merjet

Aristotle's wheel paradox

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Merlin, you didn't answer my question. After thumping your chest about what a great mathematician you are, I asked where your math was in regard to this stupid "paradox." Well? Where is it?!!!

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1 minute ago, Jonathan said:

Wow, good one!

Thank you very much.

Oh, "slipping" isn't about simple visual observation. It involves a very sophisticated measurement of a very complicated relationship. Obviously something you and J are incapable of.

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2 minutes ago, merjet said:

Thank you very much.

Oh, "slipping" isn't about simple visual observation. It involves a very sophisticated measurement of a very complicated relationship. Obviously something you and J are incapable of.

Oh, so the green "wheel" doesn't slip past it's imaginary "road" until measurements are taken and sophisticated maths employed?

You are inestimably stupid.

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3 minutes ago, Jon Letendre said:

Oh, so the green "wheel" doesn't slip past it's imaginary "road" until measurements are taken and sophisticated maths employed?

You are inestimably stupid.

Yeah, and of course I didn't have to do any measuring or deal with rotation or translation when building the animations that I've presented here. Heh.

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Alright, Merlin, you've thrown your tantrum and blown off steam. Feel better now?

Okay, so it's time to deliver some substance. Let's see the big math! Blow us out of the water. Fucking dazzle and amaze us, you math genius!

J

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9 minutes ago, Jon Letendre said:

Without showing me the math you daydream about me not understanding?

Oh yeah? Well why don't you go to your barn and eat some hay!

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"As to the actual process of measuring shapes, a vast part of higher mathematics, from geometry on up, is devoted to the task of discovering methods by which shapes can be measured – complex methods which consist of reducing the problem to the terms of a simple, primitive method, the only one available to man in this field: linear measurement. (Integral calculus, used to measure the area of circles, is just one example.) – Ayn Rand, ITOE, p. 14.

She way oversimplified it by saying only ‘area of circles’. Nevertheless, she was very insightful.

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Merlin, I have another "paradox" for you:

You have one apple, and then someone gives you two more apples, so you have a total of three apples, but it seems like four apples to you! Oh noes! How can four apples be three apples? It's a paradox! Can you solve it? Can anyone solve it?

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On 11/09/2017 at 7:23 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.

What is the problem?  I see no paradox.  I don't understand why this 'paradox' is taking so long to discuss. I have no education and I have an extremely low IQ and even I can understand this.

Aristotles_wheel.svg

(The picture from wiki does not show up for me but it is there on wiki.  picture)

The blue circle rotates once. The straight blue line is equal to the circumference of the blue circle.  The straight pink line is equal to the straight blue line which is equal to the circumference of the blue circle.

Let us assume that the diameter of the pink circle is half the diameter of the blue circle. (Even if it doesn't look like it, we can imagine.) Then the circumference of the pink circle will be half the circumference of the blue circle.

a.  The pink circle rotates independently of the blue circle.  In this case the pink circle will make 2 rotations while the blue circle makes 1 rotation.

b.  The pink circle is attached to the blue circle in such a way that that it can't rotate independently of the blue circle,  In this case it will half rotate and half skid.

This is too easy. Maybe I am too stupid to understand the problem.

 

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This is crazy. There is one wheel with an inner hub, yes? They rotate together. Both turn one revolution. The blue line represents one rev of the outer circumference -- the pink line is irrelevant.The travel distance obviously is not the circumference of the inner circle, but the outer one. Conclusion: the pink line is redundant or a red herring to confuse. Me and jts, too stupid to see the problem.

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1 hour ago, jts said:

What is the problem?  I see no paradox.  I don't understand why this 'paradox' is taking so long to discuss. I have no education and I have an extremely low IQ and even I can understand this.

Aristotles_wheel.svg

(The picture from wiki does not show up for me but it is there on wiki.  picture)

The blue circle rotates once. The straight blue line is equal to the circumference of the blue circle.  The straight pink line is equal to the straight blue line which is equal to the circumference of the blue circle.

Let us assume that the diameter of the pink circle is half the diameter of the blue circle. (Even if it doesn't look like it, we can imagine.) Then the circumference of the pink circle will be half the circumference of the blue circle.

a.  The pink circle rotates independently of the blue circle.  In this case the pink circle will make 2 rotations while the blue circle makes 1 rotation.

b.  The pink circle is attached to the blue circle in such a way that that it can't rotate independently of the blue circle,  In this case it will half rotate and half skid.

This is too easy. Maybe I am too stupid to understand the problem.

 

It appears to me that you understand it well, jts.

Your a. is understandable to me, and true, if the pink and blue were interpreted to be able to be independent. Rather, Arisistotle is proposing your b.

And your b. is correct in that any and every imaginary "rolling wheel" inside the circumference of the actual rolling wheel will execute some skidding across its imaginary road. The smaller the imaginary wheel, the more skidding.

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6 minutes ago, anthony said:

This is crazy. There is one wheel with an inner hub, yes? They rotate together. Both turn one revolution. The blue line represents one rev of the outer circumference -- the pink line is irrelevant.The travel distance obviously is not the circumference of the inner circle, but the outer one. Conclusion: the pink line is redundant or a red herring to confuse. Me and jts, too stupid to see the problem.

From what I see, you understand it just fine, Tony.

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13 hours ago, jts said:

(The picture from wiki does not show up for me but it is there on wiki.  picture)

[1] The blue circle rotates once. [2] The straight blue line is equal to the circumference of the blue circle.  The straight pink line is equal to the straight blue line which is equal to the circumference of the blue circle.

[3] Let us assume that the diameter of the pink circle is half the diameter of the blue circle. (Even if it doesn't look like it, we can imagine.) Then the circumference of the pink circle will be half the circumference of the blue circle.

[4] a.  The pink circle rotates independently of the blue circle.  In this case the pink circle will make 2 rotations while the blue circle makes 1 rotation.

[5] b.  The pink circle is attached to the blue circle in such a way that that it can't rotate independently of the blue circle, [6] In this case it will half rotate and half skid.

This is too easy. Maybe I am too stupid to understand the problem.

I added some bracketed numbers to jts’s post to aid in addressing different parts separately.

[1] True.

[2] Not as pictured. It’s only about 3/4ths of the circumference. If it equaled the circumference, then it would be an accurate representation of one revolution.

[3] Yes, circumferences are proportionate to diameters.

[4] False if the circles are meant to represent the wheel in Aristotle’s wheel paradox. To do that, the pink circle would rotate only once.

[5] Yes to accurately represent the wheel in Aristotle’s wheel paradox.

[6] It will rotate once plus move more laterally to keep pace with the blue circle.

If the blue line’s length were equal to the circumference, a radius added from the center to the 6:00 position (left and right), and the pink line changed to a dashed line, then it would be a good static representation for tackling the paradox. This video is a good dynamic representation.

One more comment, with no intention of criticizing jts. That the inner circle literally slips relative to the outer circle for an ordinary wheel is patently false. It can’t slip when it’s fixed in place. It “slips/skids” metaphorically, but that is very, very misleading. The inner circle could slip/skid literally only if it were not fixed in place. Many people utterly fail to grasp these points.

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12 hours ago, anthony said:

This is crazy. There is one wheel with an inner hub, yes? They rotate together. Both turn one revolution. The blue line represents one rev of the outer circumference -- the pink line is irrelevant.The travel distance obviously is not the circumference of the inner circle, but the outer one. Conclusion: the pink line is redundant or a red herring to confuse. Me and jts, too stupid to see the problem.

Yes, but rim would be better than hub. The pink line is very relevant in that the pink circle must roll tangent to it to serve as a model to tackle the paradox. Like in my previous post, the pink line would be better dashed.

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1 hour ago, merjet said:

[....] That the inner circle literally slips relative to the outer circle for an ordinary wheel is patently false. It can’t slip when it’s fixed in place. It “slips/skids” metaphorically, but that is very, very misleading. The inner circle could slip/skid literally only if it were not fixed in place. Many people utterly fail to grasp these points.

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

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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 his 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

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