Aristotle's wheel paradox

[Jonathan]

10 minutes ago, Brant Gaede said:

Will someone go get Roger Bissell?

--Brant

God, I hope he sides with Merlin!

 

[Brant Gaede]

12 minutes ago, Jonathan said:

Still waiting for the big math which demonstrates how wrong we all are, and why.

Merlin has said that our solutions are "wrong," that they "don't resolve the paradox," and that they are "hogwash." And yet we still have nothing from him. He has argued some semantics, and appears to have started to actually grasp some of what we've said and illustrated, but still no big reveal from the master!

It reminds me a lot of The Schoolmarm, Phil Coates, and the big build-ups that he'd hype, but then never deliver. He had the true solutions to everything, and everyone else was wrong, but he needed to be paid before delivering his brilliance, and the payment that he demanded was reverence. He wanted to reverse the normal way of doing things. Usually, a person earns the type of respect that Phil demanded by producing something of value. He wanted to be revered first, and then deliver the value later. He'd never deliver due to hiis little feelings getting hurt due to being laughed at for being such a buffoon. On this thread, Merlin has adopted a lot if his characteristics.

J

Yes, "on this thread." But Phil was all over the place. I was very sad for Phil but have never missed him. At the end he went completely off the rails. I do sincerely hope he's okay.

Merlin's fine elsewhere. My objection to his POV was the easiest presented here and he never touched it: a circle is not a wheel nor an attribute of a wheel. You can not do un-obscured wheels and ever get a "paradox." Not with real wheels.

--Brant

[merjet]

1 hour ago, Ellen Stuttle said:

Acknowledged that the inner circle doesn't literally slip relative to the outer circle, since it isn't rotating on an actual surface.  Nonetheless, the 6:00 point of the inner circle moves farther than the length of the circumference of the inner circle.  So I don't see that there is a paradox.  (As I understand the supposed paradox, it's the supposition that the red line both is and isn't equal to the circumference of the smaller circle.)

Ellen

That is the paradox. If the inner circle rolled one revolution on its own, the distance traveled would equal its own circumference. However, rolling one revolution fixed within the larger circle, it rolls a distance of the larger circle’s circumference. It’s not a contraction. The latter happens with ordinary vehicle wheels all the time. To explain why it happens is to resolve/dissolve the paradox. Trying to explain it by saying the inner circle “slips” uses a metaphor, which is confused and misleading. This is something the buffoon J doesn’t get. He goes ballistic when his metaphoric explanation is questioned in any manner. His sensitive little feelings are hurt.

The paradox is about measurements of distances. Doing/calculating said measurements is complicated. I will give my explanation in time. The impatient, impudent J will just have to wait.

 

[BaalChatzaf]

1 hour ago, merjet said:

That is the paradox. If the inner circle rolled one revolution on its own, the distance traveled would equal its own circumference. However, rolling one revolution fixed within the larger circle, it rolls a distance of the larger circle’s circumference. It’s not a contraction. The latter happens with ordinary vehicle wheels all the time. To explain why it happens is to resolve/dissolve the paradox. Trying to explain it by saying the inner circle “slips” uses a metaphor, which is confused and misleading. This is something the buffoon J doesn’t get. He goes ballistic when his metaphoric explanation is questioned in any manner. His sensitive little feelings are hurt.

The paradox is about measurements of distances. Doing/calculating said measurements is complicated. I will give my explanation in time. The impatient, impudent J will just have to wait.

 

The different magnitudes of lateral motion are also the principle  derailer gears on bicycles and planetary gears. 

[Jon Letendre]

11 minutes ago, BaalChatzaf said:

The different magnitudes of lateral motion are also the principle  derailer gears on bicycles and planetary gears. 

Those are also a deep, deep paradox.

[Jon Letendre]

1 hour ago, merjet said:

That is the paradox. If the inner circle rolled one revolution on its own, the distance traveled would equal its own circumference. However, rolling one revolution fixed within the larger circle, it rolls a distance of the larger circle’s circumference.

 

No, it does not.

The paradox invites you to imagine this, but it is not so.

[merjet]

3 minutes ago, Jon Letendre said:

No, it does not.

The paradox invites you to imagine this, but it is not so.

Look at the center, stupid.

[Jon Letendre]

A new day, and who started it today, you dimwhitted twat?

Are you capable on staying in the subject matter?

Must you always make it personal?

[Jon Letendre]

6 minutes ago, merjet said:

Look at the center, stupid.

The center is a point and cannot roll.

You have to look at the pink circumference and it does not roll over its pink line, but skids, slides over it while rotating at a rate lower than rolling.

[merjet]

The wheel in this video has 2 circles on it. There is a dot marked on each. At the start they are at 6:00 o’clock behind the two white dots on the left end of 2 horizontal wires. He rolls the wheel one revolution clockwise. The dots on the circles are then behind the two white dots on the right end of the 2 horizontal wires. During the revolution, the two black dots follow two different curves. Let’s call the dot on the bigger circle Pb (point bigger) and the dot on the smaller circle Ps (point smaller).

Please pardon my crude artwork, but the paths they took look something like the following. Pb travels the curve which is higher during most of the journey, and Ps travels the other curve. Relative to the horizontal wires, they travel an equal distance.

 

 

Next imagine this is some kind of handicapped race. Pb runs a longer distance, yet Pb and Ps run equal distances horizontally and in time. Obviously Pb runs faster. Neither runs at a uniform pace relative to the horizontal lines. Yet, pardoning my crude artwork, they remain the same distance apart throughout. They are constrained to that. Measuring how exactly far each ran along their respective curves is not easy, but it is doable with some complicated math. The path of the center of the circles is always straight.

While the arc length will vary, Pb’s curvature will be the same in all cases. Ps’s curve will change shape depending on the smaller circle’s radius relative to the larger circle’s radius. The further from the center it is, the more it will look like Pb’s curve. The closer to the center it is, the flatter it will be.

More later.

[Jonathan]

2 hours ago, merjet said:

The paradox is about measurements of distances. Doing/calculating said measurements is complicated. I will give my explanation in time. The impatient, impudent J will just have to wait.

Seriously? You STILL don't have anything? And you think that calculating and measuring is so complicated that you need to work on it some more? After all this time that we've been discussing it, you need even more time? Bah-hahahahahaha!

I'd think that a math genius of your caliber would be able to resolve the "paradox" in a matter of seconds! But, yet, in the time that I've built and posted several precise animations and still images, you still haven't had enough time to deal with this very simple issue!

Hahahaha!

[Jon Letendre]

10 minutes ago, merjet said:

The wheel in this video has 2 circles on it. There is a dot marked on each. At the start they are at 6:00 o’clock behind the two white dots on the left end of 2 horizontal wires. He rolls the wheel one revolution clockwise. The dots on the circles are then behind the two white dots on the right end of the 2 horizontal wires. During the revolution, the two black dots follow two different curves. Let’s call the dot on the bigger circle Pb (point bigger) and the dot on the smaller circle Ps (point smaller).

Please pardon my crude artwork, but the paths they took look something like the following. Pb travels the curve which is higher during most of the journey, and Ps travels the other curve. Relative to the horizontal wires, they travel an equal distance.

 

 

Next imagine this is some kind of handicapped race. Pb runs a longer distance, yet Pb and Ps run equal distances horizontally and in time. Obviously Pb runs faster. Neither runs at a uniform pace relative to the horizontal lines. Yet, pardoning my crude artwork, they remain the same distance apart throughout. They are constrained to that. Measuring how exactly far each ran along their respective curves is not easy, but it is doable with some complicated math. The path of the center of the circles is always straight.

While the arc length will vary, Pb’s curvature will be the same in all cases. Ps’s curve will change shape depending on the smaller circle’s radius relative to the larger circle’s radius. The further from the center it is, the more it will look like Pb’s curve. The closer to the center it is, the flatter it will be.

More later.

And is there a way to make Ps a flat line?

Yes. The center of the wheel goes down the road the same distance as every other wheel, even though it is a point for whom rolling is impossible, it does not roll, yet goes down the road. Amazing. A paradox.

[merjet]

28 minutes ago, Jonathan said:

Seriously? You STILL don't have anything? And you think that calculating and measuring is so complicated that you need to work on it some more? After all this time that we've been discussing it, you need even more time? Bah-hahahahahaha!

I'd think that a math genius of your caliber would be able to resolve the "paradox" in a matter of seconds! But, yet, in the time that I've built and posted several precise animations and still images, you still haven't had enough time to deal with this very simple issue!

Hahahaha!

Look at the post I made 5 minutes before yours, you stupid jerk. It should be patently obvious to anybody else that you did not have time to read and understand what I said. You behaved like I said. You went ballistic. You don't even try to understand those you want to ridicule and laugh at.

[merjet]

21 minutes ago, Jon Letendre said:

And is there a way to make Ps a flat line?

Yes. The center of the wheel goes down the road the same distance as every other wheel, even though it is a point for whom rolling is impossible, it does not roll, yet goes down the road. Amazing. A paradox.

Ps is a point. As the crow flies Ps' journey is a flat line.

[Jon Letendre]

22 minutes ago, merjet said:

Ps is a point. As the crow flies Ps' journey is a flat line.

If we put the point Ps at the center of the wheel, then roll the wheel one rotation, then Ps will travel a straight line.

Yet, Ps is a point and points cannot roll.

Therefore, Ps will travel a line equal in length to the blue line, without rolling or rotating, at all.

[Jonathan]

6 minutes ago, merjet said:

Look at the post I made 5 minutes before yours, you stupid jerk. It should be patently obvious to anybody else that you did not have time to read and understand what I said. You behaved like I said. You went ballistic.

Still no math. Lots and lots of blabber, and huffing and puffing, and sniveling and sniping, but still no math.

And remember, once the big math comes, it can't agree with anything that the rest of us has said! You've stated that our descriptions of the shapes, motions and distances are "wrong," and that they are "hogwash" which "don't resolve the paradox."

[BaalChatzaf]

56 minutes ago, Jon Letendre said:

Those are also a deep, deep paradox.

what paradox.  I  use my derailers  every time I go out riding my bike.

[Jon Letendre]

2 minutes ago, BaalChatzaf said:

what paradox.  I  use my derailers  every time I go out riding my bike.

So do I. That was sarcasm. I meant that they are not a paradox at all, just like what we are discussing.

[merjet]

 

17 minutes ago, Jonathan said:

Still no math.

.......

Lots and lots of blabber, and huffing and puffing, and sniveling and sniping, 

The bottom horizontal line and left vertical line are usable as the x-axis and y-axis of a graph (Cartesian coordinates). That's part of math, ignoramus. Did you even pass geometry?  Hahahahahahaha.

........

The pot calls the kettle black.

 

[Jon Letendre]

8 minutes ago, merjet said:

 

The bottom horizontal line and left vertical line are usable as the x-axis and y-axis of a graph (Cartesian coordinates). That's part of math, ignoramus. Did you even pass geometry?  Hahahahahahaha.

........

The pot calls the kettle black.

 

Great, it's math. You've drawn x, y coordinates and two cycloids and then commented some comparisons of the cycloids, now what? 

[Jonathan]

6 minutes ago, merjet said:

 

The bottom horizontal line and left vertical line are usable as the x-axis and y-axis of a graph (Cartesian coordinates). That's part of math, ignoramus. Did you even pass geometry?  Hahahahahahaha.

........

The pot calls the kettle black.

 

 

Heh. Did you present a mathematical resolution to the "paradox"? No. You merely mentioned some aspects of the issue in mathematical terms. You said that it was "complicated" and that you'd have "more later."

Still working on it, huh?

 

[merjet]

4 minutes ago, Jonathan said:

Heh. Did you present a mathematical resolution to the "paradox"? No. You merely mentioned some aspects of the issue in mathematical terms. You said that it was "complicated" and that you'd have "more later."

Still working on it, huh?

Have a bit of patience,  impudent jerk.

When an author gets an idea for a book or even a professional or academic journal article, do they sit down and finish writing the thing 5 minutes later?

[Jon Letendre]

2 minutes ago, merjet said:

Have a bit of patience,  impudent jerk.

When an author gets an idea for a book or even a professional or academic journal article, do they sit down and finish writing the thing 5 minutes later?

One about this paradox?

Yes, about five minutes.

[Jon Letendre]

Referring to the Wiki drawing, the blue line is equal to the blue wheel circumference.

The pink line represents the imaginary "road" the pink "wheel" is to be considered as traveling over.

The pink wheel rotates during travel over the pink road, but slower than rolling speed, so the pink wheel is skidding.

The length of honest rolling the pink wheel performs is equal to its circumference and the length of skidding it performs is equal to the length of the blue line minus the circumference of the pink "wheel."

Roughly five minutes.

 

[Jonathan]

16 minutes ago, merjet said:

Have a bit of patience,  impudent jerk.

When an author gets an idea for a book or even a professional or academic journal article, do they sit down and finish writing the thing 5 minutes later?

So, let me get this straight. You believe that your yet-to-exist solution to this non-paradox is going to be the equivalent of a professional or academic journal article? You imagine that much importance involved in this elementary school problem?

Bwah-hahahahaha!!!