merjet

Aristotle's wheel paradox

Recommended Posts

Proceeding this way, the final apparatus we are going to have will be a Heath-Robinson affair. Designed by a committee, :)

Share this post


Link to post
Share on other sites
2 hours ago, anthony said:

If I have one stance on this topic, it's (predictably, I guess) that identity and identification are the irreplaceable precursors to math, geometry, physics, mechanics, etc. Efforts to overturn that hierarchy are where the source of all the differences - and errors - arise.

Tony,

Is that really fair to what Darrell has presented?

Not implying it is for others, but Darrell has been extremely clear about the math and measurements without inverting any identity hierarchies at all.

Michael

Share this post


Link to post
Share on other sites

Michael, 

I must re-make my point from the outset, that I don't accept a track and/or slippage. For his work on the track, Darrell did very well. Out of interest, I looked into the possibilities of a physical track (and no slip) with him to see how far it could go, but up to a limit. He is clear and explicit and asks searching questions, but his and my basic premises are dissimilar. 

In counterpoint to slippage, I put forward a very simple proposition based on two facts of identity:  1. A wheel in a wheel will undeniably and logically, by definition, travel further than its circumference. Period.

(You try to 'fix' that, you either meet an impossibility, or you turn the wheel into something not "a wheel" any longer, but something else--perhaps a "machine").

At this stage, the paradox can be seen to be non-contradictory and pursued no further. "Trivial" someone said about this. But, conversely, bringing in slippage and a track becomes superfluous, requiring ponderous explanations, I believe. Saying nothing about the "equipment" you'd need...

2., Next, my ~explanation~ for this phenomenon is the long-known fact that different points within a wheel rotate at different velocities. For inner wheels and circles, dependent on their radii. A fact of identity also, one we know from experience and from logic and math.

What that shows is why the inner wheel stays in rotational synch with the outer. Whereas, if at the same [Vt] velocities a wheel, any wheel, could not maintain its integrity -- i.e. IF rotating at the same speed - the inner would turn too fast in relation to the outer, and break. AND crucially, this shows why the small wheel ( think of a little hub in very large wheel) turns only once - but travels the same distance as the big wheel circumference. There's no slip, over-spin, skid, etc. for that to happen. We'd know all about it, if it were not so.

I also ask for a "fair" consideration of all this. I'm quite a proponent of "K.I.S.S"., and in contrast to the over-complications and bulky "track-slippage" explanation --  these theories are simple and direct, based entirely on wheel identity. (And the things we see to be true). The simple route is unpopular, I think many are drawn to (unnecessary)complexity, with math, experiments, etc.

After establishing the above, the invaluable math etc.etc. comes into its own.

Share this post


Link to post
Share on other sites
12 hours ago, Darrell Hougen said:

Hi Tony,

You're really making this way more complicated than it needs to be. I've created some admittedly low quality images to help visualize what I'm talking about.

image.png

If the bottle were rolling on the floor, it would probably make contact with the floor in many places. However, we can simplify things by assuming that it only contacts two points. Imagine that they are two rails that go into the page.

In the first figure, I've shown the situation when the two rails contact the body. In this case, the diameter of the bottle at each point is equal to R, so both ends of the bottle roll at the same speed.

image.png

In the second figure, I've shown the body of the bottle supported at one point and the neck supported at one point. Again, imagine rails going into the page. Here, the large end of the bottle will roll more quickly than the small end. If the angular speed of the bottle is w, then V = Rw and v = rw. Or, after some time, the distance rolled by the big end is D = 2 * pi * R and the small end rolls 2 * pi * r. The result is that the bottle veers toward the small end because the small end doesn't go as far as the big end.

image.png

I think that what you're imagining is a situation in which the body and neck are both supported. In this case, the coefficients of friction and weight distribution do indeed matter. However, it's not necessary to consider this case. It just confuses the issue. Depending upon the friction and weight distribution, the third case will either be more like the first case or more like the second. Perhaps it will be somewhere in between and the speeds will be between the two cases. However, we need not be overly concerned with the third case. The first two cases are sufficient to illustrate Aristotle's paradox. Whoever invented "Aristotle's paradox" is saying that case 2 will behave like case 1 when they are clearly different.

I hope that clears things up for you.

Darrell

 

 

Very nice! Lucid illustrations we could have used earlier.

I've questions about veering "toward the small end because the small end doesn't go as far as the big end".

1. As I see it, R and r are now equivalent. [R] You have raised the small end to compensate for the diameters, so aren't they equal?

2. The neck's lesser rotating speed Vt compensates for its smaller diameter. If the speed was constant, - i.e. higher - yes, we'd have veering in that direction.

3. The contact made at all points (for large and small diameters between a plane and a circle) is theoretically the same.

So, will it veer?

Share this post


Link to post
Share on other sites
1 hour ago, anthony said:

Michael, 

I must re-make my point from the outset, that I don't accept a track and/or slippage.

We know. You don't like the track and slippage. You dislike the surface beneath the small wheel so much that you blank it out of existence. You tell us the lie that the original formulation of the Aristotle's Wheel Paradox did not include it (line "HK" in the original formulation). And beyond that, even if you accept the surface only for the sake of argument, you refuse to accept the reality of slippage. You don't want to hear it. You don't want to consider it.

We get it: You don't accept reality.

J

Share this post


Link to post
Share on other sites
2 hours ago, anthony said:

Thanks for the clarity. What you've led me to grasp and is being posed by all the arguments I see (for interposing a track and slippage), is not of a wheel any longer--but "a machine". Here, what you are demonstrating is a 'self-regulating machine', as I see it. 

(Quite, there will likely be an application in engineering for some machine like that, requiring electronics and sensors for its operation).

But the paradox wheel we are talking about isn't any type of purposeful - 'organism'  - which directs its own timely actions (--rolling-sliding-rolling-sliding--).

There is nothing "purposeful" in the wheel that I describe, it behaves automatically as described, with the given simple conditions.

Suppose the large wheel has perfect traction, it rolls without slipping, performing one revolution. I think everyone will agree that this is one of the conditions of Artistotle's paradox.

Now for the small wheel: suppose it has also perfect traction, then it is obvious that the whole thing will jam up (the case of the incompatible gears). But when the track of the small wheel has a finite friction coefficient that isn't too large so that it doesn't jam up the system, this wheel will slip, that is necessary to travel the forced distance (forced by the outer wheel), as its own rolling covers only part ot that distance.

Suppose now that that the friction is small, even negliglibly small, then the wheel will of course also slip, but it will still also rotate, as the rotation is forced by the outer wheel, so there is no question of free slipping. The amount of rotation that is equal to the circumference of the small wheel (2*pi*r) we call rolling, the difference 2*pi*(R-r) can only be traveled by the small wheel when it is slipping. So it is unavoidable that the movement of the small wheel is a combination of rolling and slipping. That follows inexorably from the conditions of the paradox. The amount of slippage is always the same, independent of the friction of the tracks, as long as the large wheel can roll without slipping and the friction on the track for the small wheel isn't so large as to jam up the whole system. Under those conditions its behavior is determined by pure kinematics,  by geometry alone. And if there is a difference between more or less friction, it is perhaps the difference between more and less sparks flying, more or less screeching, but the movement and the amount of slipping don't change.

 

Quote

Individuals "take in" inductively, that when something moves in a certain way, it keeps moving that way. When it begins sliding, it continues sliding. When it rolls, it goes on. Until something opposing or deliberate prevents it. We learn (affirming this) Newton's First Law of motion:  "An object at rest tends to remain at rest, and an object in motion tends to remain in motion with a constant velocity and in the same direction, unless acted upon by an unbalanced force" . 

I think then that this "machine" you have made (are envisaging) opposes both Aristotle's law of non-contradiction and the Newtonian law.

The "opposing" is here done by the movements that are forced upon the small wheel by the large wheel, due to the forces that keep the solid body together. This isn't a question of free rolling vs. free slipping, the wheel is forced to move following an exactly determined combination or rolling and slipping. You shouldn't ignore the boundary conditions. 

Share this post


Link to post
Share on other sites
On 12/6/2018 at 12:20 PM, william.scherk said:

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

 

 

 

 

Share this post


Link to post
Share on other sites
6 hours ago, anthony said:

Michael, 

I must re-make my point from the outset, that I don't accept a track and/or slippage. For his work on the track, Darrell did very well. Out of interest, I looked into the possibilities of a physical track (and no slip) with him to see how far it could go, but up to a limit. He is clear and explicit and asks searching questions, but his and my basic premises are dissimilar. 

In counterpoint to slippage, I put forward a very simple proposition based on two facts of identity:  1. A wheel in a wheel will undeniably and logically, by definition, travel further than its circumference. Period.

(You try to 'fix' that, you either meet an impossibility, or you turn the wheel into something not "a wheel" any longer, but something else--perhaps a "machine").

At this stage, the paradox can be seen to be non-contradictory and pursued no further. "Trivial" someone said about this. But, conversely, bringing in slippage and a track becomes superfluous, requiring ponderous explanations, I believe. Saying nothing about the "equipment" you'd need...

2., Next, my ~explanation~ for this phenomenon is the long-known fact that different points within a wheel rotate at different velocities. For inner wheels and circles, dependent on their radii. A fact of identity also, one we know from experience and from logic and math.

What that shows is why the inner wheel stays in rotational synch with the outer. Whereas, if at the same [Vt] velocities a wheel, any wheel, could not maintain its integrity -- i.e. IF rotating at the same speed - the inner would turn too fast in relation to the outer, and break. AND crucially, this shows why the small wheel ( think of a little hub in very large wheel) turns only once - but travels the same distance as the big wheel circumference. There's no slip, over-spin, skid, etc. for that to happen. We'd know all about it, if it were not so.

I also ask for a "fair" consideration of all this. I'm quite a proponent of "K.I.S.S"., and in contrast to the over-complications and bulky "track-slippage" explanation --  these theories are simple and direct, based entirely on wheel identity. (And the things we see to be true). The simple route is unpopular, I think many are drawn to (unnecessary)complexity, with math, experiments, etc.

After establishing the above, the invaluable math etc.etc. comes into its own.

Tony,

You know I love you. But I just can't make any sense of this.

And I have no idea where you think Darrell inverted identity...

It's all good, though...

:) 

Michael

Share this post


Link to post
Share on other sites
2 hours ago, Michael Stuart Kelly said:

Tony,

You know I love you. But I just can't make any sense of this.

And I have no idea where you think Darrell inverted identity...

It's all good, though...

:) 

Michael

No probs. 

For trying to make "any sense of this" and the paradox, I took the easy task - I simply described 2 of the wheel's attributes which, I think, together make the *apparent* contradiction null and void, when understood.

The other route is tortuous. It means imagining the inner wheel transferring from point A to B - but with a *certain amount* of skid and a *certain amount* of roll. Also, this necessitates putting in a track which it can slip/roll upon. My response is that by trying this method to answer the paradox, one tries to defeat the theory/identity of the wheel , and as for real-world demonstrations, applications and whatever  - few that I see, outside of experimentation.

I got off much easier...

(That was a general remark, I didn't single out Darrell in any manner)

Share this post


Link to post
Share on other sites
3 minutes ago, anthony said:

The other route is tortuous. It means imagining the inner wheel transferring from point A to B - but with a *certain amount* of skid and a *certain amount* of roll.

There's nothing tortuous about it for the rest of us. It's actually really simple.

 

4 minutes ago, anthony said:

Also, this necessitates putting in a track which it can slip/roll upon.

No, it doesn't necessitate putting in a track. The track is already there. It's been there all along, all the way back to Aristotle's day.

 

6 minutes ago, anthony said:

My response is that by trying this method one tries to defeat the theory of the wheel , and as for real-world applications and whatever  - few that I see.

Your response is nonsense, and reveals that you don't grasp any aspect of the alleged "paradox."

 

7 minutes ago, anthony said:

I got off much easier...

Ignorance is bliss. Stupid is easy.

J

Share this post


Link to post
Share on other sites
13 hours ago, Jonathan said:

No, it doesn't necessitate putting in a track. The track is already there. It's been there all along, all the way back to Aristotle's day.

You’re lying again, a habit of yours. Mechanica does not say track or surface or support; it says line. You lie, because you want a crutch and you’re helplessly lame-brained without a crutch. You’re quite good at faking reality.

Share this post


Link to post
Share on other sites

This whole discussion reminds me of a story I once heard about elementary school.

A teacher was trying to teach subtraction to a first grade class. She had a big "2" on blackboard and underneath was "- 2" and underneath that was a line.

She asked the class, "Who knows how much two take away two is?"

Little Johnny raised his hand and was called on.

"Two," he said.

The teacher asked, "Are you sure?"

"Yes."

So the teacher asked him to explain how he arrived at that. He walked up to the blackboard, picked up an eraser and pointed to the top number.

"Two," he said. Then he erased the "- 2" underneath the top number. "Take away two," he said.

Then he beamed at the teacher and pointed at the number he did not erase. "Is two."

Now, before you go,  "Awwwww..." there's a second part to this story, but only for adult audiences. Someone set the problem up over here on OL and a cage match ensued.

:) 

Michael

Share this post


Link to post
Share on other sites
5 hours ago, merjet said:

You’re lying again, a habit of yours. Mechanica does not say track or surface; it says line.

Heh. "It says line." The line that you intentionally omitted from your Wikipedia diagram.

Does the Mechanica say to remove the line?

 

5 hours ago, merjet said:

You lie, because you want a crutch and you’re helplessly lame-brained without a crutch. You’re quite good at faking reality.

In what way is it a crutch? You've made that accusation many times, as if it's devastating. Explain. How is reality a crutch? How is my referring to reality "faking reality"?

Does the Mechanica state that the "paradox" shall not be solved by mechanical means?

There are many substantive criticisms and questions here that I and others have put to you, but that you haven't answered. Your refusal to address them is curious. Are you incapable of answering? Stumped? You seem to think that anger and name calling is a sufficient means of countering all of the substance that we've posted. That and molesting the Wikipedia page.

It's not sufficient.

J

 

Share this post


Link to post
Share on other sites

Just for reference: here is the original text in Greek, 

Microsoft Word - ΜΗΧΑΝΙΚΑ ΤΕΛΙΚΟ-ΕΞΩΦΥΛΛΟ2.pdf

and for those whose Greek is a bit rusty is here the translation (from http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Aristotle/Mechanica*.html):

24 A difficulty arises as to how it is that a greater circle when it revolves traces out a path of the same length as a smaller circle, if the two are concentric. When they are revolved separately, then the paths along which they travel are in the same ratio as their respective sizes. Again, assuming that the two have the same centre, sometimes the path along which they revolve is the same size as the smaller circle would travel independently, and sometimes it is the size of the larger circle's path. Now it is evident that the larger circle revolves along a larger path. For an examination of the angle which each circumference makes with its own diameter shows that the angle of the larger circle is larger, and of the smaller circle smaller, Bso that they bear the same ratio as that of the paths on which they travel bear to each. Yet on the other hand it is clear that they do revolve over the same distance, when they are described about the same centre; and thus it comes about that sometimes the revolution is equal to the path which the larger circle traces out, and sometimes to that of the smaller. Le ΔΖΓ be the greater circle and  p389 ΕΗΒthe less, with Α as the centre of both. Let the line ΖΙ be the path traced by the circumference of the larger circle, when it travels independently, and ΗΚ the path travelled independently by the smaller circle, ΗΚ being equal to ΖΛ.


[image ALT: A diagram of two concentric circles, with two perpendicular diameters of the larger one drawn, one vertical and the other horizontal; a horizontal tangent is extended from the bottom of each circle.]

Fig. 13

If I move the smaller circle I am moving the same centre, namely Α; now let the larger circle be attached to it. At the moment when ΑΒ becomes perpendicular to ΗΚ, ΑΓ also becomes perpendicular to ΖΛ; so that it will have invariably travelled the same distance, that is ΗΚ, the distance over which the circumference ΗΒ has travelled, and ΖΛ that over which ΖΓ has travelled. Now if the quadrant in each case has travelled an equal distance, it is obvious that the whole circle will travel over a distance equal to the whole circumference, so that when the line ΒΗ has reached the point Κ, then the arc of the circumference  p391 ΖΓ will have travelled along ΖΛ, and the circle will have performed a complete revolution.

Similarly, if I move the large circle and fit the small one to it, the two circles being concentric as before, the line ΑΒ will be perpendicular and vertical at the same time as ΑΓ, the latter to ΖΙ, the former to ΗΘ. So that whenever the one shall have traversed a distance equal to ΗΘ, and the other to ΖΙ, and ΖΑ has again become perpendicular to ΖΛ, and ΑΗ has again to ΗΚ, the points Η and Ζ will again be in their original positions at Θ and Ι. As, then, nowhere does the greater stop and wait for the less in such a way as to remain stationary for a time at the same point (for in both cases both are moving continuously), and as the smaller does not skip any point, it is remarkable that in the one case the greater should travel over a path equal to the smaller, and in the other case the smaller equal to the larger. It is indeed remarkable that as the movement is one all the time, that the same centre should in one case travel a large path and in the other a smaller one. For the same thing travelling at the same speed should always cover an equal path; and moving anything with the same velocity implies travelling over the same distance in both cases.

To discover the cause of these things we may start with this axiom, that the same or equal forces move one mass more slowly and another more rapidly. Let us suppose that there is a body which has no natural movement of its own; if a body which has a natural movement of its own moves the former as well as itself, it will move more slowly than if it moved by itself; and it will be just the same if it naturally moves by itself, and nothing is  p393 moved with it. It is impossible for it to have a greater movement than that which moves it; for it moves not with a motion of its own, 856Abut with that of the mover.

Suppose that there are two circles, the greater Α and the lesser Β. If the lesser were to push the greater without revolving itself it is clear that the greater will travel along a straight path as far as it is pushed by the lesser. It must have been pushed as far as the small circle has moved. Therefore they have travelled over an equal amount of the straight path. So if the lesser circle were to push the larger while revolving, the latter would be revolved as well as pushed, and only so far as the smaller revolves, if it does not move at all by its own motion. For that which is moved must be moved just so far as the mover moves it; so the small circle has moved it so far and in such a way, e.g. in a circle over one foot (let this be the extent of the movement), and the greater circle has moved thus far. Similarly, if the greater circle moves the less, the small circle will move exactly as the greater does. (This will be true) whichever of the two circles is moved independently, whether fast or slowly; so the lesser circle will trace a path at the same velocity, and of the same length as the greater does. This, then, constitutes our difficulty, that they do not behave in the same way when joined together; that is to say, if one is moved by the other, not in a natural way nor by its own movement. For it makes no difference whether it is enclosed and fitted in or whether one is attached to the other. In the same way, when one produces the movement, and the other is moved by it, to whatever distance the one moves the other will also move. Now when one moves a circle which is  p395 leaning against or suspended from another, one does not move it continuously; but when they are fastened about the same centre, the one must of necessity revolve with the other. But nevertheless the other does not move with its own motion, but just as if it had no motion. This also occurs if it has a motion of its own, but does not use it. When, then, the large circle moves the small one attached to it, the smaller one moves exactly as the larger one; when the small one is the mover, the larger one moves according to the other's movement. But when separated each of them has its own movement.15 If anyone objects that the two circles trace out unequal paths though they have the same centre, and move at the same speed, his argument is erroneous. It is true that both circles have the same centre, but this fact is only accidental, just as a thing might be both "musical" and "white." For the fact of each circle having the same centre does not affect it in the same way in the two cases. When the small circle produces the movement the centre, and origin of movement belongs to the small circle, but when the large circle produces the movement, the centre belongs to it. Therefore what produces the movement is not the same in both cases, though in a sense it is.16

 

16 The ambiguity of the phrase "path of a circle" has confused the argument. It may mean (1) movement of the centre; (2) movement of a point on the circumference; (3) e.g. the impression made by a tyre on a road. Probably Aristotle usually means (3). It is not easy to be sure whether he has seen the true solution of the problem, viz.: in one case the circle revolves on ΗΘ, while the larger circle both rolls and slips in ΖΙ.

 

"both rolls and slips" again someone who says so, that must be a conspiracy! M

  • Like 1
  • Thanks 1

Share this post


Link to post
Share on other sites

This page is pretty cool. You can roll the wheels by click-hold-dragging them, you can change the ratio of the wheels, and also decouple them (the "decouple" button is in the "click here for explanation box").

Merlin and Tony, crank it all the way up to 5:1.  I doubt that you'll be able to grasp it even at that ratio, but it's your best chance.

Merlin, you might want to see if you can figure out a way to hack the site so that you can molest it like you did with Wikipedia.

J

  • Like 1

Share this post


Link to post
Share on other sites
57 minutes ago, Jonathan said:

This page is pretty cool. You can roll the wheels by click-hold-dragging them, you can change the ratio of the wheels, and also decouple them (the "decouple" button is in the "click here for explanation box").

And what do we read on that site?

  1. The inner wheel must slip as it rotates, i.e. it is not a a pure rotation, but is also being dragged along by the rotation of the outer wheel.

Et tu, Brute?

Share this post


Link to post
Share on other sites
4 hours ago, Max said:

And what do we read on that site?

  1. The inner wheel must slip as it rotates, i.e. it is not a a pure rotation, but is also being dragged along by the rotation of the outer wheel.

Et tu, Brute?

Yep, more hawrgwarshing con art scammers!

Loozhuns!

J

Share this post


Link to post
Share on other sites
8 hours ago, Max said:

And what do we read on that site?

  1. The inner wheel must slip as it rotates, i.e. it is not a a pure rotation, but is also being dragged along by the rotation of the outer wheel.

Et tu, Brute?

Can M go there and modify--I mean correct--it?

--Brant

oh, you beat me to it; I read these threads from the last post

Share this post


Link to post
Share on other sites
On December 13, 2018 at 5:09 AM, anthony said:

An object in flight is either level, rising or falling.

Too late to tell Newton he goofed.  The planets, not having been informed that they can't orbit, are still orbiting.

Ellen

Share this post


Link to post
Share on other sites
5 minutes ago, Ellen Stuttle said:

Too late to tell Newton he goofed.  The planets, not having been informed that they can't orbit, are still orbiting.

Ellen

Constantly “level” and constantly falling. His head is going to explode!

Share this post


Link to post
Share on other sites
17 hours ago, Jonathan said:

Merlin and Tony, crank it all the way up to 5:1.  I doubt that you'll be able to grasp it even at that ratio, but it's your best chance.

Sure, J, crank it up to 5:1. I doubt that you'll be able to grasp my explanation even at that ratio, but it's your best chance. Here goes. Watch the two points where the radius meets the circles when the wheel rolls. The point on the small circle traverses the same horizontal distance as the point on the large circle, because the former’s path is shorter and more efficient than the latter’s path. I bet most 10-year-old's could grasp that. Of course, what holds for these two points holds for any two points, one from each circle.

If you still don’t get it, i.e. continue to blank out reality, then I suggest you write Mr. Wessen and request he add another ratio, e.g. 20:1.

Share this post


Link to post
Share on other sites
19 hours ago, Max said:

And what do we read on that site?

  1. The inner wheel must slip as it rotates, i.e. it is not a a pure rotation, but is also being dragged along by the rotation of the outer wheel.

Et tu, Brute?

Mr. Wessen didn’t have a better explanation, as given in my previous post.

Why isn’t 'forced along by the rolling of the wheel' enough of an explanation?

He used an incorrect title, Aristotle’s Wheels. He also says, “All circles have the same circumference!” Oh, my! What an authoritative site you and J appeal to.  😃

Share this post


Link to post
Share on other sites
7 hours ago, merjet said:

The point on the small circle traverses the same horizontal distance as the point on the large circle, because the former’s path is shorter and more efficient than the latter’s path.

That doesn't get anywhere toward addressing the ancient eddheads' concerns. It only adds more burdens to the "paradox." They couldn't grasp how two wheels of different circumferences could unroll without each slipping on its own path of the same length as the other's. Informing them of cycloids would only provide them with another set of facts that contradicted their false premise. "Now we not only have a circumference that unrolls, without slipping, the other wheel's length instead of its own, but we also now have two differing cycloids, which despite not being the same length or shape, nevertheless travel the same horizontal distance and have one-to-one corresponding points at any and all moments during travel. Therefore, are there voids and other eggheaded infinity stuff?"

Cycloids are pretty, and they're interesting, but they don't address the actual problem of the "paradox." The actual problem is that it contains the mistaken premise of both wheels rolling freely over their surfaces with neither slipping.

That premise is just as mistaken as the premise in the three gear challenge that I posted. One cannot solve a "paradox" which contains a false premise without addressing and correcting the false premise. Observing that there are different cycloids while maintaining the false belief that there is no slippage of either wheel on its own surface is failure. The only success involved with cycloids would be to take the position that they are one of many means of confirming slippage and of confirming which single circumference is rolling freely and not slipping.

J

  • Like 1

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now