started this thread. It provides a quick little test for visual/spatial/mechanical incompetence. 

[BaalChatzaf]

2 hours ago, Jonathan said:

......It provides a quick little test for visual/spatial/mechanical incompetence. 

Or more positively,  it provides an opportunity to appreciate how essential infinitesimal methods  are necessary to do the physical mechanics of revolution and rotation.  The physics or revolution and rotation are a necessary preliminary to developing a theory of cyclic  heat engines. Hold on!  Here come the pumps and locomotives.........

 

[Ellen Stuttle]

12 hours ago, merjet said:

Your have it backward to what I said. On Sep 23 I wrote, "The circular path's length is obviously the circumference and ignores the fact that the circle is moving. The curved paths do not ignore the circles moving."

The circular path is only rotation with no horizontal movement. Since the circles do move horizontally in the experiment, the circular path is not part of the experiment. The circular path is part of the paradox.

 

Below is the paragraph with the sentence I quoted from your Sep 23 "Resolving the Paradox" post.  (I've italicized the sentence.)

 

On September 23, 2017 at 9:05 AM, merjet said:

[itlaics added]

 

Resolving the Paradox

[....]

The circular paths are not part of the rolling experiment and make a "red herring." Let Rs denote the radius of the smaller circle and Rb the radius of the bigger circle. The lengths of their paths follow, along with whether or not they are actually traveled. I show a range for Ps's curved path because the formula is very complicated. The closer Ps is to Pb, the closer the length is to 8*Rb. The closer Ps is to the center of the circles, the closer it is to Rb*2*pi.

[....]

You elide specifications in the third sentence, since what you mean is the length of the path a point follows, not the length of the radius' paths.

I agree with Jonathan's response:

9 hours ago, Jonathan said:

Actually, curved paths do ignore the circles moving. You eliminate the circles from the setup, and replace them with three points on a line segment which undergoes rotation and translation. The "paradox" setup is about circles/wheels traveling on lines/surfaces, not about a line/stick with points/dots on it flipping through the air

Ellen

[Ellen Stuttle]

7 hours ago, BaalChatzaf said:

Or more positively,  it provides an opportunity to appreciate how essential infinitesimal methods  are necessary to do the physical mechanics of revolution and rotation.  The physics or revolution and rotation are a necessary preliminary to developing a theory of cyclic  heat engines. Hold on!  Here come the pumps and locomotives.........

 

Granted that developments in mathematics were essential for later working out of "the physical mechanics of revolution and rotation."  However,  I've been thinking that even with the limited math available in classical times, the "paradox" came from ignoring reality and thinking in a sloppy fashion.

As I wrote yesterday - here:

"Seems to me that this wheel problem differs from the special-relativity paradoxes in that those come from reasonable assumptions whereas the wheel problem doesn't."

The question raised by the "paradox" in effect asks, "Why doesn't something which can't move independently move independently?"

Answer:  It doesn't because it can't.

Ellen

 

[Ellen Stuttle]

[duplicate deleted]

[BaalChatzaf]

8 minutes ago, Ellen Stuttle said:

Granted that developments in mathematics were essential for later working out of "the physical mechanics of revolution and rotation."  However,  I've been thinking that even with the limited math available in classical times, the "paradox" came from ignoring reality and thinking in a sloppy fashion.

As I wrote yesterday - here:

"Seems to me that this wheel problem differs from the special-relativity paradoxes in that those come from reasonable assumptions whereas the wheel problem doesn't."

The question raised by the "paradox" in effect asks, "Why doesn't something which can't move independently move independently?"

Answer:  It doesn't because it can't.

Ellen

 

Math Think and Symbolic Logic Think are the only truly precise methods of thinking I am certain of. 

[Ellen Stuttle]

19 minutes ago, BaalChatzaf said:

Math Think and Symbolic Logic Think are the only truly precise methods of thinking I am certain of. 

LOL.  You do like to play the fool.

In what year did precise thinking become possible?

Ellen