I will delete Boingboing from my RSS and bookmarks forever if this happens.
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Paradoxical coins are by definition non-rational, and thus have no place in our economic system. They should be removed from circulation.
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Exactly! Itâs all about the coinâs orientation relative to the red line, not to how we see it.
Yep. I feel this is one of those problems that also functions as a litmus test for visual thinkers. I realize Iâm revealing my own lack of confidence when I say this, but the speed at which I figured this out only made me doubt that Iâd even understood the prompt. Because any problem labeled a âparadoxâ (itâs not) must have a trick to itâŚright?
It reminds me of another problem, also kindling for visual thinkers: Imagine a mountain with only one walkable path. In the morning (7am) you start out at the bottom and make your way up the path (at varying speed), reaching the top by nightfall. The next morning (again, 7am), you descend the mountain along this same path towards the bottom (again at varying speed) and arrive by sunset. Is there a point along this path that you reached at the exact same time of day on both trips? (Another way this question is asked: is there a time of day when you were at the same point along the path on both of your trips?)
Yes of course. Just a simple grade school astronomy/navigation, at least for this old 1960âs Cub Scout. It is for the same reason that a sidereal day (a day with respect to the stars rather than the Sun) is ~236 seconds days shorter than a normal (Solar) day. Hint if you donât get it yet: The Moon never revolves with respect to the earth during its one lunar month-long orbit. Thus, in one half lunar month, a moon-based observer would see the firmament (i.e. distant & therefore "fixed"stars) rotate exactly 180 degrees. Make sure you understand that before moving on. Now, imagine that the Moon were to also to revolve exactly once per lunar month: then the same observer would see the stars rotate a full 360 degrees during any HALF orbit. The rotating and orbiting coin exactly models the once-per-month orbiting and revolving Moon.
Try it with two coins. Now you can finally understand why stars rise above the horizon at different times on successive days. (There is one MORE day each normal year with respect to the stars than to the Sun due to Earth orbit.) Neat, huh?
If your speed varies, then itâs possible, but not guaranteed that thereâs a point where you would have been at the same time of day both days. If you moved at constant speed, of course there would be: the half-way point, as long as you assume that sunrise and sunset were at the same time each day (which they arenât on this planet.)
This is what makes intuitive sense to me:
The outer circle rotating about the inner circle is undergoing two kinds of rotation: One from its spin, and one from its rotation about the inner circle. Therefore it completes a revolution before its traveled the same distance.
Howâs that?
Imagine you firmly attach an arrow to the end of a rod, and then swing that rod 180 degrees (like a baton). The arrow will be pointing exactly the other direction. This is true no matter where you attach the arrow and no matter where you are rotating the rod about. By rotating the rod, you are also rotating the arrow by the same amount.
Now imagine we firmly fix the outer circle to an invisible rod that runs through the center of the inner circle, and we dot the outer circle where it touches the inner one. As we rotate that rod, the outer circle rotates by the same amount, after rotating 180 degrees, the dot is now on the âopposite sideâ of the outer circle.
Now imagine the outer circle spinning so that it doesnât ârubâ against the inner circle at the point of contact but instead rolls around it. Whether or not the outer circle is spinning, it still travels the same distance along the inner circleâs circumference (which is equal to its own circumference). But that extra spin makes it complete a revolution from our perspective before its traveled the full circumference.
Sounds fuzzy. Lets put math to this.
Letâs nail down the rate of spin. For every degree that the outer circle rotates, How much must the outer circle spin so that it doesnât rub against the inner circle? Well if its not rubbing or slipping then by definition the distance the outer circle travels along the inner circleâs circumference must be equal to the distance the outer circle spins through.
Let R_inner be the radius of the inner circle, and θ_about_inner be the angle of rotation of the outer circle about the inner circle.
Let R_outer be the radius of the outer circle, and θ_about_outer be the angle of the outer circle about its own center.
So 2(pi) * R_inner * θ_about_inner / 360 = 2(pi) * R_outer * θ_about_outer / 360
Or circumference of inner circle traveled = circumference of outer circle spun through
We cancel out the constants so that its just:
R_inner * θ_about_inner = R_outer * θ_about_outer
Since we know R_inner and R_outer are equal (since inner and outer circle are the same size), we can cancel those out and are left with:
θ_about_inner = θ_about_outer
For every degree the outer circle travels/rotates along the inner circle, it also spins a degree. So after traveling 180 degrees along the inner circle it will spin an additional 180 degrees and appear to complete a revolution.
As is evident in the math, this only works out like this when the circles are the same size. If the inner circleâs radius was twice as large as the outer circleâs radius, then for every degree the outer circle traveled along the inner circleâs circumference it would rotate 2 degrees. After traveling 120 degrees along the inner circle, the outer circle would spin an additional 240 degrees for a total of 1 revolution.
In that scenario the inner circle wouldâve traveled:
2(pi) * (R_inner) * 120/360 = 2(pi) * (2 * R_outer) * 1/3 = 2/3 of the outer circles circumference
There are two other ways Iâve seen this underlying math expressed:
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If you rotate your head along the point of contact (effectively removing the rotation about the inner circle), you clearly see that at the 180 degree mark its only spun 180 degrees and still needs to spin 180 more as @RConBB and @Nelsie point out.
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As the radius of the inner circle approaches infinity, the rotation due to rotation about the inner circle disappears and it essentially becomes a circle spinning along a line. As the radius of the inner circle approaches zero, the distance it travels through in one revolution also approaches zero, as @Dave_Craig pointed out.
The trick, of course, is imagining both journeys traced along the same spatial path, in different directions, but with a jerky (in time) line. Even if you canât determine where exactly the two paths intersect in time, you know, since they completely overlap in space, that at some point âat the exact same time of dayâ, they did indeed intersect.
Since I canât be arsed right now, exercise for the reader**: Does the fact that âthe same time of dayâ will shift from the day of ascent to the âday of descentâ (dibs on band name) invalidate the reasoning?
The two frames of reference may be different, but do the paths not still intersect at some point in a description of time, no matter how it might be described relative to different frames of reference?
** typical pontificator speech for âI think I know the answer, but I really am tired, and not convinced that I could come up with the answer even when well restedâ
Youâre right of course. Iâm terrible with visual-spatial thinking. Ever since I was a little kid even. I canât âimagine imagesâ. Recognizing something visually feels entirely automatic. When I try to visualize a clock for instance, at best my mental image can hold onto a circle. If I start adding numbers to the face of the clock, the circle disappears.
The original question is why is the short arc half as long as the red at the bottom. The bottom line is a bit of a red herring. Move it up so that the ends are in the centers of the coins. Now draw an arc from the beginning and ending centers at the top illustration. They both travel the same distance with one revolution, so the center-to-center measurements are the same.
The smaller arc is half the size because it traces half the radius of the longer.
More of an exercise in grade-school geometry than Kessel Run.
The speed limit to light isnât really âproposedâ, but that aside, you are kind of misunderstanding it. Suppose you and your infinite friends all accelerated to half the speed of light (to me). So from my point of view you are all going 1.5x108 m/s. The speed of light would still be, from your perspective, 3x108 m/s faster than you. So you stay at that speed and all of your friends start traveling half the speed of light again, that is, they are all travelling 1.5x108 m/s relative to you, but the speed of light is still the same to them. Since you have infinite friends, this can continue forever. Itâs Xenoâs paradox made reality. If you accelerate infinitely from your own perspective but you will never achieve that âfiniteâ speed from mine.
It helps me to remember that distance and time are fundamentally measured in the same units, so the speed of light is merely a unit conversion. This may not be helpful to anyone else.
And mathematicians.
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If this is an example of a âparadoxâ, I donât know what all these quantam physicists are having such problems explaining the universe. The distance traveled at any clock postion of either coin would be a parabolic graph⌠in A, itâs amplitude would be higher, thus it would require less x axis distance to equal the same length as a lower amplitude parabola.
The paradox is in your grammar.
Welcome to BB.
The trick with this puzzle is that weâre asked to look at the rotation of the coin relative to our point of view. What you need to do instead is look at the rotation of the coin relative to the path itâs rolling on.
With the bottom picture, the coin does a full 360° rotation relative to its path - a straight line.
In the top picture, relative to its path (the perimeter of the other coin), the coin only rotates 180°, but the path itself also rotates 180°, so the two add up to the full 360° that we see in the bottom picture.
Similarily, if the coin was rolling inside a concave curve (like inside a bowl), it would actually need to roll further to achieve the same rotation*.
*I think. I havenât actually tested that thought experiment.
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I donât know whether to be more impressed by your math knowledge or your markdown skillsâŚ
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Does this mean the earlier tracks on an LP would sound better, since theyâre using more âgroove-inchesâ per second of audio?
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Now youâre getting into âaudiophileâ territory, and you know that the only thing that matters is how much you spent to upgrade every single component in your system. And after that, if you didnât bother to have your headphones professionally burned in, then, well, we might as well be talking about wax cylinders.
Tinfoil has a much better bass response than wax. Do you only listen to spoken word albums for the deaf?
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the degree of hypotenuse is mirrored on a circle [of equal radius] vs. a flat line so in due course it is twice as long.
