Aristotle's wheel paradox

[Darrell Hougen]

4 hours ago, Jonathan said:

I agree with the above.

No, it's not at all about being impolite because of someone's not understanding something. It's completely about their condescension, their evading questions, immediately rejecting substantive content without even taking the time to absorb it. And their name-calling.

Initiating bad. Retaliating good.

J

Point taken.

[Darrell Hougen]

13 minutes ago, anthony said:

Okay, obviously not what I intended, but true - I take your point. I guess you take mine.

How bout simplicity: You swing a rope around your head to which are attached a weight half-way along and a weight at the end. Which moves faster?

The one at the end, obviously. Now go back and look at the post where I put numbers to your theory.

Darrell

[anthony]

59 minutes ago, Darrell Hougen said:

Hi Tony,

Okay, let's put some numbers to our theory.

Let's say we have a bottle with a neck that is 1 inch in diameter and a body that is 3 inches in diameter. Then, if the bottle rolls completely around 10 times, how far does the body travel? How far does the neck travel?

distance traveled = number of revolutions * pi * diameter = N * pi * D

So, for the body of the bottle,

distance traveled = 10 * pi * 3 =~ 94.2 inches

For the neck of the bottle,

distance traveled = 10 * pi * 1 =~ 31.4 inches

Darrell

Distance traveled = 94.2in for the bottle AND, 94.2 for the neck. The (inner wheel) neck's distance entirely depends on the (outer wheel) bottle's travel distance, not its own circumference .

Applied back to a tire and wheel rim. 10 revolutions by the tire = 10 by the rim. Completed distance - equal for both. Circumference of tire - probably twice that of the circumference of rim. The rims travel well exceeds its circumference-length. I rev, or 10.

But a car's wheelrim and tire stay together, slip-lessly. That, one knows. How it happens without slip/skid, is another question.

What I know is one can't ignore tangential speed. 

[Darrell Hougen]

 

20 hours ago, anthony said:

Briefly, I maintain that placing the bottle/combined cylinders onto two tracks (one beneath the neck, the other beneath the bottle, and compensating for their radii difference) is not going to, in essence, change anything about their combined motion. The "neck" or smaller wheel, turns slower (as we know)  in the same time as the main bottle turns slightly faster, therefore without slippage, and the two finish at the same endpoint after a rotation by each.

 

13 minutes ago, anthony said:

Distance traveled = 94.2in for the bottle AND, 94.2 for the neck. The (inner wheel) neck's distance entirely depends on the (outer wheel) bottle's travel distance, not its own circumference .

Hi Tony,

But, if the neck of the bottle rolls on a separate track (or book, for example) then according to my calculation above, it only travels 31.4 inches.

distance traveled = N * pi * d = 10 * pi * 1 =~ 31.4 inches.

If there is something wrong with my calculation, what is the problem?

The tangential velocity doesn't really matter in this case because the bottle could be rolled fast or slowly. The only important thing is the total distance traveled.

Darrell

[Jonathan]

1 hour ago, anthony said:

Before he buries the evidence, this was his post.

Heh. Why would I bury evidence that supports my position????

 

1 hour ago, anthony said:

I - "don't see tracks or ledges out on the road .."etc.

But yay! lookee here. Wiki proves there are curbs! "...the conditions that he demands exist..."

Yes. Exactly. You said that the conditions don't exist out there in reality. I gave an example of where they do, an example provided by someone in a Wikipdedia quote. And now you're trying to mock it?

Sorry, but your mental functioning continues to astound me every time that I encounter it. I should know by now to expect it.

 

1 hour ago, anthony said:

(Obviously the 'slipping" there can't be anything but between tire and 'inner hubcap".

No. The opposite true. It's what everyone has been trying to get you to understand. But, your mind can't handle it, so it keeps going back to the wrong idea of slippage between the tire and the hub.

 

1 hour ago, anthony said:

Slipping on the curb makes no sense, in the context of the wheel paradox).

False, it makes perfect sense. You are not capable of grasping the setup, nor what the alleged paradox is.

J

[anthony]

5 hours ago, william.scherk said:

Maybe if I called Tony a moronic scumbag I could get some traction ...

 

 

Now, William, you haven't joined the J pack have you? Not like you. I don't know, centrifugal force pushes the flanges against the rails, I guess?

[anthony]

2 hours ago, Darrell Hougen said:

 

 

Hi Tony,

But, if the neck of the bottle rolls on a separate track (or book, for example) then according to my calculation above, it only travels 31.4 inches.

distance traveled = N * pi * d = 10 * pi * 1 =~ 31.4 inches.

If there is something wrong with my calculation, what is the problem?

The tangential velocity doesn't really matter in this case because the bottle could be rolled fast or slowly. The only important thing is the total distance traveled.

Darrell

As before, and on off-set tracks which are almost frictionless like blades (to equalize drag), the distance moved is the same for both - 94.2in. That means the different tangential velocities, slower for the neck and slightly faster for the big wheel (bottle) ~do~ matter. The small wheel laterally moves the large wheel's circumference/distance in an equal 10 revolutions ~because~ it has less rotation speed. As before, the circumference of the small wheel plays no part in travel distance, that's determined only by the outer wheel (like a rolling cartwheel - an inner hub has no relation to distance). 

The bottle "could be rolled fast or slowly", yes, that's not relevant. What matters is the relative revolving speeds of outer and inner wheels, bottle and neck.

[Darrell Hougen]

3 minutes ago, anthony said:

As before, and on off-set tracks which are almost frictionless like blades (to equalize drag), the distance moved is the same for both - 94.2in. That means the different tangential velocities, slower for the neck and slightly faster for the big wheel (bottle) ~do~ matter. The small wheel laterally moves the large wheel's circumference/distance in an equal 10 revolutions ~because~ it has less rotation speed. As before, the circumference of the small wheel plays no part in travel distance, that's determined by the big one (like a rolling cartwheel in relation to its inner hub). 

The bottle "could be rolled fast or slowly", yes, that's not relevant. What matters is the relative revolving speeds of outer and inner wheels, bottle and neck.

Hi Tony,

Okay, so the offset track is almost frictionless so that the small wheel veritably slides across the surface, barely disturbing the path of the body of the bottle. Is that what you're saying?

If that's what you're saying, then you are correct that the neck of the bottle will move with the body and that the distance moved will be 94.2 inches for both.

Darrell

 

[Darrell Hougen]

On 11/27/2018 at 3:22 AM, merjet said:

Also, the journal article ‘Aristotle's Wheel: Notes on the History of a Paradox’ by Israel E. Drabkin, which is referenced 6 times (!) in the Wikipedia Article includes the following:  “For though the smaller circle traverses a distance equal to that traversed by the larger, it does not keep pace with the larger by sliding over the tangent, if by ‘sliding’ we mean that a point on the circumference is at any time in contact with a finite segment of the tangent” [my bold].

Hi Merlin,

This is a technical objection that has very little to do with the current discussion. The small wheel rolls and slides simultaneously so that the tangent point changes continuously and no point on the circumference of the wheel ever contacts more than a single point on the track.

We haven't even gotten to the finer points of the discussion. Based on the description on Mathworld, Aristotle's paradox may have to do with both the mapping between points on circles of differing sizes and on the differing lengths of the circumference. For example, any radial line intersects both the big circle and small circle at exactly one point. That shows that there is a one-to-one mapping between points on the small circle and points on the big circle. How is it then that the two circles have different circumferences?

Darrell

[merjet]

8 hours ago, Darrell Hougen said:

Hi Merlin,

This is a technical objection that has very little to do with the current discussion.

Huh?

[Jonathan]

9 hours ago, Darrell Hougen said:

Hi Merlin,

This is a technical objection that has very little to do with the current discussion. The small wheel rolls and slides simultaneously so that the tangent point changes continuously and no point on the circumference of the wheel ever contacts more than a single point on the track.

We haven't even gotten to the finer points of the discussion. Based on the description on Mathworld, Aristotle's paradox may have to do with both the mapping between points on circles of differing sizes and on the differing lengths of the circumference. For example, any radial line intersects both the big circle and small circle at exactly one point. That shows that there is a one-to-one mapping between points on the small circle and points on the big circle. How is it then that the two circles have different circumferences?

Darrell

Exactly.

Anything and everything is a "paradox" if one eggheads it enough.

Merlin likes cycloids, and thinks that they take him somewhere, but the same point-mapping issue, using the same eggheaded standard, can be applied to cycloids: Despite the measurably different lengths of the proper cycloid created by the large wheel and the curtate cycloid created by the small wheel, there is a one-to-one mapping of points on each: Any point along either cycloid at any moment corresponds to a point, and only one point, on the other! Then how is it that the cycloids can have different kegths?, OMG, it's a paradox! Cycloids don't resolve the paradox, but make it worse by revealing another one!

Eggheadery.

J

[Brant Gaede]

16 hours ago, anthony said:

Now, William, you haven't joined the J pack have you? Not like you. I don't know, centrifugal force pushes the flanges against the rails, I guess?

The wheels want to go straight but the rail won't let them so one edge of the wheel hits the rail as opposed to merely the center of the flange which is shaped to accommodate turns.

--Brant

[Pelagius160]

On 11/20/2018 at 11:05 PM, Peter said:

My parents bought me and my older brother Schwinn’s one Christmas and mine was too big for me.

Eye could knot agree Moore.

When eye was stay shunned at the navel academy, wee wood ride are bike’s two wear whee worked on a paradox bye the see, one witch was hire, and the other witch was lore. Wee had two role bear alls on them too lode are ship. Won thyme, a bear all got aweigh from me and eye saw it role on both doc’s at wants, and they’re was know slip age, at leased knot that eye could sea. Sew, their for, if sum of the four most authorities inn the whirled, such as Toe Knee oar Myrrh Linn, say their is know slip age on a paradox, eye wood knot halve grounds on witch two Kant test it.

R.P.

[Jonathan]

Ah, reminiscing!

Remember back when this thread started? Those were the good ol' days.

Here's the position that Merlin took back then, prior to messing up the Wikipedia page:

On 9/13/2017 at 10:07 AM, merjet said:

The paradox is described on the Wikipedia page in my first post. Better yet, watch this video as the wheel makes one full revolution. The marked point (at the 6:00 o'clock position at both start and end) on each circle appears to travel its circle's circumference, but the two circumferences are not the same length. On the other hand, the straight horizontal lines the two circles travel along are the same length. Thus the contradiction or paradox.  

 

He refers to the two horizontal lines on which the circles travel. Now he wants there to be only one line. Heh.

His position back then was that of the ancient eggheads -- that each circle appears to travel the distance of its own circumference.

And that's where the "paradox" exists: Only in the minds of those to whom it appears that each circle travels its own circumference.

That illusion does not happen in my mind, nor in the minds of others here. Both circles do not appear to travel their own circumferences. Judging by the comments at various sites online which discuss the "paradox," hardly anyone agrees with the ancient eggheads and Merlin that each circle appears to travel the distance of its circumference.

J

[Brant Gaede]

2 hours ago, Pelagius160 said:

Eye could knot agree Moore.

When eye was stay shunned at the navel academy, wee wood ride are bike’s two wear whee worked on a paradox bye the see, one witch was hire, and the other witch was lore. Wee had two role bear alls on them too lode are ship. Won thyme, a bear all got aweigh from me and eye saw it role on both doc’s at wants, and they’re was know slip age, at leased knot that eye could sea. Sew, their for, if sum of the four most authorities inn the whirled, such as Toe Knee oar Myrrh Linn, say their is know slip age on a paradox, eye wood knot halve grounds on witch two Kant test it.

R.P.

Aw. Yes. Good ole Ellen Moore.

She thought she was the bomb.

Bombed every time.

RIP

--Brant

[anthony]

He who can't see in the video that the smaller circle is turning slower is vision-impaired or illogical.

BOTH circles traverse the same circumference-length - yes? (the larger one's)

Both circles begin and finish their rotation at the same time and same point - yes?

Therefore, what is the only differentiation?

The turning speed of the small vs the big circle.

I realize great investment has been made by some in 'tracks and slippage'. Preconceptions can lead to fixation or rationalizing.

[anthony]

On 12/6/2018 at 6:45 PM, Brant Gaede said:

The locus of the focus is the paradox. Don't go off in all directions.

--Brant

I think they all connect. Covered by the identity/causation of a wheel/circle.

Another direction I'd like to go in: the wheel and the wheel centre have their fundaments in the lever and fulcrum, which everyone - implicitly - knows, not often consciously. I envisage the perimeter of the hand wheel as an infinity of levers. So, a small wheel (like a faucet) could be too hard to untighten, but replaced with a large wheel, turned easily.

Means a wheel is not only about velocity, it is also force. The manifestations of the wheel are nearly endless. Same as the rotational (tangential) velocity, we also gathered the knowledge inductively from childhood. 

It is funny when I read snarky comments about "wheel identity", the law of identity- et al.  They don't know they presume on "identity" and identification in order to belittle them.

Something for this 'paradox' debate, a word from our sponsor - Aristotle's law of non-contradiction: "Nothing can be A and non-A at the same time and in the same respect".

[Darrell Hougen]

1 hour ago, anthony said:

He who can't see in the video that the smaller circle is turning slower is vision-impaired or illogical.

BOTH circles traverse the same circumference-length - yes? (the larger one's)

Both circles begin and finish their rotation at the same time and same point - yes?

Therefore, what is the only differentiation?

The turning speed of the small vs the big circle.

I realize great investment has been made by some in 'tracks and slippage'. Preconceptions can lead to fixation or rationalizing.

Tony,

What is the "turning speed"?

Also, you didn't respond to my last comment. It's hard to maintain the continuity of a conversation if you don't respond.

Darrell

[Max]

5 minutes ago, Darrell Hougen said:

Also, you didn't respond to my last comment. It's hard to maintain the continuity of a conversation if you don't respond.

Ah, how many words haven't I already written here, with detailed explanations. He'll never answer the concrete details, never say "there you make an error, because...". If he replies at all, it is with vague generalities, like:

"A is A"

"preconceptions can lead to fixation or rationalizing"

"You guys have introduced mechanics to an abstract exercise which does not need or ask for concretist explanations"

" "Tracks" , slippage and stuff, however specially "defined" (as some have insisted) are in self-contradiction to the identity of a wheel"

"this wasn't meant to be resolved by a simple mechanical or a complex mathematical solution. And those applications come after identification"

"But you all need to have a tangible, physical "track" to fulfil the "slippage solution" - so, track it must be...You are "destroying" reality, not solving the paradox"

"haha. A problem with the young guys is to not identify before they calculate. Floating math abstractions."

"You take reality from animations. Experiments online. Any and all can have a bias to what the maker wishes - "Explain why there is, apparently, 'slippage' in this depiction of the inner wheel's motion. Or unequal contact"

Of course I gave then again a detailed explanation, but the only reply I got was:

"Right. I see it. Good effort. One helluva investment for so little return. (I do not think Aristotle was looking for solutions to the phenomenon, it puzzled him, that's all).

 

You could as well explain Aristotle's paradox to a sea cucumber.

 

[Brant Gaede]

2 hours ago, anthony said:

I think they all connect. Covered by the identity/causation of a wheel/circle.

Another direction I'd like to go in: the wheel and the wheel centre have their fundaments in the lever and fulcrum, which everyone - implicitly - knows, not often consciously. I envisage the perimeter of the hand wheel as an infinity of levers. So, a small wheel (like a faucet) could be too hard to untighten, but replaced with a large wheel, turned easily.

Means a wheel is not only about velocity, it is also force. The manifestations of the wheel are nearly endless. Same as the rotational (tangential) velocity, we also gathered the knowledge inductively from childhood. 

It is funny when I read snarky comments about "wheel identity", the law of identity- et al.  They don't know they presume on "identity" and identification in order to belittle them.

Something for this 'paradox' debate, a word from our sponsor - Aristotle's law of non-contradiction: "Nothing can be A and non-A at the same time and in the same respect".

I think you're full of helium.

--Brant 

not, however, hydrogen

[anthony]

On 12/7/2018 at 1:01 AM, Darrell Hougen said:

Hi Tony,

Okay, so the offset track is almost frictionless so that the small wheel veritably slides across the surface, barely disturbing the path of the body of the bottle. Is that what you're saying?

If that's what you're saying, then you are correct that the neck of the bottle will move with the body and that the distance moved will be 94.2 inches for both.

Darrell

 

Darrel, This discussion revolves round matching theory and practice. I think one has to continuously move between them, comparing one with the other.

Having said that, I take reality/practice as the only guide, and if theory doesn't concur with that, I dump the theory.

 I began with a bottle simply rolling on a floor (of course rolling presupposes a degree of friction). From this alone you can see that the bottle is a total entity which covers a distance y after x rolls - and its neck rolls correspondingly with it. It would be ridiculous to insist on measuring the neck's circumference, 3.14in, and deduce - 10 rolls of the neck = 31.4 in ... therefore the bottle rolled 31.4 in!

Obviously not. The bottle's circumference is the determinant of distance covered -- and the neck's circum is superfluous.

So far this practical test is keeping with the theory of the paradox diagram and validating that. ie.The inner wheel has nothing to do with distance.

Trying to now reproduce this action on a second track. I said earlier that the conditions of the bottle on the floor, with its neck moving and turning unimpeded without contact, must be precisely adhered to, in order to conduct an experiment.

A proper experiment has to faithfully re-create reality, so to speak, not add more variables. That brings up an error I made about "frictionless" supports. This is wrong. We need friction and weight for rolling to take place. But - this must be equalised for the whole bottle.

I argue that the bottle can be balanced to roll on a 'track' as it did on the floor. Straight. With rolling, i.e., no slip/slide of the neck (inner wheel) -- and because of their different tangential speeds, the neck turns slower. By your example, the bottle will cover 94.2in.here, too

If a bottle will roll on its own, what difference do 'tracks' make? Why should the neck change the behavior of the bottle? It is the 'inner wheel', so that depends completely on the outer.

(But I can see why 'track-slippage is so important to everyone. All the arguments here rely on it exclusively - no track, no slippage). 

Similar to a penny-farthing bicycle, while not on the same axle and unequal revs, a larger (drive) wheel is the only wheel that matters for distance. The little one runs along with it (and doesn't 'slip')

[anthony]

1 hour ago, Darrell Hougen said:

Tony,

What is the "turning speed"?

Also, you didn't respond to my last comment. It's hard to maintain the continuity of a conversation if you don't respond.

Darrell

It is the rotational ("tangential") speeds I can visibly perceive made by the circles in that video. 

Who doesn't see a speed difference?

[anthony]

22 minutes ago, Brant Gaede said:

I think you're full of helium.

--Brant 

not, however, hydrogen

Ha, better than lead.

[anthony]

And will someone please give a real world example of an inner wheel slipping on a track? Not the silly one of jamming a tire against a curb. Time we saw this theory in practice, not experiments and animations. 

[william.scherk]

31 minutes ago, anthony said:

And will someone please give a real world example of an inner wheel slipping on a track?

Flange squeal on tight curves.