Max

1 hour ago, Jon Letendre said:

Sorry, I was mistaken above.

When X is 1, A is half a radian and B is a quarter mile.

A is 1 divided by (1+X) radians.

B is A radians multiplied by (X divided by (1+X))

So B equals 1 – (2*X*pi) equals X divided by ((1+X) squared)

Ah, that makes it definitely less pleasant. Your previous version was a quadratic equation, easy to solve. Now you get a third degree equation. This can be solved, but doing that is as pleasant as a third degree interrogation. You'll find many methods on the Internet, but whatever method you use, the law of conservation of misery applies.

Perhaps the easiest solution in practice is to use a numerical method like Newton-Raphson. It does also have its difficulties, but these are not too bad in comparison.

I haven't looked further at your equations, as I'm now busy with my own calculations, and I'm already confused enough... 

9 minutes ago, Jon Letendre said:

Once around the South Pole and enough extra to end exactly one mile west of where you started?

Isnt’t  it 

1 plus X miles north of the South Pole such that:

1 minus (pi2X) equals X divided by (1 plus X) ?

Could you tell me the meaning of your symbols? "1" = "1 mile: , or just "1"? pi2X: 2*pi*X? X = the "overshoot" on the small circle? Is 1 + X miles an approximation? Certainly justified in the case of 1 mile trips, but I'd like to know... At least your equation can be solved if I understand your notation correctly... I tried to be too general in my calculations, resulting in quite complex equations that can't be solved analytically.

I've tried to calculate the second problem, but I get always implicit functions that cannot be solved analytically, only numerically, so I guess this problem has no explicit solution. For generality I've used a variable for the "walked" distance instead of "1 mile".

We're talking here about the geometric north, ignore the magnetic poles.

33 minutes ago, Ellen Stuttle said:

I think that Max meant what I meant in asking if you meant "actual geographic locations."  

Places where one can literally walk in compliance with the instructions.  Out in the ocean isn't such a place. Or in irregular terrain. And the going would be a bitch around the South Pole even if there was a smooth ice sheet.

 

 

No, that would be like objecting to Aristotle's paradox by insisting that his wheel is not a good car wheel, it's not in the spirit of the puzzle. It's just the conversion from miles to latitudes etc. We can always assume that we're talking about Jesus or St Francis of Paola, who allegedly could walk on water.

41 minutes ago, Jon Letendre said:

There are more.

Similar to the solutions around the South Pole in the original puzzle, but now with the small circle with a circumference < 1 mile, such that after traversing 1 mile west you cross the meridian that crosses the larger circle 1 mile west of the starting point. 

14 minutes ago, Jon Letendre said:

Yes, that’s one.

I don’t know what you mean by corresponding location.

How many degrees, minutes, seconds,  north... I'm not familiar with miles (nautical? statute?). 

32 minutes ago, Jon Letendre said:

Extra points for some of the places where one can follow the walking instructions and end precisely one mile west of the start point.

0.5 mile north of the equator. I'm too lazy to calculate the corresponding geographic location...

1 hour ago, BaalChatzaf said:

"benevolent" and "malevolent" should only be applied to entities capable of  intentions.   The better words to  use  are   "beneficial"  and "harmful" (resp.)  which can be applied to any entities  capable of producing  beneficial (harmful)  results or effects.    "volent"  indicates will or intent.  
 

I agree, those terms were badly chosen. Further, the idea of a beneficial (let alone "benevolent") universe is a bit of a tautology: man evolved in such a way that he could survive in his environment. It's the anthropic principle again: we shouldn't be surprised that the universe makes intelligent life possible, we wouldn't be there to be surprised if that had not been the case.  

Bacteria in those hot springs could also wonder that their local universe is so beneficial to them. while they of course evolved in such a way that they could survive in that environment (which would be lethal for humans).

But beneficial or not, it won't continue endlessly, one day, when that big asteroid comes, we're finished, probably long before the sun finally kills us. Even the dinosaurs (OK, with exception of the birds) were wiped out, while they had existed for some 100 million years, so they were exceptionally well adapted to their environment, and yet the universe decided one day to be no longer benevolent to them, to borrow for a moment the anthropic view of the universe. 

49 minutes ago, BaalChatzaf said:

Nope.  The only places where two longitudes intersect are at the poles.  Heading North means moving on a line of longitude  in the direction of the North geographic pole. 

 

You ignore the possibility that going south and going north can be done on the same line of longitude, while going west between these two displacements.

7 hours ago, anthony said:

Max: "...an inner track *is* given".

But I'm quite happy to accept an inner track, and such.  

The problems it brings in become greater than the original 'problem'. As my last post, this wheel isn't going anywhere, let alone solve anything.

But the inner track is part of the original problem, as you easily can verify by reading the original text. After all this is Aristotle's paradox and not a "What I Find Intersting About Wheels" discussion. Aristotle wasn't trying to design some new wheel or commenting on the quality of the Greek wheels of his time, this is about kinematics, about circles rolling over tracks, not about dynamics. These may become relevant when constructing functioning wheels, but they are completely irrelevant for solving the paradox. That can be done by purely geometrical/mathematical methods. If you remove that second track,  you throw the paradox-baby out with the track-bathwater. That is not the same as solving it.

2 minutes ago, Jon Letendre said:

I didnt even realize there were more solutions beside North Pole until after starting the thread!

I think the puzzle originally was supposed to have only 1 solution, until people discovered more solutions.

2 minutes ago, Jon Letendre said:

I haven’t checked any resources, but Jonathan seems to see the same additional solutions that I do.

Dont spoil it or hint too much!!

Mum's the word.

Well, one suggestion: make a few sketches, that may help.

That's an old riddle, I remember seeing it around 1960. Probably even much older.

I've no idea where this fancy border comes from...

5 hours ago, anthony said:

There's that elephant in the room, again, which everyone is slipping past, going unrecognized.

You cannot even begin to consider the option, sliding/slip, until you first factor in the (proportionately) reduced speeds of inner radii (e.g. an inner wheel). That is a given. (An inner track is not).

Nobody ignores the fact that a point on the smaller wheel has a lower tangential speed than a point on the larger wheel. And an inner track is given, read the original text. 

Quote

Ask yourself, could it be the case that its ~lesser~ rotating velocity (Vtangential) *cancels out* any potential slippage, for the inner wheel? Shouldn't this inherent property of the wheel be considered way above all else?

No, it does not cancel out slippage, as I've shown in my proof (you should really read that some time). I've given the exact amounts of the various speeds, instead of your vague conjecture.

Quote

Following, one erroneous explanation that I found - "Satisfying Explanation..." - by one fellow who comprehends the apparent 'problem', but tied himself in complicated knots and got his conclusion wrong, because he took into account only angular velocity.

E.g. "...both rotate with equal velocity..." [!]

He is correct, they do rotate with equal velocity, rotation velocity or angular velocity is the vector ω, with magnitude = number of revolutions/ time unit. The tangential velocity = ω x r  and the tangential speed is ω*r.

With boldface I indicate vectors, instead of trying to put an arrow over the variable (see also below). "x" indicates a vector product.

5 hours ago, anthony said:

The velocity of any point PP on a wheel can be written as the sum of two velocities: the velocity V→V→ of the center OO and the velocity ω→×OP→ω→×OP→ of rotation about the center, where ω→ω→ is angular velocity (perpendicular to the plane of the wheel).

A wheel turns without sliding with respect to a given path if the velocity of the contact point between wheel and path vanishes. Let then CC and C′ be the contact points of the two wheels. We have

v→C=V→+ω→×OC→andv→C′=V→+ω→×OC′→

If v→C=0 then V→=−ω→×OC→ and

v→C′=−ω→×OC→+ω→×OC′→=ω→×(OC′→−OC→)=ω→×(CC′→).

This cannot vanish, unless C=C′. So the assumption that both circles turn without sliding is false.

 

The formatting has gone awry, but the explanation is correct, it is in fact the same as my proof, only using velocity vectors instead of speed scalars+direction. I've reformatted it, avoiding those pesky arrows by indicating vectors with boldface (and "x" for the vector product).

v_C = V + ω x OC and v_C’= V + ω x OC’  [I see that I now somehow have used different fonts, ignore that; quirks of this editor.]

if v_C = 0 then

V = - ω x OC [the translation velocity and the tangential velocity at the contact point cancel when there is no slipping] and

v_C’ = - ω x OC + ω x OC’ = ω x (OC’ - OC) = ω x CC’

This cannot vanish, unless C = C’. So the assumption that both circles turn without sliding is false.

[ v_C and v_C’ are the velocity vectors of the contact points (6 o’clock positions) of respectively the larger and the smaller circle. v_C = 0 is the condition that the large circle rotates without slipping, v_C’ = 0 is the condition that the small circle rotates without slipping.]

1 hour ago, Michael Stuart Kelly said:

Max,

Here is the problem. You have parents who take their kids to be vaccinated. When they arrive the kids are vibrant. After the vaccination, the kids are zombies.

This is not ONE story or a fabrication of some conspiracy theorists. This is the story or TENS OF THOUSANDS of people. They exist.

Then people like you come along and dismiss it all citing some scientific paper or other. And even affect some kind or posture of superiority. Thus insinuating they all lived a coincidence and were too stupid to know it.

Don't you see where there would be a credibility issue with the scientific side, and especially when the scientific side keeps talking about "settled science" with vehemence even though they don't use that term?

Tens of thousands of cases is a lot to blank out.

That is religion, not science.

A peer reviewed magic wand will not make all those people go away.

Michael

It isn't rational to just dismiss "some paper" as the product of those awful scientists. The Danish study I referenced followed all children born in Denmark in the period  January 1, 1991 to December 31, 1998, a total of 537,303 children followed for a total of 2,129,864 person-years. Read the paper to see how careful this study was set up, how painstakingly and meticulously all kinds of possible factors were taken into account. If you think you can just dismiss the study, you should point out the errors therein. Follow also the references in that study to see the results of other studies that come to the same conclusion. 

As I said before, "data" is not the plural of "anecdote" and "post hoc ergo propter hoc" is a common fallacy. With many millions of people it is statistically unavoidable that there will be "remarkable" coincidences. How impressive these might seem, in themselves they don't prove anything. Therefore you need large and carefully designed scientific studies, not a collection of anecdotes. In such cases I trust only scientific data. Not that these are automatically correct (far from it!), but at least I have some possibility to check the accuracy and the soundness of the methods used.

9 minutes ago, merjet said:

Yes, it is. Saying "it slips" is merely another way of saying the smaller circle rolls further than its own circumference.

We now know that this is the mechanism behind the further rolling, but Aristotle didn't understand it, as I've shown in one of my previous posts. Therefore it is no longer a real puzzle for us, while it was an enigma for those guys in the past. But I'm glad to know that you now have also been converted to the Slipping School.

32 minutes ago, merjet said:

LOL. I didn't know cycloids were given by the paradox either. They are certainly not in Mechanica.  But somehow Max believes they are (by ESP?).

As I've shown before, cycloids are a completely unnecessary element added by you, allegedly "proving" that both wheels travel the same distance. Well, that they do, Aristotle already knew, you can read that in his text. So in that regard you don't prove anything that isn't already in Aristotle's text. The cycloids are just an irrelevant extra.

1 hour ago, merjet said:

LOL. Slippage is not a solution; it is a mere restatement of the paradox.

No, it isn't. Artistotle wrote: "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" and "the smaller does not skip any point", so he considers only stopping of the large circle and skipping of the small circle as possible explanations for the problem. As he rejects these possibilities, he cannot solve the paradox, because he is not aware of the possibility of slipping (forwards for the small circle, forced by the large one, and backwards for the large circle, forced by the small one), which enables a continuous movement that explains the problem. 

10 hours ago, BaalChatzaf said:

Have a look here  https://www.vox.com/2018/8/21/17588032/vaccination-rates-united-states    The article indicates 92 percent of the population is vaccinated. If vaccination caused autism (it doesn't) then over 80 percent of the population would be autistic.  However only one in ninety is diagnosed with autism.  Clear proof that the hypothesis  immunization causes autism is poppycock.  

But even if you suppose that in only one of 1000 cases vaccination would cause autism, this would show up in the statistics if your sample is big enough. The question is not how many vaccinated children become autistic, but: is there a difference in the percentage of children diagnosed with autism between vaccinated and unvaccinated children? If there is no difference, then there is no evidence for the hypothesis that vaccination causes autism, that is elementary statistics. Now "data" is not the plural of "anecdote", you need a large sample to get reliable results. Such studies have been done, and the conclusion of all of them was that there is no evidence that vaccination causes autism. 

For example, there has been a large Danish study wherein more than half a million children were followed for 8 years ( https://www.nejm.org/doi/full/10.1056/NEJMoa021134 ). There was found no difference between vaccinated and unvaccinated children. In fact among the vaccinated children there was less autism diagnosed (but the difference was not significant). Further "There was no association between the age at the time of vaccination, the time since vaccination, or the date of vaccination and the development of autistic disorder." 

This doesn't prove that vaccination cannot cause autism, but it does prove that if that were the case, it would be exceedingly rare and certainly no cause for concern.

1 hour ago, BaalChatzaf said:

Bullshit. If that were true the majority of Americans would be mentally crippled.  

You mean they aren't? 

2 hours ago, merjet said:

Do you agree that all 3 solutions I put on Wikipedia are correct?

These are not solutions. "Solution" 1 is nothing else than a short recapitulation of the paradox:

Aristotle: If I move the smaller circle I am moving the same centre, namely Α; now let the larger circle be attached to it [...] it will have invariably travelled the same distance [i.e. case 1: the smaller circle forces the larger circle to travel only the distance of the circumference of the small circle]

[...]Similarly, if I move the large circle and fit the small one to it [case 2: the large circle forces the smaller circle to travel the distance of the circumference of the large circle]

[...] 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 [Aristotle doesn't understand how in case 1 the large circle is forced to travel the smaller distance]

the smaller does not skip any point [neither does he understand how in case 2 the smaller circle is forced to travel the larger distance]

[...]When, then, the large circle moves the small one attached to it, [in other words, when the large circle forces the small one]

the smaller one moves exactly as the larger one; when the small one is the mover, [that is, the small circle forces the large one]

the larger one moves according to the other's movement.

Compare that with your "solution" 1: If the smaller circle depends on the larger one (Case I), then the larger circle forces the smaller one to traverse the larger circle’s circumference. If the larger circle depends on the smaller one (Case II), then the smaller circle forces the larger one to traverse the smaller circle’s circumference. This is the simplest solution.

It is clear that this doesn't tell us anything new that Aristotle hadn't written already. 

That "solution" 2 and "solution" 3 are not solutions, I've already shown in earlier posts. In fact they are also just recapitulations of the paradox.

2 hours ago, merjet said:

Max has no solution without his crutch – a second surface, track, or support. He needs a crutch so he can claim slippage.

When the problem conditions are met, the smaller circle horizontally moves the circumference of the larger circle (or vice-versa) with or without a second surface, track, or support. The paradox does not appear and disappear depending on whether or not there is a second surface, real or imagined. My solutions hold with or without a second surface, track, or support. Max has refused to acknowledge my solutions as solutions. He has even denied they are. No good reason given; it’s mere ego and stubbornness.

I'm stubborn in that I keep referring to the original formulation by Aristotle (or whoever it was). That seems to me to be appropriate if we're talking about Aristotle's wheel paradox. And in that original formulation the second track, ΗΘ, is an essential element. To fresh up your memory, from the original text (in translation): 

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's obvious that Aristotle's essential problem was that he couldn't understand how the smaller circle could traverse the greater distance without skipping somewhere. The solution is that the smaller circle is not skipping but slipping, i.e. that the point of that circle in the 6 o'clock position has a translation speed > 0. We now know that a 1-1 mapping from a smaller circle or line segment onto a larger line segment is possible, so that is no longer an objection and with calculus we can give an exact description. The interaction between the circle and its track is the crux of the paradox, removing that track is not giving a solution, but explaining the paradox away. Your argument is like that of someone who "solves" Einstein's twin paradox by saying that there are no twins from whom one travels with extremely large speeds through space, they are just a crutch!

1 hour ago, Jonathan said:

Well, okay, then, I'll just believe that guy's opinion rather than my own observations of what happened at the time. Yeah, his charts and graphs nullify the things that were being said at the time. I didn't hear those things because your Doctor Kimball C. Atwood didn't look for them, but instead presented charts and graphs. And I haven't seen interviews of Marshall or Warren in which they discussed the dogmatic mindset that they faced, because your Kimball C. Atwood didn't see such interviews. If he didn't look for them, find them or see them, then they never happened. It's settled science that it's just a silly myth. Everyone was actually very nice and open minded to the ideas of Drs. Marshall and Warren. It was all smooth sailing.

And it's all very "nuanced" to take a subject which, by its nature, involved mostly oral expressions of resistance to new ideas, then to not look for any evidence of those expressions, then to not find any of them, and to conclude that it was all a myth!

Were absolutely certain that it was a myth. Atwood's comments prove it. Let's say "myth" some more just to make it even more true.

 

Max, you're demonstrating the illogic and pompous stupidity that we're criticizing. Your linked source doesn't address the actual issue, but attempts to bypass it with non sequiturs, obfuscation, equivocation, and sloppy assumptions.

Huh? Did we read the same article? I think Atwood is very fair to Marshall in his article, he gives simply the facts that demolish the notion of Marshall as a lone fighter against those stubborn people from "settled science". He mentions all the studies that were done at the time to check the bacterium theory. He also tried to contact Marshall to comment for his article, but Marshall had not replied. Interesting is that Marshall in 1991 wrote: “In my naïveté I expected H. pylori to be immediately accepted as the cause of duodenal ulcer,” [but] “the presence of H. pylori in many apparently healthy persons has made its pathogenic role harder to understand and has delayed wide acceptance of the new bacterium as an important pathogen” (Marshall 1991). 

Already before I'd read this article, I had my doubts, as I'd found how soon after his first publication a large study was conducted to test the theory, and how soon Marshall's views were vindicated, and how many awards he has won since then:  the Warren Alpert Prize, the Australian Medical Association Award, the Albert Lasker Award for Clinical Medical Research, the Gairdner Foundation International Award, de Paul Ehrlich and Ludwig Darmstaedter Prize, the Dr. A.H. Heineken Prize for Medicine, the Florey Medal, the Buchanan Medal of the Royal Society, and last but not least the Nobel prize, to name just a few. He can hardly complain about lack of recognition.

But perhaps you can show me some of all those non sequiturs, obfuscation, equivocation and sloppy assumptions in that article?

16 hours ago, Jonathan said:

Back in the 80s, the medical and scientific communities ridiculed the hell out of doctors Barry Marshall and Robin Warren for going against the "settled science" that no bacterium could live in the acids of the human stomach, and that ulcers were caused by spicy foods and stress. Even after proving their case on helicobacter pylori, it took the snarky authorities years to accept reality, let go of their pissy mindsets and judgments of Marshall and Warren as kooks and snake oilers.

Which mindset is more disturbing and dangerous, that of kooks who make false speculations and come to mistaken, unsupported conclusions, or that of people who pose as authorities and impede or shut down advancements because of their own brand of kookiness? Whose ulterior motives are worse, the creep trying to make a buck off of families of affected children, or established authorities whose reputations and pocketbooks will suffer when, say, Tagamet is instantly no longer bringing in billions easing ulcers?

J

I think it isn't quite so black-and-white as that. For an article that brings some nuance to that story, see https://www.csicop.org/si/show/bacteria_ulcers_and_ostracism_h._pylori_and_the_making_of_a_myth