Aristotle's wheel paradox

[Jonathan]

6 minutes ago, Jon Letendre said:

Tony and Merlin think they understand what happens when the wheel rolls, but they plainly do not.

It will take me about 15 minutes to make the setup with gears, but I can’t get to it ‘till next week.

Tony might get that, but Merlin won't. His objection will be that the initial setup didn't include gears, and that they are therefore irrelevant. He doesn't understand why you would be showing gears, and what gears reveal about the physics of the wheels and surfaces.

8 minutes ago, Jon Letendre said:

They reject the truth of the above description because they think they are visualizing the motions correctly, but they are not.

Correct. The issue is nothing but certain people not having the visuospatial/mechanical-reasoning cognitive capacity to grasp the physics of what's happening. Tony at least seems to be somewhat open to accepting the possibility he might not be fully understanding everything. Merlin is not open to considering that at all. (Remember that he also imagined seeing a ledge behind the wheel in the youtube video that he had posted, a ledge that wasn't there, and he also believed that he saw a regular cycloid being created by the larger circle on that wheel, when in reality it was a curtate cycloid.)

All that this thread is is a study in Merlin's retardation and stubbornness.

J

 

[Max]

4 hours ago, merjet said:

Real rims don't slip or skid or screech on imaginary roads.

A thought experiment is not made invalid by the fact that there is no real world equivalent of that experiment (yet). That is in fact irrelevant, as long as there in principle could be a real world equivalent. And it wouldn’t be difficult for an instrument maker to make a model to illustrate Aristotle‘s paradox. It could for example be a dual rail system, one higher rail for the small wheel and a lower rail for the large wheel (like an adaption of a train wheel). Those wheel-rail combinations could be made exchangeable, so that one can choose for a gear teeth combination to ensure rotation without slipping, and a smooth combination that enables slipping. Such a system would show that the wheels will be locked if both combinations have gear teeth: rotating of both wheels without slipping is impossible, contrary to the premise in definition of the paradox in the Wikipedia article).

 

I’ve demonstrated before that this slipping can be unequivocally described mathematically and that it follows automatically from the description of the system. It is a very real effect, even if you might not encounter such systems in daily life. After all we’re talking about a thought experiment, not about what’s happening in the streets.

[Jonathan]

3 minutes ago, Max said:

A thought experiment is not made invalid by the fact that there is no real world equivalent of that experiment (yet). That is in fact irrelevant, as long as there in principle could be a real world equivalent. And it wouldn’t be difficult for an instrument maker to make a model to illustrate Aristotle‘s paradox. It could for example be a dual rail system, one higher rail for the small wheel and a lower rail for the large wheel (like an adaption of a train wheel). Those wheel-rail combinations could be made exchangeable, so that one can choose for a gear teeth combination to ensure rotation without slipping, and a smooth combination that enables slipping. Such a system would show that the wheels will be locked if both combinations have gear teeth: rotating of both wheels without slipping is impossible, contrary to the premise in definition of the paradox in the Wikipedia article).

 

I’ve demonstrated before that this slipping can be unequivocally described mathematically and that it follows automatically from the description of the system. It is a very real effect, even if you might not encounter such systems in daily life. After all we’re talking about a thought experiment, not about what’s happening in the streets.

We've been through this already with Merlin. He doesn't understand any of the above. He's not capable of understanding it.

If you propose adding gear teeth to the wheels and surfaces, Merlin doesn't understand why, and thinks that you're just adding them to try to trick him with an optical illusion or something.

He's that stupid.

J

[Jonathan]

5 hours ago, merjet said:

Slipping is irrelevant. There is no relying on a vague, confused, incoherent, ad hoc, and half-baked metaphor. Ditto my first solution. 

By the way, I have not denied that the inner circle "slips" (by some big stretch of imagination). I have said the inner circle doesn't slip (literally). If the obnoxious ignoranus had grasped this difference early on, this thread would be much, much shorter than it is. Despite my making the distinction many times, he still doesn't get it. Ironically, he sarcastically asked, "Do you know what scare quotes are?"

Heh. The retard says that "slipping is irrelevant," and that it is a "metaphor," and then he says that he has not denied that the inner circle slips, but uses scare quotes on the word "slips," and then says that he has said that "the inner circle doesn't slip (literally)."

So, what's his retarded position? I can't decipher the above nonsense. He says that it does't slip, but he doesn't deny that it slips??? What in the fuck is a "metaphorical" slip versus a literal slip???

[Rodney]

Hold a flower by the stem and twirl it in your fingers. Notice how slowly you can move your fingers while the petals whirl around.

[Jonathan]

34 minutes ago, Rodney said:

Hold a flower by the stem and twirl it in your fingers. Notice how slowly you can move your fingers while the petals whirl around.

Merlin would reply that the wheel "paradox" isn't about flowers and their stems, so, therefore, your suggestion that people should try twirling a flower is irrelevant and a con job scam. And it's an optical illusion, and the slowness that you mention of the finger motion is metaphical but not literal.

J

[Rodney]

Looking over this thread I see that all the points I would want to make have already been made. The paradox is easily resolved in reason. The resolution obviously would be able to be mathematically expressed; but it would be a mathematical expression of slippage, and quite unnecessary to the curious mind. 

So I am out.

[anthony]

5 hours ago, Rodney said:

Hold a flower by the stem and twirl it in your fingers. Notice how slowly you can move your fingers while the petals whirl around.

[anthony]

What I called "rotational velocity" is properly "linear velocity". This decreases from maximum on the wheel rim for any point going toward to the center (for obvious reasons: greater and lesser circular distance travelled in the same time.) 

Angular velocity is the measure of a circle's 'angle per second' speed, measured in radians/sec, and that IS equal - at every point - for a wheel turning at constant speed (reducing as it slows or starts).

However, the varied "linear" velocity is what matters in this argument. That's why any inner circle rotates slower than the rim - and does so without slip and spin - rotating through an identical one to one revolution to the rim.

Rodney : Your flower makes a good example. btw, The wheel and axle is actually of the lever and fulcrum family (I roughly recall from a long while back) - seems logical.

 

[Rodney]

I hope that wasn't to argue against me, because I know all that you said of course. The slip is not between the wheels, but between one of the wheels and its tangential path. Which one depends on which wheel is doing the actual rolling.

[anthony]

Sigh. Can we get back on point?

Here below, the original terms of the 'paradox'. Dead simple. No slippage, nothing about an inner track for the inner wheel to run on, no load on it. Etc.

"The paths traced by the bottom of the wheels...etc." "But the two lines have the same length..."etc. There is all we need to brief us.

Best, in fact, just to consider revolving "circles" which equally imply the 'paradox'/contradiction, which was the full purpose for its designer, evidently. There would hardly need be an inner "wheel" rolling on anything, since that'd be an irrelevant diversion in this case of circle circumference-to-distance travelled.

What this is, is more a theoretical abstract, hardly at all a mechanical model. Forget "wheels" and focus on circles, my advice. ;)

 

Different sizes of circles *cannot* have the same circumference. Self contradiction. The paths of inner circles are redundant to the path the main circle travels. Circumferences of inner circles are redundant to the circumference of the outer circle. To equate lines traversed from the bottoms of the inner circles with their relevant circumferences is fallacious.

"Attributes cannot exist by themselves, they are merely the characteristics of entities; motions are motions of entities; relationships are relationships among entities". AR

Aristotle's wheel paradox

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Aristotle's wheel paradox is a paradox appearing in the Greek work Mechanica traditionally attributed to Aristotle.^[1] 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. The paths traced by the bottoms of the wheels are straight lines, which are apparently 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.

Contents

• 1History of the paradox
  • 1.1In antiquity
  • 1.2In the Scientific Revolution
  • 1.3In the 19th Century
• 2Analysis
• 3References
• 4Further reading

[anthony]

2 hours ago, Rodney said:

I hope that wasn't to argue against me, because I know all that you said of course. 

1

Nope, nothing against anyone. Trying to be pleasant, which is a waste of energy in this thread. Spinning flower - - Ferris Wheel? See the imagery?

[KorbenDallas]

Quote

Aristotle's wheel paradox is a paradox appearing in the Greek work Mechanica traditionally attributed to Aristotle.^[1] 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. The paths traced by the bottoms of the wheels are straight lines, which are apparently 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.

Criteria 1)  "The wheels roll without slipping for a full revolution."

Criteria 2)  "The paths traced by the bottoms of the wheels are straight lines"

Criteria 3)  "The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. "

Criteria 4)  "the two lines have the same length"

Criteria 5)  "the wheels must have the same circumference"

____

I'm going to comment on these, but I'll just say first that the entire paradox is set up fallaciously.  First the wheels do not have the same circumference as Criteria 5 says.  Criteria 4, however, is correct that the lines do have the same length.  But what the two lines represent are the distance between the midpoints of the circles from point A to point B, (Criteria 1 does say the wheels roll).  So these two lines do not represent circumferences like Criteria 3 says, "the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences."

Criteria 1, "The wheels roll without slipping for a full revolution."----What point are they referring to that do not slip?  It's ambiguous as written.  So:

• If Criteria 1 is saying the point is the midpoint of the circles that the wheels do not slip on, then that part is true.  They rotate on their axis without any slippage from point A to point B.
• If Criteria 1 is saying the point they do not slip on is on the lines drawn at the bottom of the circles and that the wheels are fixed along their tracks, then this is wrong in saying that they can move from point A to point B without slippage.  One wheel can have a fixed track at a time and not have slippage, but they cannot both have fixed tracks at the same time and not have the inner wheel slip, to explain:
  • If the outer wheel were fixed along its track the inner wheel has slippage along its track.
  • If the inner wheel were fixed along its track, the outer wheel would not make a full revolution.  Why?  Because the inner wheel's circumference is less than the outer wheel's circumference.  (Which again, Criteria 5 is fallacious saying that, "the wheels must have the same circumference.")

So to summarize:

Criteria 1)  "The wheels roll without slipping for a full revolution." -- FALLACIOUS

Criteria 2)  "The paths traced by the bottoms of the wheels are straight lines"  -- TRUTH

Criteria 3)  "The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. "  -- FALLACIOUS

Criteria 4)  "the two lines have the same length" -- TRUTH

Criteria 5)  "the wheels must have the same circumference" -- FALLACIOUS

 

The only thing paradoxical to me is how this thread keeps going!

[Max]

4 hours ago, KorbenDallas said:

Criteria 1)  "The wheels roll without slipping for a full revolution." -- FALLACIOUS

In previous posts I’ve already indicated at least six times that this is the source of the apparent contradiction. What you call criterion 5 is merely the logical consequence of criterion 1 in the given setup. We know that this cannot be true by definition, so we must conclude that criterion 1 is false.

4 hours ago, KorbenDallas said:

The only thing paradoxical to me is how this thread keeps going!

The reason is that some people still don't get it, although the solution of the paradox has already been explained in detail many times. Reading must be difficult for some people.

[Jonathan]

12 hours ago, anthony said:

Tony, the part that you're missing is that the scenario's premise includes the idea that the concentric wheels are rolling. They are not stationary as you show above. The are not mounted on a fixed axel which holds them in position. They experience rotation and translation. Understand?

[Jonathan]

8 hours ago, KorbenDallas said:

Criteria 1)  "The wheels roll without slipping for a full revolution."

Criteria 2)  "The paths traced by the bottoms of the wheels are straight lines"

Criteria 3)  "The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. "

Criteria 4)  "the two lines have the same length"

Criteria 5)  "the wheels must have the same circumference"

____

I'm going to comment on these, but I'll just say first that the entire paradox is set up fallaciously.  First the wheels do not have the same circumference as Criteria 5 says.  Criteria 4, however, is correct that the lines do have the same length.  But what the two lines represent are the distance between the midpoints of the circles from point A to point B, (Criteria 1 does say the wheels roll).  So these two lines do not represent circumferences like Criteria 3 says, "the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences."

Criteria 1, "The wheels roll without slipping for a full revolution."----What point are they referring to that do not slip?  It's ambiguous as written.  So:

• If Criteria 1 is saying the point is the midpoint of the circles that the wheels do not slip on, then that part is true.  They rotate on their axis without any slippage from point A to point B.
• If Criteria 1 is saying the point they do not slip on is on the lines drawn at the bottom of the circles and that the wheels are fixed along their tracks, then this is wrong in saying that they can move from point A to point B without slippage.  One wheel can have a fixed track at a time and not have slippage, but they cannot both have fixed tracks at the same time and not have the inner wheel slip, to explain:
  • If the outer wheel were fixed along its track the inner wheel has slippage along its track.
  • If the inner wheel were fixed along its track, the outer wheel would not make a full revolution.  Why?  Because the inner wheel's circumference is less than the outer wheel's circumference.  (Which again, Criteria 5 is fallacious saying that, "the wheels must have the same circumference.")

So to summarize:

Criteria 1)  "The wheels roll without slipping for a full revolution." -- FALLACIOUS

Criteria 2)  "The paths traced by the bottoms of the wheels are straight lines"  -- TRUTH

Criteria 3)  "The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. "  -- FALLACIOUS

Criteria 4)  "the two lines have the same length" -- TRUTH

Criteria 5)  "the wheels must have the same circumference" -- FALLACIOUS

 

The only thing paradoxical to me is how this thread keeps going!

You got it.

What keeps this thread going? Retardation keeps it going. It's fueled by Merlin's visuospatial/mechanical-reasoning incompetence and stubbornness.

J

[anthony]

The "wheels" - plural - roll without slipping ... is explained ambiguously. a. are the wheels autonomous and rolling separately on different tracks (not mentioned)? in which case, slipping isn' t feasible - or b. are the wheels fixed together, the outer one rolling, the inner one complying? We can't have it both ways.

 I've been assured that everyone thinks they are fixed, I agree and that's how I proceeded. If b., then the inner wheel matches exactly the rotation of the outer one, albeit at a slower linear speed. And so far there's no problem. Inner circles (and points) conform to the motion of the outer one.

The visual take-home by way of unnecessary lines in the diagram is where the confusion enters. 

I.e. The "path" traced by the smaller wheel extends impossibly past its circumference.

A so-called paradox. Is "fixing" the paradox (with gears) the objective? It can't be. This was not set up as a mechanical problem. No, looking at the diagram one can see the circles turn true to reality : 1. that a fixed circle within a larger circle will indeed act that way - without slip (like a wheel and tire) BUT, 2. the coincidence of line-lengths, measured (to confuse viewers) at BOTH the wheels' bottoms is extraneous  and superfluous- ONLY the outside wheel and its circumference can determine the distance, and this is the line that matters. The ~attached~ inner wheel must comply.

There can't be "a solution" to a contradiction, so one must give it up.

[jts]

I marvel at the length of the discussion about the wheel paradox so-called. Is it possible to be that stupid?

 

[anthony]

1 hour ago, Jonathan said:

Tony, the part that you're missing is that the scenario's premise includes the idea that the concentric wheels are rolling. They are not stationary as you show above. The are not mounted on a fixed axel which holds them in position. They experience rotation and translation. Understand?

Sure. Like the flower example this picture was only to affirm the varying linear speeds in a wheel.

[Peter]

Youse guys could invent a better 18 wheeler.

[william.scherk]

1 hour ago, jts said:

Is it possible to be that stupid?

Maybe everyone has a few soft spots in their reasoning ... nothing a round of water fasting can't fix.

[Jonathan]

42 minutes ago, william.scherk said:

Maybe everyone has a few soft spots in their reasoning ... nothing a round of water fasting can't fix.

 

[Jonathan]

2 hours ago, jts said:

I marvel at the length of the discussion about the wheel paradox so-called. Is it possible to be that stupid?

 

Hard to believe, isn't it? But, yes, Merlin is that stupid. So were previous eggheads, some of them famous, who were visuospatially deficient enough to accept the self-contradictory premise of the so-called "paradox."

[jts]

59 minutes ago, william.scherk said:

Maybe everyone has a few soft spots in their reasoning ... nothing a round of water fasting can't fix.

Stupidity can be genetic, can be a health problem, can be temporary from sleep deficiency.

Health is a combination of genetics and total life style. Life style includes diet, sleep, exercise, sunshine, avoiding poisons, etc. There is no one thing, not even fasting, that can by itself totally reverse the effects of a bad life style. Fasting is usually to assist the transition from the old life style to the new life style. If you go back to the old life style, you lose the benefit.

But sometimes fasting does seem to at least temporarily improve mental function. Perhaps this is exceptional.

Shelton writes:  In detailing his experiences during his forty days' fast, taken some years since, Dr. Tanner said: "My mental powers were greatly augmented, to the very great surprise of my medical attendants, who were constantly on watch for mental collapse, which was freely predicted, if I persisted in the experiment until the tenth day.

Fry went to a chess club once a week and he fasted that day because he could play chess far better fasting.

Many years ago I took a correspondence course. I got 100% on only one assignment in that course and it was by far the most difficult assignment in the course. I did that entire assignment during a 15 day fast (water only).

I understand that these examples are only anecdotal and don't conclusively prove much. Even if fasting does sometimes help mental function, it might sometimes have the opposite effect.

 

[william.scherk]

5 minutes ago, jts said:

1 hour ago, william.scherk said:

Maybe everyone has a few soft spots in their reasoning ... nothing a round of water fasting can't fix.

Stupidity can be genetic

DNA variants that are bad for health may also make you stupid