Aristotle's wheel paradox

[Ellen Stuttle]

On October 5, 2017 at 5:37 AM, merjet said:

[1] [first part]  You did [change focus from the curved to the linear path]. On Sep 30 you wrote “all points of all circles of the "paradox" setup .....travel farther than the outermost circle's/wheel's circumference.["] What does the latter phrase mean? If it means the straight horizontal path – you did change the focus mid-sentence, which you deny – then the statement is trivially true and gratuitous.

I suppose what you mean by "the latter phrase" is "the outermost circle's/wheel's circumference."  Using the notation you used in your chart, the circumference of the outermost circle/wheel is "Rb*2*pi."

Rb*2*pi is also the length of the straight horizontal path in the formulation of the supposed paradox.  The outer circle/wheel is revolved once around its circumference in the "paradox" formulation, but by "circumference" I meant "circumference," not horizontal distance.

 

On October 5, 2017 at 5:37 AM, merjet said:

[1] [continued] If it means the curved path of a point on the outermost circumference – you didn’t change the focus mid-sentence like you insist – then your claim was clearly false. Shorter paths do not exceed the longest path.

What are you calling a shorter path and what are you calling the longest path?  According to your chart, all points of all circles (or wheels) travel farther than Rb*2*pi, which is the minimum not the maximum.  A point on the outer circle's/wheel's circumference (according to your chart) travels 8*Rb - which is certainly longer than the outer circle's/wheel's circumference.

 

 

On October 5, 2017 at 5:37 AM, merjet said:

[2] You earlier confused the difference between describing a paradox and resolving a paradox. You did it again.

I think what you said earlier (I have to leave in a few minutes and don't have time to check) was that I'm confusing there being a paradox with resolving a paradox. [See PS.]

I'm not making either confusion but instead disagreeing with you that there is a genuine paradox to be resolved.

 

On October 5, 2017 at 5:37 AM, merjet said:

[4] So what? The cycloids are of significance to resolving the paradox. Moreover, I said so a mere two sentences after what you quoted.

Repeat, I don't think that there is a genuine paradox.  A paradox that doesn't exist can't be resolved.  Furthermore, although I find cycloids interesting and I think that your chart neatly compares the varying distances of respective measures, I don't think that cycloids are relevant to addressing the issue raised in the setup of the supposed paradox.

I expect we'll continue to disagree and I think that there isn't anything to be gained by retreading the same ground.  I hope you understand now the sentence producing trouble in [1].

Ellen

PS - added on edit:

I checked.  You did say as I remembered - here, on Sep. 27:

[MJ to ES]  "It seems to me you are confusing there being a paradox and resolving a paradox."

[Brant Gaede]

On 9/11/2017 at 6:23 PM, merjet said:

Can you resolve this paradox?  

It may help to imagine a point at the 6:00 o'clock position on each circle and then rolling the wheel one revolution. 

I will give my solution later.

This is Merlin's first post starting this thread.

Note the implied superior position (for he's superior?).

Note where he's ended up.

--Brant

OL is the wrong place to push shit

it's the only reason I'm here

 

[merjet]

On 10/4/2017 at 1:25 PM, Jonathan said:

Here's a new video:

Now, will Merlin still not get it? What moronic evasive tactic will he come up with next? What new method will he use to deny reality?

 

On 10/4/2017 at 3:30 PM, Jonathan said:

Hey, thanks! But, the question is, is it good enough of a demonstration to reach someone as mechanically inept and stubborn as Merlin!

Thanks for posting that video.  It was helpful. 

I have never not gotten the reality, moron. It’s translation plus rotation. Also, when a wheel rotates it is plain that the length of an arc further away from the axle is proportionately longer than the length of an arc closer to the axle for the same angular movement. These basic facts suffice to explain the reality. With a real wheel, and a real axle instead of your protrusion, there is no rock ledge for the axle to “ride on” or “slip on.” There is also no rock ledge to hide any of the wheel’s motion below the axle and obscure a real wheel’s look. Without the ledge, the area of the wheel immediately below the protrusion moves alongside the protrusion. With the ledge a stationary object is imposed next to the protrusion and hides the alongside movement. That makes a big visual difference, greatly exaggerating the appearance of “slipping.” Such is the nature of Jonathan’s animation design.

To Jonathan the “slipping” was “as obvious as hell” (his phrase). He was probably oblivious to the exaggeration that he created. According to him I was “blind” because I wasn’t duped by his scam.

It took me a while, but now I have identified Jonathan’s true character: scam artist.

What new method will the self-deluded scam artist try to use to deny the above reality? 

It seems I have progressed from “visual/spatial/mechanical inept” to only “mechanically inept.” Maybe I am not “incredibly lost” (Jon’s words) after all. 

[merjet]

3 hours ago, Brant Gaede said:

This is Merlin's first post starting this thread.

Note the implied superior position (for he's superior?).

Note where he's ended up.

--Brant

OL is the wrong place to push shit

it's the only reason I'm here

 

Oh, the irony that Brant fails to recognize. 

[Jon Letendre]

2 hours ago, merjet said:

 

Thanks for posting that video.  It was helpful. 

I have never not gotten the reality, moron. It’s translation plus rotation. Also, when a wheel rotates it is plain that the length of an arc further away from the axle is proportionately longer than the length of an arc closer to the axle for the same angular movement. These basic facts suffice to explain the reality. With a real wheel, and a real axle instead of your protrusion, there is no rock ledge for the axle to “ride on” or “slip on.” There is also no rock ledge to hide any of the wheel’s motion below the axle and obscure a real wheel’s look. Without the ledge, the area of the wheel immediately below the protrusion moves alongside the protrusion. With the ledge a stationary object is imposed next to the protrusion and hides the alongside movement. That makes a big visual difference, greatly exaggerating the appearance of “slipping.” Such is the nature of Jonathan’s animation design.

To Jonathan the “slipping” was “as obvious as hell” (his phrase). He was probably oblivious to the exaggeration that he created. According to him I was “blind” because I wasn’t duped by his scam.

It took me a while, but now I have identified Jonathan’s true character: scam artist.

What new method will the self-deluded scam artist try to use to deny the above reality? 

It seems I have progressed from “visual/spatial/mechanical inept” to only “mechanically inept.” Maybe I am not “incredibly lost” (Jon’s words) after all. 

Incredibly lost is putting it mildly. Moron. Liar.

If you were grasping accurately what a wheel actually does as it rolls, then you would have zero objections to Jonathan's 100% accurate rock wheel. I happen to find your objections indecipherable, but that's beside the point since there is nothing to object to and "greatly exaggerating the appearance of “slipping.”" Is sufficient to establish that you are still completely and utterly confused.

[Jonathan]

3 hours ago, merjet said:

With a real wheel, and a real axle instead of your protrusion, there is no rock ledge for the axle to “ride on” or “slip on.”

Why not? Of course there is a rock ledge if we put one there, or a simple ruler or tape measure if we choose to put either in the scene. That upper line or plane, and it's relevance, doesn't vanish from the "paradox" setup just because you personally don't like it or understand its importance.

Go back to your initial post and follow the link that you posted which explains the "paradox." Notice that it includes a line which is in constant contact with the smaller circle. Do you see it now? Well, see, that line is what I was representing with the upper rock ledge in my video! Neat, huh? Get it now? The entire point of the "paradox" involves the relationship of the small circle to the upper line. That's why the upper line is included in the drawings and descriptions of the "paradox."

Youve been insisting that the smaller circle must be represented in reality as a hole in the larger wheel. There's nothing about the "paradox" or about logic which would require it to be represented by a recess rather than a protrusion. You're just imposing your own preferences and stupidity onto the setup.

Quote

There is also no rock ledge to hide any of the wheel’s motion below the axle and obscure a real wheel’s look. Without the ledge, the area of the wheel immediately below the protrusion moves alongside the protrusion. With the ledge a stationary object is imposed next to the protrusion and hides the alongside movement. That makes a big visual difference, greatly exaggerating the appearance of “slipping.” Such is the nature of Jonathan’s animation design.

You're talking out of your ass. You're describing nothing but your own personal experiences of visual ineptitude. The rest of us are not limited to what you can or can't see and understand. There is nothing exaggerated about the video that I posted. It is precisely mathematically correct. Measure the entities and their motions for yourself, and apply the math that you found online. Oh, yeah, I forgot, you can't do that, can you? Heh. 

[Max]

3 hours ago, merjet said:

 

I have never not gotten the reality, moron. It’s translation plus rotation. Also, when a wheel rotates it is plain that the length of an arc further away from the axle is proportionately longer than the length of an arc closer to the axle for the same angular movement. These basic facts suffice to explain the reality. With a real wheel, and a real axle instead of your protrusion, there is no rock ledge for the axle to “ride on” or “slip on.” There is also no rock ledge to hide any of the wheel’s motion below the axle and obscure a real wheel’s look. Without the ledge, the area of the wheel immediately below the protrusion moves alongside the protrusion. With the ledge a stationary object is imposed next to the protrusion and hides the alongside movement. That makes a big visual difference, greatly exaggerating the appearance of “slipping.” Such is the nature of Jonathan’s animation design.

To Jonathan the “slipping” was “as obvious as hell” (his phrase). He was probably oblivious to the exaggeration that he created. According to him I was “blind” because I wasn’t duped by his scam.

 

I can't make head nor tail of this. You suggest that the slipping on Jonathan’s animation is exaggerated. Does that mean that you accept just a little bit of slipping, as long as it isn’t too much? In fact it is really easy to see how much the inner wheel/protrusion is slipping if we assume that the outer wheel/the rim is rolling without slipping: for one revolution it’s just the difference between the circumference of the outer wheel and the circumference of the inner wheel. It’s easy as that, no need for cycloids to solve the paradox, and it’s perfectly illustrated by Jonathan’s video.

 

[merjet]

It occurred to me that a fairly simple graphic of three cycloids can be used to deal with the paradox. Take a roll of duct tape (or painter's tape, etc.). Draw a radius on it. Call the two points where the radius meets the outer circle and inner circle Pb and Ps as before. Set them at the 6:00 position and observe them as you roll the tape one revolution (360 degrees) along a straight line. Each point traces a curve. Call them curve #1 for the bigger circle and #2 for the smaller circle. Curve #1 is higher than curve #2, except very near the start and very near the end. Yet the straight horizontal line distance between start and end are identical. It follows that curve #2 is a more efficient path to travel the same horizontal distance. It is also longer than the tape's inner circumference. It must be to cover the same horizontal distance as curve #1 does.

Such curves are similar to the two shown in Mathematical Fallacies and Paradoxes. (The relevant pages can be seen on Google Books.) Curve #1 is much like the curve traced by point A on the half-dollar (p. 6). Curve #2 is much like the curve traced by point B on the dime (p. 8). The book doesn't show a third curve, but I will. It will depict what would be traced by point B on the dime if the dime were unglued and rolled one rotation by itself. Such curve would be much like the half-dollar's curve, but smaller.

Next find or make a circular disc with the same diameter as the hole of the duct tape. (Or perhaps use a second roll of tape whose outer diameter equals the diameter of the duct tape's hole.) Mark a spot on its circumference. Call it Psa. Setting Psa at the 6:00 position, generate a cycloid with the edge of the disc rolling along a straight line. Align this curve so that its peak is vertically aligned with the peaks of #1 and #2, and its horizontal line atop #2's. Call this curve #3. Then curve #3 will fit within #2 (identical at their peaks), also showing very clearly that curve #2 is longer than curve #3. Curve #3's length is 8*pi*r, which is longer than its circumference. In more mathematical looking notation: Length(Curve #2) > Length(Curve #3) > Length(smaller circle's circumference).

Like the adage says, a picture is worth a 1000 words. The following drawing is a static representation analogous to what dynamically occurs in this video, plus curve #3. The drawing is crude, but the message should be clear enough. The star-shaped dots are Ps, Pb, and Psa (Point, small, alone). It only shows the center's start and end for the tape. The center's start and end for the disc are omitted to lessen clutter, but they would be straight above the ends of the lowest horizontal line. The author of Mathematical Fallacies and Paradoxes shows cycloids, but not all in one graphic as shown below.

The length of curve #2 minus the length of curve #3 could be regarded as how much the inner circle "slips" during one revolution. So could the length of #1 minus the length of #2. I gave another interpretation on Sep 21. As Baal pointed out here a point "slips" behind the axle/hub part of the time and "slips" ahead of the axle/hub part of the time. One might compare different arc lengths at different distances from the center swept by the same angle. That gives six different meanings of "slips" and shows what a slippery (ha!) word it is.

This graphic illuminates the paradox and its solution. The false part of the paradox is assuming that the length of curve #2 equals the length of curve #3. The fact, hence the solution, is that the length of curve #2 exceeds the length of curve #3. 

[merjet]

1 hour ago, Max said:

In fact it is really easy to see how much the inner wheel/protrusion is slipping if we assume that the outer wheel/the rim is rolling without slipping: for one revolution it’s just the difference between the circumference of the outer wheel and the circumference of the inner wheel. It’s easy as that,

 

39 minutes ago, merjet said:

That gives six different meanings of "slips" and shows what a slippery (ha!) word it is.

Make that seven.

[Jon Letendre]

8 minutes ago, merjet said:

 

Make that seven.

Everyone on the thread is using slip and skid in the same, one way.

Baal does not use it the way you imagine he does, not in the post you link to above, and not in any of his other posts.

This resolution does work, as soon as you prove that #2 length exceeds #3 length. I know it does, but you've asserted it without proof.

[merjet]

4 minutes ago, Jon Letendre said:

This resolution does work, as soon as you prove that #2 length exceeds #3 length. I know it does, but you've asserted it without proof.

How do you know it? What proved it to you?

I have asserted w/o proof??   

[Jon Letendre]

3 minutes ago, merjet said:

How do you know it? What proved it to you?

I have asserted w/o proof??   

Yes, asserted without proof.

[Jon Letendre]

1 hour ago, merjet said:

 The length of curve #2 minus the length of curve #3 could be regarded as how much the inner circle "slips" during one revolution. So could the length of #1 minus the length of #2.

Both statements are incorrect.

Neither of those is a measure of the extent of slip or skid that the inner "wheel" performs over its "road'," (which is the one and only meaning of slip/skid used by everyone participating in this thread.)

[Jon Letendre]

And you've also left out a lot that is needed to complete your resolution.

After proving the lengths exceed, you still have to relate what you've demonstrated about cycloids back to Aristotle's actual alleged equality, which is not about cycloids, but wheels, circles, circumferences and rolling distances.

[Jonathan]

Here's an accurate tracing of the paths of the relevant points in the video that Merlin posted way back at the beginning, and to which he refers constantly (and in which he imagined seeing a shelf which isn't there):

 

[Max]

It's interesting to note that many commenters on that video give the correct and simple solution of the paradox.

[Jon Letendre]

1 hour ago, Jonathan said:

Here's an accurate tracing of the paths of the relevant points in the video that Merlin posted way back at the beginning, and to which he refers constantly (and in which he imagined seeing a shelf which isn't there):

 

Yes, very good.

Merlin's #2 cycloid has straight portions at beginning and end.

No cycloid can have any straight portion at all.

[Jonathan]

35 minutes ago, Max said:

It's interesting to note that many commenters on that video give the correct and simple solution of the paradox.

Yeah, it appears that almost everyone but Merlin gets it.

[Jonathan]

28 minutes ago, Jon Letendre said:

Yes, very good.

Merlin's #2 cycloid has straight portions at beginning and end.

No cycloid can have any straight portion at all.

Exactly.

[merjet]

4 hours ago, Jon Letendre said:

Merlin's #2 cycloid has straight portions at beginning and end.

No cycloid can have any straight portion at all.

Dumb ass. I said my drawing was crude. I am not a pro graphic artist.

The scam artist and his stupid sidekick, both capable of nothing more than nit-picking and ad hominem, can entertain one another. 

Bye.

[Jon Letendre]

41 minutes ago, merjet said:

Dumb ass. I said my drawing was crude. I am not a pro graphic artist.

The scam artist and his stupid sidekick, both capable of nothing more than nit-picking, can entertain one another. 

Bye.

Nice try, shit-for-brains.

Picking up your toys and leaving again, like every other time reason fails you.

You post crap and when called on it, you run away.

Because you are a dishonest idiot loser.

[Jonathan]

57 minutes ago, merjet said:

I am not a pro graphic artist.

Not only are you not a "pro" graphic artist,  you're not even a kindergarten-level graphic artist.

Quote

The scam artist and his stupid sidekick...

Haha, Jon, you're just my sidekick! I'm the Batman of scam artists, and you're the Robin.

[Jonathan]

2 minutes ago, Jon Letendre said:

Nice try, shit-for-brains.

Picking up your toys and leaving again, like every other time reason fails you.

You post crap and when called on it, you run away.

Because you are a dishonest idiot loser.

What's weird is that Merin didn't used to seem to be retarded. He seemed like a fairly intelligent dude. Now he's a fricken stubborn idiot. I've been critical of a lot of Objectivish-types for what I call their "visual incompetence," which is pretty common among O-types, but I've never seen anything at this level. Merlin is the king retard of Objectivish visual incompetence.

[Jon Letendre]

2 minutes ago, Jonathan said:

What's weird is that Merin didn't used to seem to be retarded. He seemed like a fairly intelligent dude. Now he's a fricken stubborn idiot. I've been critical of a lot of Objectivish-types for what I call their "visual incompetence," which is pretty common among O-types, but I've never seen anything at this level. Merlin is the king retard of Objectivish visual incompetence.

And he's failing at objectivity quite badly. He feels free to reject, prior to even understanding first, whatever comes from "scam artists."

Willful blindness is not one of the virtues, as I recall them.

[Jon Letendre]

7 minutes ago, Jonathan said:

He seemed like a fairly intelligent dude.

If you say so.

I've seen him posting, SOLO, Rebirth, SoloP, OL, at least for ten years and never observed that.