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.
Satellites in sun-synchronous orbits need to precess (their orbital plane must slowly rotate) because of this situation to actually stay sun-synchronous.
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.
> 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.
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.
> 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.)
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?
> 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?
[+] [-] GuB-42|2 years ago|reply
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.
[+] [-] _Microft|2 years ago|reply
https://en.wikipedia.org/wiki/Sun-synchronous_orbit
[+] [-] mannykannot|2 years ago|reply
https://en.wikipedia.org/wiki/Solar_time
This is one of the causes of the analemma.
https://en.wikipedia.org/wiki/Analemma
[+] [-] thunderbong|2 years ago|reply
I guess I'm one of today's lucky 10,000!
https://xkcd.com/1053/
[+] [-] defvar|2 years ago|reply
[+] [-] jojobas|2 years ago|reply
[+] [-] xpe|2 years ago|reply
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|2 years ago|reply
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.
[+] [-] dahart|2 years ago|reply
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|2 years ago|reply
[+] [-] symmetricsaurus|2 years ago|reply
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?
[+] [-] xpe|2 years ago|reply
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?
[+] [-] jojobas|2 years ago|reply
[+] [-] seesawtron|2 years ago|reply
[0] https://math.stackexchange.com/questions/1351058/circle-revo...