Aristotle's wheel paradox

[Brant Gaede]

12 hours ago, merjet said:

Resolving the Paradox

Observing this video helped me to conclude that the paradox comes from confounding magnitudes of three different paths -- a straight one, a circular one, and a curved one. The curved one is a cycloid. Let Ps denote the point/dash at the  6:00 o'clock position on the circumference of the smaller circle with the disc at rest in the video. Let Pb denote the point/dash at the 6:00 o'clock position on the circumference of the bigger circle with the disc at rest. Roll the conjoined circles one revolution like in the video. How far either point moves is measurable in three ways. The straight path is the simplest one – as the crow flies -- but it is one that Pb or Pb does not travel. Its length is the same for Ps and Pb (all points really). 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 curved paths show that Pb travels farther than Ps, the distance for Pb being 8 times its circle's radius (more than the circumference). Also, clearly Pb travels farther away from its horizontal line than Ps does (midway for both).

What I call 'straight', 'circular', and 'curved' correspond to translation, rotation, and rolling in this excellent video that Baal posted earlier.

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.

       Point                      Straight                       Circular                                     Curved

         Ps                   No, Rb*2*pi                      No, Rs*2*pi                Yes, min= Rb*2*pi,  max= 8*Rb

         Pb                   No, Rb*2*pi                      No, Rb*2*pi                Yes, 8*Rb

       Center              Yes, Rb*2*pi                         n/a                                  n/a

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. The different curved lengths, resulting from two different shaped paths, which many people don't consider, are the ones actually traveled. Many people know that circumference C = 2*pi*r, but few people know about the path lengths or even think about it. Considering them resolves the paradox.

If a vehicle wheel rolls many revolutions, common sense says that an inner circle travels the same distance as the outer circle. It's also true – based on the straight path. The same holds for a given point or arc on the inner circle based on the straight path, but it doesn't hold for the curved path.

There are many common objects that fit the wheel's basic shape, for example, a roll of duct tape. Saying the smaller circle formed by the duct tape's hole slips/slides/skids relative to the tape's largest circumference is bizarre to me.

The circle can do what you think it would/could, for it really isn't doing anything except an imagination. A hole is not a circle. The duct tape didn't make a circle. You made a circle. You can make a million of them all doing nothing; none doing something. Wheels roll, not any circle except for a line of a shape you made. The duct tape hole could form a triangle. Made by you. The math should be more difficult, but I bet you could make your ersatz points using any shapes you want as long as the line you start with ends at the same point with an encompassed space inside. Even try a peanut (shape).

--Brant

the triumph of philosophy over physics?--I don't think soooo

 

[Brant Gaede]

On 9/22/2017 at 6:12 AM, Jonathan said:

No, I got it.

He was lashing out by targeting what he hopes is a sore spot for me, which is my suckiness at math. The thing is, I freely admit that I suck at math, and that it takes a hell of a lot of effort for me to grasp math that others understand with ease. I understand and accept my deficiency. I don't have a psychological need to tell seasoned math professionals that their math which is way over my head is wrong. I don't identify with that type of insecurity, nor do I have any sympathy for it. 

 I like carrying on the old Atlantis traditions. If someone's looking to play rough, he'll be accommodated. I'll be his huckleberry.

J

--Brant

[merjet]

7 hours ago, Brant Gaede said:

Things no one has to know.

--Brant

and Drucker is full of it

 

7 hours ago, Brant Gaede said:

The circle can do what you think it would/could, for it really isn't doing anything except an imagination. A hole is not a circle. The duct tape didn't make a circle. You made a circle. You can make a million of them all doing nothing; none doing something. Wheels roll, not any circle except for a line of a shape you made. The duct tape hole could form a triangle. Made by you. The math should be more difficult, but I bet you could make your ersatz points using any shapes you want as long as the line you start with ends at the same point with an encompassed space inside. Even try a peanut (shape).

--Brant

the triumph of philosophy over physics?--I don't think soooo

 

You  are full of it.

[Ellen Stuttle]

On September 22, 2017 at 12:12 PM, merjet said:

"[It] is the recipient who communicates. The so-called communicator, the one who emits the communication, does not communicate. He utters. Unless there is someone who hears, there is no communication. There is only noise. The communicator speaks or writes or sings – but he does not communicate. Indeed, he cannot communicate. He can only make it possible, or impossible, for a recipient – or rather, "percipient" – to perceive." - Peter F. Drucker

I.e., a person who plugs his ears doesn't hear well.

Ellen

[Ellen Stuttle]

22 hours ago, merjet said:

Resolving the Paradox

Observing this video helped me to conclude that the paradox comes from confounding magnitudes of three different paths -- a straight one, a circular one, and a curved one. The curved one is a cycloid. Let Ps denote the point/dash at the  6:00 o'clock position on the circumference of the smaller circle with the disc at rest in the video. Let Pb denote the point/dash at the 6:00 o'clock position on the circumference of the bigger circle with the disc at rest. Roll the conjoined circles one revolution like in the video. How far either point moves is measurable in three ways. The straight path is the simplest one – as the crow flies -- but it is one that Pb or Pb does not travel. Its length is the same for Ps and Pb (all points really). 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 curved paths show that Pb travels farther than Ps, the distance for Pb being 8 times its circle's radius (more than the circumference). Also, clearly Pb travels farther away from its horizontal line than Ps does (midway for both).

What I call 'straight', 'circular', and 'curved' correspond to translation, rotation, and rolling in this excellent video that Baal posted earlier.

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.

       Point                      Straight                       Circular                                     Curved

         Ps                   No, Rb*2*pi                      No, Rs*2*pi                Yes, min= Rb*2*pi,  max= 8*Rb

         Pb                   No, Rb*2*pi                      No, Rb*2*pi                Yes, 8*Rb

       Center              Yes, Rb*2*pi                         n/a                                  n/a

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. The different curved lengths, resulting from two different shaped paths, which many people don't consider, are the ones actually traveled. Many people know that circumference C = 2*pi*r, but few people know about the path lengths or even think about it. Considering them resolves the paradox.

If a vehicle wheel rolls many revolutions, common sense says that an inner circle travels the same distance as the outer circle. It's also true – based on the straight path. The same holds for a given point or arc on the inner circle based on the straight path, but it doesn't hold for the curved path.

There are many common objects that fit the wheel's basic shape, for example, a roll of duct tape. Saying the smaller circle formed by the duct tape's hole slips/slides/skids relative to the tape's largest circumference is bizarre to me.

 

 

So you've figured out or looked up how to calculate the respective distances traveled by the respective 6:00 o'clock points (or any respective points) of the outer circumference and of any inner circle.  Regarding the supposed paradox, the math confirms what the physics demonstrations have shown - and what's eyeball visible to begin with in a simple diagram.  There isn't a real paradox, since it isn't the case that a point on an inner circle's circumference both is and isn't traveling a distance equal to that circle's circumference.  All points on all circles from the outer circumference inward are traveling farther than their circle's respective circumferences.

Regarding your last sentence, who has said that "the smaller circle formed by [for instance] the duct tape's hole slips/slides/skids relative to the tape's largest circumference"? In the posts I've seen which talk about slippage/sliding/skidding, the "slipping"/"sliding"/"skidding" is said to occur relative to an imagined "road" (a horizontal tangential line) on which the inner circle is rolling.

Ellen

PS: typo in "but it is one that Pb or Pb does not travel."

Pb or Ps

[merjet]

6 hours ago, Ellen Stuttle said:

There isn't a real paradox, since it isn't the case that a point on an inner circle's circumference both is and isn't traveling a distance equal to that circle's circumference.  

....

Regarding your last sentence, who has said that "the smaller circle formed by [for instance] the duct tape's hole slips/slides/skids relative to the tape's largest circumference"? In the posts I've seen which talk about slippage/sliding/skidding, the "slipping"/"sliding"/"skidding" is said to occur relative to an imagined "road" (a horizontal tangential line) on which the inner circle is rolling.

You say it as if a paradox requires there be a real contradiction. That mischaracterizes a paradox. “A paradox is a statement that, despite apparently sound reasoning from true premises, leads to an apparently self-contradictory or logically unacceptable conclusion” (Wikipedia, my bold).

Pages 3-9 of Mathematical Fallacies and Paradoxes is about the paradox we are addressing. Said pages can be seen on Google Books. The author's "wheel" consists of an axle, dime, and half-dollar, such that the axle passes through the centers of both the dime and half-dollar glued together. On page 9 he presents his solution to the paradox as follows. By the way, this is the source cited on Wikipedia to justify "slipping."

"[The] "point" tracing the line segment at the foot of the dime is both rotating and slipping all the time. It is slipping with respect to the tabletop. Since the dime does not touch the tabletop, you do not notice the slipping."

That describes the "slipping" relative to the "real road," not an "imaginary road" tangent to the dime. So pardon my pun, but it doesn't make a dime's worth of difference whether the "slipping" is relative to the horizontal tangent of the larger circle or the horizontal tangent of the smaller circle. Moreover, he says that it is not noticeable, as opposed to you and others here that it is clearly visible.

My contention from day 1 has been that "slipping" or "skidding" is a metaphor, and a very poor one as a substitute for translation or horizontal motion (ref.1, ref. 2).

Anyway, thanks for spotting the typo, now fixed.

[Jon Letendre]

Don't even need videos or diagrams, it is clearly visible as I turn the wheel, in my head.

[Jon Letendre]

Slippage  is an engineering term, not a metaphor.

[Jon Letendre]

Skid is not a metaphor, but an auto handling dynamics term.

https://en.m.wikipedia.org/wiki/Skid_(automobile)

[merjet]

1 hour ago, Jon Letendre said:

Slippage  is an engineering term, not a metaphor.

 

30 minutes ago, Jon Letendre said:

Skid is not a metaphor, but an auto handling dynamics term.

https://en.m.wikipedia.org/wiki/Skid_(automobile)

Duh! Many terms have both literal and metaphoric uses. The 'or' in 'literal or metaphoric" is not a mutually- exclusive 'or.'

[Jon Letendre]

The skidding the inside "wheels" perform over their imaginary roads is literal,

[Jon Letendre]

12 minutes ago, merjet said:

 

Duh! Many terms have both literal and metaphoric uses. The 'or' in 'literal or metaphoric" is not a mutually- exclusive 'or.'

You feel compelled to write, "Duh!" 

You cannot discuss without personal insult.

[merjet]

5 minutes ago, Jon Letendre said:

The skidding the inside "wheels" perform over their imaginary roads is literal,

No.

9 hours ago, merjet said:

My contention from day 1 has been that "slipping" or "skidding" is a metaphor, and a very poor one as a substitute for translation or horizontal motion (ref.1, ref. 2).

 

[merjet]

6 minutes ago, Jon Letendre said:

You feel compelled to write, "Duh!" 

You cannot discuss without personal insult.

LOL. You invited it with all the personal insults you have made and profanity. Since when is "duh" the sort of ad hominem you use?

[Jon Letendre]

4 minutes ago, merjet said:

No.

 

Actually, it's yes.

And in the setup of the paradox, Arisistotle suggests that it does not skid, but stays in constant rolling contact with the imaginary road. Falling for that suggestion is the essence of the paradox's trap.

[Jon Letendre]

4 minutes ago, merjet said:

You invited it with all the personal insults you have made and profanity.

You started it each time. Called me stupid out of nowhere, when I was staying in the subject. Just like today. I am talking respectfully about wheels and circles, and you feel compulsion to be personal.

[Jon Letendre]

12 minutes ago, merjet said:

LOL. You invited it with all the personal insults you have made and profanity. Since when is "duh" the sort of ad hominem you use?

It's not about sorts but you starting it every time.

[merjet]

16 minutes ago, Jon Letendre said:

You started it each time. Called me stupid out of nowhere, when I was staying in the subject. Just like today. I am talking respectfully about wheels and circles, and you feel compulsion to be personal.

The facts prove you wrong again. You called me "inestimably stupid" Sep 18 (link). I called you stupid one time on Sep 19 (link). Other times I said it, it was to Jonathan after he had called me that or worse.  

[merjet]

17 minutes ago, Jon Letendre said:

It's not about sorts but you starting it every time.

Wrong again. Jonathan has been the primary initiating culprit. You agreed with him and piled on.

[Jon Letendre]

21 minutes ago, merjet said:

The facts prove you wrong again. You called me "inestimably stupid" Sep 18 (link). I called you stupid one time on Sep 19 (link). Other times I said it, it was to Jonathan after he had called me that or worse.  

You are wrong.

You called me an "evader" on Sept 16.

[Jon Letendre]

23 minutes ago, merjet said:

Wrong again. Jonathan has been the primary initiating culprit. You agreed with him and piled on.

Every day is the same trying to discuss something with you.

I am polite and on topic.

Then you get personal, describe my points that you don't grasp as "Mumbo-jumbo" that "is waste of [your] time."

Exactly like today.

You cannot avoid being personal, you fall into it every time.

[Jon Letendre]

If we do as Aristotle asks and think of the inside "wheel" as real wheel in rolling contact with its imaginary road, then yes, said "wheel" literally skids across its road, while rotating.

[merjet]

2 hours ago, Jon Letendre said:

You called me an "evader" on Sept 16.

Yes. It was in response to this post by Jon, in which he was "so polite and on topic." Not! His idea of being polite is him saying to me "it's you incapable of getting out of your erroneous mental constructs" and "f___k off". He addressed me using the f-word two other times. His idea of "on topic" was invoking irrelevant gears. There are no gears in Aristotle's wheel or a roll of duct tape. He even said to me, "Meaning, no, you cannot see the things I asked about", which is misstated. He should have said you cannot see the things you asked about. I asked for an example or two, and he refused to provide any. That is why I called him "evader." 

 

[Jon Letendre]

24 minutes ago, merjet said:

Yes. It was in response to this post by Jon, in which he was "so polite and on topic." Not! His idea of being polite is him saying to me "it's you incapable of getting out of your erroneous mental constructs" and "f___k off". He addressed me using the f-word two other times. His idea of "on topic" was invoking irrelevant gears. There are no gears in Aristotle's wheel or a roll of duct tape. He even said to me, "Meaning, no, you cannot see the things I asked about", which is misstated. He should have said you cannot see the things you asked about. I asked for an example or two, and he refused to provide any. That is why I called him "evader." 

 

So you CAN stay on topic and leave out personal insults?

I'm delighted and hopeful.

[Jon Letendre]

Aristotle asks us to consider an inner circle as being a wheel in contact with and rolling on an imaginary road.

The "wheel" Aristotle asks us to consider skids over said road.