This, by the way, is why there is a thing called "sidereal time", with days that are slightly shorter than 24h.<p>The commonly used 24h days are solar days, defined relative to the sun, but since the earth goes around the sun too, that makes an extra rotation relative to the star background, which means a year has 366.25 sidereal days instead of the usual 365.25.
Here is a nice animation of this puzzle. <a href="https://www.geogebra.org/m/v3a437ux" rel="nofollow">https://www.geogebra.org/m/v3a437ux</a>
> Each of the above explanations describes the circle's movement as a decomposition into rotation and revolution, but in reality no such decomposition is taking place.<p>Comments:<p>1. This is a specific instance of a widely taught principle from Buddhism: “Concepts are not real things; a conceptualized world is a dead world. Living actualities lose their life when put into concepts.” ― Gyomay M. Kubose, Everyday Suchness: Buddhist Essays on Everyday Living<p>2. For a broader audience, I'd probably rephrase the above as: "Concepts are human representations; they are <i>different</i> than the actual phenomena."<p>3. The above decomposition is represented in many of our brains. In that sense it is "real" as any other form of physical matter. Why? The concept is (somehow) encoded in the structure and relationships of neurons (as I understand it).<p>4. I'm torn: saying that "decomposition" isn't "taking place" is simultaneously insightful and obvious. In any case, as phrased, for a modern audience, it risks missing the point; namely, a decomposition is a useful way of understanding the world. For example, the idea of decomposing motion into {rotation and translation} is similar to decomposing the position of a point by referring to its {position in a coordinate system}, whether it be Cartesian, polar, barycentric, or otherwise. Doing so helps us bring analytic methods to bear.
> I figured the answer must be four revolutions. So imagine my surprise when I saw that the answer was given to be five!<p>The answer is four from the reference frame of the small moving circle (the fifth rotation belongs now to the big circle). Imagine two circles fixed and both rotating together, like connected gears. The question is fun but only surprising because it’s ambiguous and assuming a specific reference frame without saying it (which would be a clue to what’s really being asked.)
Interesting case!<p>You can see it is R/r local revolutions of the small circle. Then you need to add one global revolution from going around the large circle. So R/r + 1.<p>This is of course what the article is saying pretty much.<p>Are there other situations that require a similar reasoning?
Despite the visualizations, I found the mathematical solution described here [0] quite convincing.<p>[0] <a href="https://math.stackexchange.com/questions/1351058/circle-revolutions-rolling-around-another-circle" rel="nofollow">https://math.stackexchange.com/questions/1351058/circle-revo...</a>