4 ms·

Circles rolling on circles (2014)

57 points by montalbano 3y ago

defvar 3y ago
Here is a nice animation of this puzzle. https://www.geogebra.org/m/v3a437ux https://www.geogebra.org/m/v3a437ux
symmetricsaurus 3y ago
Interesting case! 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. This is of course what the article is saying pretty much. Are there other situations that require a similar reasoning?
- jojobas 3y ago
  Sidereal day vs solar day.
- xpe 3y ago
  > 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. I understand the article, but I don't I think I quite get or buy your explanation. I'm curious about your linguistic separation of "local" and "global". What is local and what is global in this situation? I wonder if you are the frame of reference concept? Could you unpack what you mean? I don't think the terms quite fit. You did when you wrote it; do you still? By this I mean: do you think the way you're using the terms local and global would be intuitive to, say, an audience with a high-school level background in geometry? This is an empirical question, but my inclination would be to say 'probably not'. I find the article's emphasis on decomposing sliding (i.e. translation) from rotating to be much more intuitive. (Of course people will vary.) Still, I'm curious. I'm searching for a sense in which this linguistic global/local distinction adds explanatory power. Care to elaborate?
  - adammarples 3y ago
    Yes, the missing phrase from the article was clearly "frame of reference". Ie the coin makes 4 rotations if we stretch the larger circle out into a straight line, which we do implicitly when doing our mathematical calculation, but we should be aware that this frame of reference is also making a full rotation.
  - symmetricsaurus 3y ago
    Yes I’m thinking in terms of different frames of reference. Imagine that you’re a tiny ant that lives on the large circle. When you push the small circle around the large circle you will see it make four rotations before you get back to the starting point. This is the local frame of reference. Now imagine you’re a giant living in the space with the circles. You see the small circle do five full rotations. Four are the same that the ant sees but you also see the ant itself do one rotation simultaneously as it walks along the large circle making five in total. It would be more accurate to say that you have 4 local rotations and that the local frame rotates one full turn in the global frame. Does this make sense to someone with a high-school level of geometry knowledge? Not as I wrote it I initially (but there was no such goal). The analogy with the giant and the ant together with some nice illustrations maybe?
  - mannykannot 3y ago
    Here's another way of putting it, without explicitly mentioning frames of reference: Before starting, imagine drawing a radius on the moving coin from its center to its point of contact with the stationary coin. It is pointing down. As the moving coin goes clockwise around the fixed coin, this radius will also rotate clockwise around the center of the moving coin. The next time the end of the radius is in contact with the fixed coin is after the moving coin has gone 1/4 of the way around the fixed one - but now the radius is pointing to the left. It has passed the point where it is pointing down again (which is when the moving coin has made one full revolution) and gone a 1/4 turn beyond that. Repeat until you are back at the top again.
  - xpe 3y ago
    typo fix: "I wonder if you are using the frame of reference concept? Could you unpack what you mean?"
xpe 3y ago
> 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. Comments: 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 2. For a broader audience, I'd probably rephrase the above as: "Concepts are human representations; they are different than the actual phenomena." 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). 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.
- rhn_mk1 3y ago
  You're reading too much into it. "Reality" here is used as a metaphor to aid in learning. Given equivalent mathematical descriptions of something, they are all just as true, or just as false. None of them is more "real". The metaphor uses that word, but it doesn't make them unequal in any other way than comprehension by humans. As humans, especially in mathematics, concepts are all we have. Dig deep enough, and you'll hit the wall of the unknown rather than the real. What is an electron, really? Is this question even answerable, or is it more refined concepts all the way down? Can you even perceive anything real, raw, without interpreting it into a high level concept like a color or a neuron firing (Kant comes to mind). The decomposition is taking place. The decomposition isn't taking place. It's all the same.
  - xpe 3y ago
    > You're reading too much into it. What do you mean by this? I'm getting it wrong? I'm mistaking the author's meaning? (How do you know what the author's meaning is?) I shouldn't reflect on this? Something else?
jojobas 3y ago
Rolling on the inside is funner. At R/r = 2, it takes one revolution of the smaller circle to ride the entire larger circle.
GuB-42 3y ago
This, by the way, is why there is a thing called "sidereal time", with days that are slightly shorter than 24h. 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.
- thunderbong 3y ago
  That's incredible! I've never thought of it that way! Thanks a lot. I guess I'm one of today's lucky 10,000! https://xkcd.com/1053/ https://xkcd.com/1053/
  - Natsu 3y ago
    It's funny, I usually only think about circles this way when trying to construct mutually tangent circular arcs, because if you have one circle inside (or outside) the other, their arcs are tangent at the point where they touch. I used circles a lot when doing CADs because they let you measure all the points a certain distance away from a given point, so intersecting two of them lets you do a lot of construction, like finding the two possibilities for the center of a circular arc that passes through two points of a given radius, assuming that they're not too far apart for that to exist.
- _Microft 3y ago
  Satellites in sun-synchronous orbits need to precess (their orbital plane must slowly rotate) because of this situation to actually stay sun-synchronous. https://en.wikipedia.org/wiki/Sun-synchronous_orbit https://en.wikipedia.org/wiki/Sun-synchronous_orbit
- mannykannot 3y ago
  the 24-hour day is a construct of mean solar time. Because the speed of the Earth varies through the year (fastest at perihelion - see Kepler I and II), the apparent or true solar day varies in duration throughout the year. https://en.wikipedia.org/wiki/Solar_time https://en.wikipedia.org/wiki/Solar_time This is one of the causes of the analemma. https://en.wikipedia.org/wiki/Analemma https://en.wikipedia.org/wiki/Analemma
dahart 3y ago
> I figured the answer must be four revolutions. So imagine my surprise when I saw that the answer was given to be five! 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.)
- learn_more 3y ago
  If the smaller circle rolled along the INSIDE of the larger circle, it would be one LESS rotation rather than one MORE.
seesawtron 3y ago
Despite the visualizations, I found the mathematical solution described here [0] quite convincing. [0] https://math.stackexchange.com/questions/1351058/circle-revolutions-rolling-around-another-circle https://math.stackexchange.com/questions/1351058/circle-revo...

57 points by montalbano 3y ago

Here is a nice animation of this puzzle. https://www.geogebra.org/m/v3a437ux https://www.geogebra.org/m/v3a437ux

Interesting case! 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. This is of course what the article is saying pretty much. Are there other situations that require a similar reasoning?

jojobas 3y ago
Sidereal day vs solar day.
xpe 3y ago
> 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. I understand the article, but I don't I think I quite get or buy your explanation. I'm curious about your linguistic separation of "local" and "global". What is local and what is global in this situation? I wonder if you are the frame of reference concept? Could you unpack what you mean? I don't think the terms quite fit. You did when you wrote it; do you still? By this I mean: do you think the way you're using the terms local and global would be intuitive to, say, an audience with a high-school level background in geometry? This is an empirical question, but my inclination would be to say 'probably not'. I find the article's emphasis on decomposing sliding (i.e. translation) from rotating to be much more intuitive. (Of course people will vary.) Still, I'm curious. I'm searching for a sense in which this linguistic global/local distinction adds explanatory power. Care to elaborate?
- adammarples 3y ago
  Yes, the missing phrase from the article was clearly "frame of reference". Ie the coin makes 4 rotations if we stretch the larger circle out into a straight line, which we do implicitly when doing our mathematical calculation, but we should be aware that this frame of reference is also making a full rotation.
- symmetricsaurus 3y ago
  Yes I’m thinking in terms of different frames of reference. Imagine that you’re a tiny ant that lives on the large circle. When you push the small circle around the large circle you will see it make four rotations before you get back to the starting point. This is the local frame of reference. Now imagine you’re a giant living in the space with the circles. You see the small circle do five full rotations. Four are the same that the ant sees but you also see the ant itself do one rotation simultaneously as it walks along the large circle making five in total. It would be more accurate to say that you have 4 local rotations and that the local frame rotates one full turn in the global frame. Does this make sense to someone with a high-school level of geometry knowledge? Not as I wrote it I initially (but there was no such goal). The analogy with the giant and the ant together with some nice illustrations maybe?
- mannykannot 3y ago
  Here's another way of putting it, without explicitly mentioning frames of reference: Before starting, imagine drawing a radius on the moving coin from its center to its point of contact with the stationary coin. It is pointing down. As the moving coin goes clockwise around the fixed coin, this radius will also rotate clockwise around the center of the moving coin. The next time the end of the radius is in contact with the fixed coin is after the moving coin has gone 1/4 of the way around the fixed one - but now the radius is pointing to the left. It has passed the point where it is pointing down again (which is when the moving coin has made one full revolution) and gone a 1/4 turn beyond that. Repeat until you are back at the top again.
- xpe 3y ago
  typo fix: "I wonder if you are using the frame of reference concept? Could you unpack what you mean?"

xpe 3y ago

> 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. Comments: 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 2. For a broader audience, I'd probably rephrase the above as: "Concepts are human representations; they are different than the actual phenomena." 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). 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.

rhn_mk1 3y ago
You're reading too much into it. "Reality" here is used as a metaphor to aid in learning. Given equivalent mathematical descriptions of something, they are all just as true, or just as false. None of them is more "real". The metaphor uses that word, but it doesn't make them unequal in any other way than comprehension by humans. As humans, especially in mathematics, concepts are all we have. Dig deep enough, and you'll hit the wall of the unknown rather than the real. What is an electron, really? Is this question even answerable, or is it more refined concepts all the way down? Can you even perceive anything real, raw, without interpreting it into a high level concept like a color or a neuron firing (Kant comes to mind). The decomposition is taking place. The decomposition isn't taking place. It's all the same.
- xpe 3y ago
  > You're reading too much into it. What do you mean by this? I'm getting it wrong? I'm mistaking the author's meaning? (How do you know what the author's meaning is?) I shouldn't reflect on this? Something else?

jojobas 3y ago

Rolling on the inside is funner. At R/r = 2, it takes one revolution of the smaller circle to ride the entire larger circle.

GuB-42 3y ago

This, by the way, is why there is a thing called "sidereal time", with days that are slightly shorter than 24h. 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.

thunderbong 3y ago
That's incredible! I've never thought of it that way! Thanks a lot. I guess I'm one of today's lucky 10,000! https://xkcd.com/1053/ https://xkcd.com/1053/
- Natsu 3y ago
  It's funny, I usually only think about circles this way when trying to construct mutually tangent circular arcs, because if you have one circle inside (or outside) the other, their arcs are tangent at the point where they touch. I used circles a lot when doing CADs because they let you measure all the points a certain distance away from a given point, so intersecting two of them lets you do a lot of construction, like finding the two possibilities for the center of a circular arc that passes through two points of a given radius, assuming that they're not too far apart for that to exist.
_Microft 3y ago
Satellites in sun-synchronous orbits need to precess (their orbital plane must slowly rotate) because of this situation to actually stay sun-synchronous. https://en.wikipedia.org/wiki/Sun-synchronous_orbit https://en.wikipedia.org/wiki/Sun-synchronous_orbit
mannykannot 3y ago
the 24-hour day is a construct of mean solar time. Because the speed of the Earth varies through the year (fastest at perihelion - see Kepler I and II), the apparent or true solar day varies in duration throughout the year. https://en.wikipedia.org/wiki/Solar_time https://en.wikipedia.org/wiki/Solar_time This is one of the causes of the analemma. https://en.wikipedia.org/wiki/Analemma https://en.wikipedia.org/wiki/Analemma

dahart 3y ago

> I figured the answer must be four revolutions. So imagine my surprise when I saw that the answer was given to be five! 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.)

learn_more 3y ago
If the smaller circle rolled along the INSIDE of the larger circle, it would be one LESS rotation rather than one MORE.

seesawtron 3y ago

Despite the visualizations, I found the mathematical solution described here [0] quite convincing. [0] https://math.stackexchange.com/questions/1351058/circle-revolutions-rolling-around-another-circle https://math.stackexchange.com/questions/1351058/circle-revo...