Aristotle's wheel paradox

[Jon Letendre]

8 hours ago, merjet said:

There being only one tangent line is a figment of your imagination. It takes less than 12 seconds of this video to prove it. There is an unlimited number of tangent lines to the cycloid. The center of the circle in the video is what Baal called the hub. What Baal refers to as "skid" – a metaphor -- is how the slope of the tangent lines changes along different points on the curve . "Shift" describes it better than "skid." He used "skid" in a much different way than you did. The two usages are about as much alike as day and night. So his using it does not support your notion of "skid" one iota. Your comment and alleging Baal agrees with you doesn't even begin to pass muster.

I'd bet the measurement implications of all this is not even on your radar screen. 

Merlin, we are all,discussing the topic of the thread that you started. We're not much discussing cycloids.

My answer to you, that there is only one line of tangency is correct given I thought you were talking about a rolling wheel, which is, again, the topic of the thread and current discussion,

 

[Jon Letendre]

On 9/25/2017 at 11:37 AM, merjet said:

Suppose the point on the rim you refer to is at 6:00 o'clock before the roll begins. The wheel rotates 90 degrees clockwise to put the point at 9:00. Where is the point's tangent line?

Knowing now you mean NOT the rolling wheel, the line of tangency you seek is the line that intersects your point and touches the end point of the cycloid being drawn.

I am trying to get a still from your video, but Photobucket is not cooperating...

[Jon Letendre]

A still from your video.

The tangent line you ask about is in red. See the "T" composed of the red line and the black line? Their perpendicularity holds throughout rotation.

The tangent line of the rolling wheel, which Baal and the rest of us and the thread title are all talking about, is the road. And the point of tangency is the one point of the wheel in contact with the road.

[Jon Letendre]

9 hours ago, merjet said:

There being only one tangent line is a figment of your imagination. It takes less than 12 seconds of this video to prove it. There is an unlimited number of tangent lines to the cycloid. The center of the circle in the video is what Baal called the hub. What Baal refers to as "skid" – a metaphor -- is how the slope of the tangent lines changes along different points on the curve . "Shift" describes it better than "skid." He used "skid" in a much different way than you did. The two usages are about as much alike as day and night. So his using it does not support your notion of "skid" one iota. Your comment and alleging Baal agrees with you doesn't even begin to pass muster.

I'd bet the measurement implications of all this is not even on your radar screen. 

Skid is a literal dynamics term with perfectly clear meaning. It is not a metaphor.

What you say Baal was referring to is in fact not what he was referring to. He was not talking about the varying slope of the cycloid's line of tangency. He was talking about a rolling wheel whose line of tangency is always the road.

Baal's usage of "skid" is precisely the same as everyone else's on the thread. He means the same skidding that everyone else here means. He simply describes another method of detecting its reality.

[Jon Letendre]

The inner, green "wheel" rotates and also skids across its "road."

Hit the middle-bottom button for replays.

Aristotle suggests the green "wheel" truly rolls over its "road"." He was wrong. Therefore, no paradox.

 

[Jon Letendre]

20 hours ago, merjet said:

Where did you get that notion? It's not on the Wikipedia page for Aristotle's wheel paradox following "The problem is then stated."

The alleged equality of road lengths truly rolled over by the wheel and truly rolled over by the smaller circle "wheel"  drawn on the wheel is the very essence of Aristotle's Wheel Paradox.

For it is that equality, contrasted with the inequal circumferences of wheel and "wheel" that gives rise to the paradox.

Without that "notion" the paradox can't even be expressed.

[Jon Letendre]

On 9/25/2017 at 9:14 AM, BaalChatzaf said:

Any translation of the center (hub) that is not accompanied by an instantaneous roll round the point of tangency  at the rim   is by definition a skid.  Of course, we are assuming the rigidity of the wheel/circle.  At no point does the wheel/circle become deformed.

Baal is talking about a rolling wheel.

By "point of tangency" he means the point of the wheel in contact with the road. Said point is at 6 o'clock.

 

Said point does indeed rotate around to 9 o'clock, and keeps rotating (and being translated,) drawing a cycloid.

But Baal is not talking about any of that.

[Jon Letendre]

10 hours ago, merjet said:

The obnoxious numbskull also "bet" that I would never present a solution to the "non-paradox." That "bet" has been proven dead wrong.

I recall you posted something, calling it your resolution/solution, but now I can't find it.

Would you post it again, please?

[merjet]

8 hours ago, Jon Letendre said:

A figment of my imagination. You feel compelled to talk about me.

The pot calls the kettle black big time, while feeling compelled to pretend he is so innocent.  

8 hours ago, Jon Letendre said:

Resolving the paradox is as simple as grasping those two sentences. Amazing as it seems, Merlin does not yet grasp them.

 

[Jon Letendre]

15 minutes ago, merjet said:

The pot calls the kettle black big time, while feeling compelled to pretend he is so innocent.  

 

Amazing as it seems.

I see how that could hit the other side and feel personal. I'm sorry.

To rephrase without the personal, "Merlin disputes both of those sentences."

[Jon Letendre]

19 hours ago, merjet said:

There being only one tangent line is a figment of your imagination. It takes less than 12 seconds of this video to prove it. There is an unlimited number of tangent lines to the cycloid. The center of the circle in the video is what Baal called the hub. What Baal refers to as "skid" – a metaphor -- is how the slope of the tangent lines changes along different points on the curve . "Shift" describes it better than "skid." He used "skid" in a much different way than you did. The two usages are about as much alike as day and night. So his using it does not support your notion of "skid" one iota. Your comment and alleging Baal agrees with you doesn't even begin to pass muster.

I'd bet the measurement implications of all this is not even on your radar screen. 

I think this, from you, hours earlier, is what got me irritated and feeling you were first in getting personal.

Because, I am happy to see and consider all the measurement implications and high math you allude to but never present much of. You've said I don't consider it, wouldn't understand it, but steadfastly avoid ever presenting it.

I have commented on the vast majority of what you have presented and I understand all of what you have presented. If I've missed anything, let me know.

Perhaps getting personal is not the right way for me to put it, I think simply being unfair is more accurate.

Please Show Me now whatever it is you think I have failed to consider, name it, illustrate it, something. It isn't in your bit above, because I understand every word of that. It has some errors, but I understand it all. 

Tell me what measurement implications I haven't considered.

[Jon Letendre]

Here are the measurements I consider as most readily demonstrative of the skidding that Aristotle's circle drawn on a wheel performs as it travels the imaginary road Aristotle asks us to consider.

We can see true rolling evidenced in the equal lengths of the pink arc and pink line.

We can see that the green, blue and yellow circles travel the same length of road as the whole (pink) wheel, evidenced in the equal lengths of all the lines.

And, contra Aristotle, we can see that the lengths of the green, blue and yellow circles' circumferences contributed to rolling (the arcs) are shorter than the length of road they travelled (the lines,) which is called skidding. The paradox directly implies that all the arcs are EQUAL in length and also equal to road traversed length (the lines.) But they are NOT equal. The 3-to-1 ratio of road-traversed to circumference-put-through-rolling is evidenced in the disparity between the yellow line length and the yellow arc length. The yellow circle "wheel"* rotates  while it travels down the road*, but not fast enough to avoid significant skidding.

*Aristotle asks us to consider and think of it as such. If we comply with his request, the skidding is unavoidable and that term is very precise, literal, exact, given his request to think of the circles drawn on a wheel as actual wheels themselves, rolling down an imaginary road they are in contact with.

Thus, there is no equality to explain and the paradox evaporates.

[merjet]

9 hours ago, Jon Letendre said:

Amazing as it seems.

I see how that could hit the other side and feel personal. I'm sorry.

To rephrase without the personal, "Merlin disputes both of those sentences."

The two sentences you refer to are:

On 9/26/2017 at 2:21 AM, Ellen Stuttle said:

The initial supposed paradox attributed to Aristotle includes as part of the setup the false premise that the inner circle is rolling in true contact with its imagined road.  It couldn't be doing that.

I replied:

On 9/26/2017 at 4:58 AM, merjet said:

Where did you get that notion? ....

Pardon me, but my question was not disputing what Ellen said. I asked the question because what she said wasn't clear to me.  What does "the inner circle is rolling in true contact with its imagined road", especially "rolling in true contact", mean? The latter was a first-time appearance on this thread.  There is plenty of misunderstanding of one another's terminology on this thread already, so I didn't want to guess what it meant and reply based on what I guessed. I have a hypothesis about what it meant, and if it is accurate, then I agree with those two sentences.

"Rolling" does not have a unique meaning for me. Consider a bowling ball thrown down the lane. Its manner of rolling changes between foul line and pins. It can skid a lot more than it rotates. It can rotate faster than its movement on the lane. It may do a "true roll" where rotation and movement on the lane are in sync. 

A bowling ball's motion is more complicated than that of a wheel. Still, the rolling = rotation + translation can vary. Two different cases could have the same rate of rotation but different rates of translation. That results in somewhat different examples of rolling.

Imagine we have two rolls of duct tape, with the same inner diameter (hole) of 3 inches, but different outer diameters of 3.5 inches and 4.5 inches due to different amounts of tape remaining. We roll them one rotation. One circumference is 10 inches; the other is 14.14 inches. So in one case the inner circle travels 10 inches, the other 14.14 inches. Same rotation, different translation (horizontal movement). From your perspective, I think, there must have been different amounts of "slipping". How would you quantify those different amounts? From my perspective there were different amounts of translation. Quantifying that is easy -- 10 inches versus 14.14 inches.

[merjet]

14 hours ago, Jon Letendre said:

I recall you posted something, calling it your resolution/solution, but now I can't find it.

Would you post it again, please?

Resolving the Paradox

[Jonathan]

On 9/15/2017 at 2:01 PM, Jonathan said:

...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:

 

Hey, Merlin, take another look at the video which you claimed showed "skidding" that wasn't real but an optical illusion based on the wagon wheel effect. Watch one point on the circle starting at the 6:00 position. What do you see? What is the path that it traces? Holy smokes, is that a curtate cycloid?!! Measure it and apply the math that you found online. What are the results?

Heh.

Oh, and if you missed it earlier, here's what your trusted Baal had to say about the video:

On 9/15/2017 at 7:51 PM, BaalChatzaf said:

That is an excellent display. 

Thanks, Bob!

J

[merjet]

Hey, Jonathan, imagine a real world experiment with this tire. The radius looks at least 6 feet tall, making the tire circumference at least 37.7 feet. Put a visible mark on the rim and observe the wheel roll very slowly, e.g. 37 seconds for one rotation. (Your video is 8 seconds.) With the tire covering only about one foot per second, could you really see the rim "skidding"? If you believe so, make it rotate even slower.

[Brant Gaede]

3 hours ago, merjet said:

The two sentences you refer to are:

I replied:

Pardon me, but my question was not disputing what Ellen said. I asked the question because what she said wasn't clear to me.  What does "the inner circle is rolling in true contact with its imagined road", especially "rolling in true contact", mean? The latter was a first-time appearance on this thread.  There is plenty of misunderstanding of one another's terminology on this thread already, so I didn't want to guess what it meant and reply based on what I guessed. I have a hypothesis about what it meant, and if it is accurate, then I agree with those two sentences.

"Rolling" does not have a unique meaning for me. Consider a bowling ball thrown down the lane. Its manner of rolling changes between foul line and pins. It can skid a lot more than it rotates. It can rotate faster than its movement on the lane. It may do a "true roll" where rotation and movement on the lane are in sync. 

A bowling ball's motion is more complicated than that of a wheel. Still, the rolling = rotation + translation can vary. Two different cases could have the same rate of rotation but different rates of translation. That results in somewhat different examples of rolling.

Imagine we have two rolls of duct tape, with the same inner diameter (hole) of 3 inches, but different outer diameters of 3.5 inches and 4.5 inches due to different amounts of tape remaining. We roll them one rotation. One circumference is 10 inches; the other is 14.14 inches. So in one case the inner circle travels 10 inches, the other 14.14 inches. Same rotation, different translation (horizontal movement). From your perspective, I think, there must have been different amounts of "slipping". How would you quantify those different amounts? From my perspective there were different amounts of translation. Quantifying that is easy -- 10 inches versus 14.14 inches.

The inner circle doesn't travel except in the sense of being carried. When you hike you travel. When you get on an airplane you are carried, aka traveled as in a shorthand expression. A real estate agent doesn't sell a listing--the owner of the property sells it, but the common parlance in real estate is that the agent sells it.

--Brant

[merjet]

20 minutes ago, Brant Gaede said:

The inner circle doesn't travel except in the sense of being carried.

Wow! I think maybe you got it. "Being carried" is a kind of translation.    Hands clapping.

[Jonathan]

2 hours ago, merjet said:

Hey, Jonathan, imagine a real world experiment with this tire. The radius looks at least 6 feet tall, making the tire circumference at least 37.7 feet. Put a visible mark on the rim and observe the wheel roll very slowly, e.g. 37 seconds for one rotation. (Your video is 8 seconds.) With the tire covering only about one foot per second, could you really see the rim "skidding"?

Yes.

 

Quote

If you believe so, make it rotate even slower.

I can still see the skidding. Lowering the speed of travel doesn't change my ability to see it.

And your inability to see it at any speed has no bearing on my or anyone else's ability to see it. Other people are not limited to your personal limitations just because you need to believe that they share your limitations and feel insulted that they can easily grasp what you can't.

Now, you didn't answer my questions. You evaded my challenge that you measure the objects and their movements in my video. You're incapable of doing so, aren't you? You can't actually apply to reality the math that you found online.

It's really a good thing that you at least have some mathematical abilities to help you semi-understand some things, what with your total ineptitude with visual/spatial/mechanical reasoning!

 

Quote

(Your video is 8 seconds.)

Indeed it is. The wheel in my video makes one full rotation in 8 seconds. The frame rate of the video is 30 frames per second. Do the math: How many rotations per second would have to occur in order for the "wagon-wheel-effect" optical illusion to appear when using a wheel which is equally divided into 8 distinct sections or "slices"?

J

[Jonathan]

2 hours ago, merjet said:

Wow! I think maybe you got it. "Being carried" is a kind of translation.    Hands clapping.

Skidding is also a kind of translation, dork.   No hand claps for dorks.

[merjet]

23 hours ago, Jonathan said:

Skidding is also a kind of translation, dork.   No hand claps for dorks.

Hey, numbskull dork, I didn't say "being carried" was the only kind of translation.  Are you trying to tell us that skidding is the only kind of translation, numbskull dork? You are pathetic at logic.

[merjet]

49 minutes ago, Jonathan said:

I can still see the skidding.

I didn't mean seeing it some concocted video animation, nitwit.  Jeesh, you could screw up a junkyard.

[Jonathan]

1 hour ago, merjet said:

I didn't mean seeing it some concocted video animation, nitwit.  Jeesh, you could screw up a junkyard.

Heh. In the post of mine that you just quoted, I wasn't talking about seeing skidding in an animation, but in the scenario that you presented. I can see it in reality. As I said, your inability to see it doesn't affect my ability to see it. Your ineptitude applies to you, not to me. Your mere incredulity based on your own experience of incompetence is not a valid means of measuring and determining others' capabilities. Your personal limitations are not the universal limits of all of mankind.

[Jonathan]

And...still no post in which Merlin applies his borrowed math to the objects and motions in the video that I posted. Hahaha!

And also no math to back up his assertion that my video is an example of the optical illusion known as the "wagon-wheel effect."

He's evading!!!

[merjet]

4 minutes ago, Jonathan said:

I can see it in reality. 

Hahahahahahahaha!! Your credibility is zero.