From Wikipedia:
Aristotle's wheel paradox is a paradox from the Greek work Mechanica traditionally attributed to Aristotle. There are two wheels, one within the other, whose rims take the shape of two circles with different diameters. The wheels roll without slipping for a full revolution. The paths traced by the bottoms of the wheels are straight lines, which are apparently the wheels' circumferences. But the two lines have the same length, so the wheels must have the same circumference, contradicting the assumption that they have different sizes: a paradox.
From Wolfram Mathworld:
A paradox mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric circles of different diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large circle with points on the small circle, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two circumferences of different sized circles are equal, which is impossible.
Do you think you know the Explanation?
I'll tell you in a sec, but see if you can figure it out first.
Just a second. You have a second, don't you? Ok, Ok, fifteen seconds. Thirty seconds? Naw, too much thinking.
No cheating now ...
... here it comes ...
From Wikipedia:
The fallacy is the assumption that the smaller wheel indeed traces out its circumference, without ensuring that it, too, rolls without slipping on a fixed surface. In fact, it is impossible for both wheels to perform such motion. Physically, if two joined concentric wheels with different radii were rolled along parallel lines then at least one would slip; if a system of cogs was used to prevent slippage then the wheels would jam. A modern approximation of such an experiment is often performed by car drivers who park too close to a curb. The car's outer tire rolls without slipping on the road surface while the inner hubcap both rolls and slips across the curb; the slipping is evidenced by a screeching noise.[1]
From Wolfram Wathworld:
The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length (or even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same:
(the cardinality of the continuum), so the points of any of these can be put in a one-to-one correspondence with those of any other.From Paul Simon:
Slip sliding away .... you know the nearer your destination the more likely you're slip sliding away ....
It's T-5 days until I "celebrate" the 2nd anniversary of this weblog, which makes me cringe because of the way I was raised, it is the height of hubris to call attention to one's self.
ReplyDeleteOr in a nutshell: I'm no Charlie Sheen.
I really can't wait for that day to pass so much as come. I don't know what I'll write, probably me being me, to do a shoutout to those who have helped me SO much in the last two years commit myself to this War on Ignorance I am waging, notably Andrew Thomas of Swansea Wales, Phil Warnell, Stephen D. "Plato"hagel, Sabine "Bee" Hossenfelder and Stefan Scherer, Christine Dantas, Garrett Lisi, Jérôme Chauvet, Pat Ballew, Dave Richeson, Tomasso Dorigo, Chad Orzel and Emmy and Steelykid and "Good job!" Kate, Neil Bates, Bob Park, Phil Plait, whomever runs Universe Today, Paul Glister, Feng Luo, Doyle Knight, Weinberg and 't Hooft and Veltman, Lee Smolin, Peter Woit, Renate Loll, Carlo Rovelli, Hawking and Penrose, and John Baez and all the fine people at n-Category Cafe... and my apologies to the dozens of others I've unintentionally left off, who inspire me, and continue to inspire me, nevertheless.
And yeah you too Lubos, you utter Big Energy capitalist tool you, God forbid anyone mention Math Phys and not mention YOU, huh?
The War on Ignorance must be fought, and here's hoping we win, because the other side is very organized, albeit in a random sort of way, heh.
Hi Steven,
ReplyDeleteI said : ”I tend to look at your wheel problem in considering the center of motion as the only commonality and to envision the travel of each wheel from a point on each circumference mapping out respective waves, as having the frequency the same and yet the amplitude of each vary although the total forward journeyed length being the same. In other words it’s a magician’s trick, where an arbitrary redirection of focused attention has us fail to recognize the true symmetry as it relates to invariance.”
You said: “Sorry Phil, it's not a magician's trick, it's a real paradox that vexed thousands across centuries until friction and slipping and sliding were found to be the explanation. It's one of the rare times that Engineering trumped Mathematics than the other way around. I explain at my latest weblog entry,..” .”
The first quote is what I wrote on Bee’s and Stefan’s blog and the second your response. What I gave was a mathematical explanation fully laying out the situation. That is as the frequency of the wave and the forward distance travelled respective of each circumference remain same, the amplitudes which maps out their respective two dimensional distance of travel and area as they relate to the centre point encompassed are different. The paradox only arises if one places one dimensional (mathematical) restrictions onto a two dimensional situation (reality). Actually I believe the Wolfram explanation is essentially saying the same only its meaning is easy lost amongst all the jargon.
Best,
Phil
Yeah, I know what you're saying Phil, my critique was about the way you expressed it, to whit:
ReplyDeleteIt's all about "cardinality", Phil, that the infinite numbers between 0 and 1 have a direct one-to-one correspondence between the infinite numbers on the complete number line.
Which makes NO SENSE the first time you hear it, but yah, after thought, it does. There is no need to bring amplitudes into the discussion, although, being the expert Philosopher that you are (and yes, you are IMO, zero sarcasm my friend, you're awesome), I'm more than willing to bring imaginary numbers and complex arithmetic from first principles into the discussion, if you are.
Hi Steven,
ReplyDeleteI have some level of comfort when it comes to matters of the Aleph and yet that would be as much as to say I’m fairly comfortable as to finding myself in reality more generally. The reason I chose to us waves as they contrast to straight lines, is I find such a concept in itself a more everyday familiar way to express the manifestation and consequences related to cardinality.
That is there are mappings one can do on a one on one basis and others that you simply can’t; such as the difference between the countables and uncountables from the Cantorian perspective. The question to be asked is how can all this extraness be accounted for and explained, which is to attempt to have it expressed mathematically in terms of dimensions (degrees of freedom) or physically if you prefer. Then of course we can ask if there truly is any distinction to be made between the two conceptualizations.
Best,
Phil
It comes from the uncountability of the real line, so directly from Cantor's diagonal theorem.
ReplyDeleteNice article Steven. It got amazed and confused for (little) while :)
Best,
OK Phil, I gotcha. Thanks to you too, Jerome.
ReplyDeleteWhat this also points out to me is how an animation in this case, or a mental construct in the case of the ancients (and ourselves today ... were we not so dependent on 'puters), conflicts with the REAL world.
In other words, if those were rubber band lines, the band on the top wheel would stretch, the one on the bottom, not.
Alternately, if they were gears, say a rack and pinion such that no stretching or slipping and sliding were allowed, the wheels would lock and nothing would move.
Hi Steven,
ReplyDeleteProbably the easiest way to state it is that physical space requires dimension(s) while numbers don’t. Never the less, with Cantor’s revelations respective of the Aleph, although numbers don’t have dimensions, they do have cardinality, which can accommodate their existence.
“Thus Descartes was not so far from the truth when he believed he must exclude the existence of an empty space. The notion indeed appears absurd, as long as physical reality is seen exclusively in ponderable bodies. It requires the idea of the field as the representative of reality, in combination with the general principle of relativity, to show the true kernel of Descartes' idea; there exists no space "empty of field".”
-Albert Einstein, “Relativity: The Special and the General Theory”, Crown Publishing (fifth edition, 1954)
Best,
Phil
👏👍
DeleteHi Phil,
ReplyDeleteYeah I know. Thanks for the Einstein quote too, you know what a sucker I am for those.
Btw, I was thinking what a cool science project this would be (math fair project ? ... WHO runs those?!), or as an exhibit in a Science Museum. I was going to say Mathematics Museum but where are those, exactly? :-(
In any event, the exhibit would first show the animation, at the risk of breaking brains, natch. THEN, use springs where the lines are, and it becomes self-evident when the smaller wheel's spring stretches more than the larger one's.
Hey, I'll build that if anyone wishes to send me money. :-)