A group of people decided to seat together and talk about some casual math problems
The one problem I keep remembering is a bet about 1000 men walking by on the street in a row. Random chance is not guaranteed - especially when it's suddenly a parade :)
I'm probably butchering this, but in my mind it is something like:
1. From the squirrels frame of reference and local coordinate system, the man has remained "in front" of the squirrel. The squirrel is orienting and rotating in sync with the man and therefore has not observed that the man has "gone round" it.
2. From our perspective (and on reflection from the man), the man has circled the squirrel in the global coordinate system of the scene.
As the reader we assume that our perspective is the authoritative one, but I am sure the squirrel disagrees.
One could argue that the moon is orbiting the sun. The fact that it's orbit is a little wobbly because of interferes from the earth is a rounding detail, no?
> An enduring myth about the Moon is that it doesn't rotate. While it's true that the Moon keeps the same face to us, this only happens because the Moon rotates at the same rate as its orbital motion, a special case of tidal locking called synchronous rotation.
My colleagues once spent a good hour trying to explain this fact to me and I still really struggle to accept it. I can see that the moon is rotating on its own axis from the point of view of a space that is external to the system it forms with the earth. But then isn’t everything on earth rotating about its own axis with respect to that external space? It seems arbitrary to isolate the moon from all this other stuff and make a special case of it…
1. Unlike position and velocity, which are relative (there is no given "origin" for them, no way to say where a thing is or how fast it's moving except relative to other things), rotation is absolute. A thing is either rotating or not, regardless of its relation to other things. Objects that rotate "experience (centrifugal) forces as a result" or "require (centripetal) forces to hold them together" depending on how you choose to describe it. This is detectable: hook two weights together with a newton-meter in space and the newton-meter will read non-zero when the assemblage is rotating, zero when not. The reading tells you how fast it is rotating regardless of any external reference point. (An equivalent device to detect position or velocity is not possible, but it is for acceleration.)
2. Yes, everything "at rest" on earth is in fact rotating at the rate the earth rotates. If you stand on the equator at midday and do not rotate you will be standing on your head at midnight.
>no way to say where a thing is except relative to other things
This is always true. The origin is just a thing that other things are relative to. It's just as possible to define an origin in the real world as it is on a piece of graph paper.
Thanks for this explanation. If I understand correctly then, the moon requires some centripetal force in order not to dissipate due to its rotation whereas e.g. my head or the Eiffel Tower do not because they are not subject to absolute rotation.
Indeed. The Eiffel tower and your head do both have some (extremely small) centripetal force compensating for their rotation along with the earth.
(You can break that down in different ways, i.e. use various choices of generalised coordinates to describe it, so exactly what constitutes "centripetal", "centrifugal", "gravitational", "tidal", etc. forces depends on that. I'm being pretty vague in how I decribe it. Regardless, rotation is absolute, or in other words the equations of physics take a different form in a rotating frame of reference than in a non-rotating one.)
> But then isn’t everything on earth rotating about its own axis with respect to that external space?
From the point of view of the moon, for the purposes of action due to gravity, anything on Earth is essentially part of the Earth, not an entity that is massive enough to be considered separately. The aggregate centre of mass is what counts. Similar for the Sun looking at the Earth/Moon system: from that PoV Earth+Moon is it a single mass with a centre somewhere between the two major masses that form it.
If the Moon where sufficiently consistent in its shape and density, it could rotate freely in any direction while orbiting the Earth, that fact that it is more dense on one side means that it is more energy efficient for it to spin in step with its orbit such that the dense side keeps facing us. If something massive hit the moon (let's assume this somehow happens without significantly affecting its orbit or causing significant problems for Earth too!) it might push the rotation off for a bit, but it would slowly be pulled back into sync. If something sufficiently massive simply landed on the moon, that would affect the mass distribution and the exact face that points at us would slowly change to reach a new equilibrium.
Pick the sun as reference: the moon rotates. Pick the earth as reference: the moon rotates. Stand on the moon and pick any star as reference: the moon rotates.
Yes but from the point of view of the earth the moon does not appear to be rotating around its own axis since it is tidally locked. In that sense it’s confusing to me to distinguish it from everything else on earth but the comment above about centripetal force clarifies this for me I think
I remember this from a Martin Gardner “aha!” book (or one in that series) which my parents gave to me somewhere around the age of 10. Those books had a profound effect on me.
(Of course, the text in the linked article predates Gardner’s work.)
I remember this being mentioned by Charles Peirce as an argument for pragmatism (the philosophical kind): It's a nonsensical question unless you can phrase it in terms where the answer has some practical consequence.
That's true. The man did not go around the squirrel. They both were orbiting some point near the center of the tree trunk. Otherwise, one could say that the farther point on the Moon's surface is going around the point that is facing Earth.
I take it that the squirrel didn't circle the man?
Two squirrels running around the same tree, are they circling each other? Or is it that when two bodies are orbiting the same center, then the body with the larger orbit is circling the one with the smaller?
What is the definition of "circling"?
Say instead of just walking, the man was laying down a net/barricade around the tree. As soon as the man completes the circumference, the squirrel must admit that it has been gone around.
Now let us suppose the squirrel is at the same distance as the man.
Has the man have gone around the squirrel and the squirrel around the man?
If it's only radii less than the other, where is the limit?
To get it I think I have to re-frame it like this:
If you hold out an object toward the centre, you clearly go around it when completing an orbit.
If you keep extending that to the origin but then go beyond, so your arm is longer than the radius, then you still go around it, until your arm reaches twice the radius.
It is still not clear to me. The periodicity of their orbit around the tree is the same. I think this is an instance of us meaning different things by “go around”
> Some years ago, being with a camping party in the mountains, I returned from a solitary ramble to find everyone engaged in a ferocious metaphysical dispute. The corpus of the dispute was a squirrel—a live squirrel supposed to be clinging to one side of a tree-trunk; while over against the tree’s opposite side a human being was imagined to stand. This human witness tries to get sight of the squirrel by moving rapidly round the tree, but no matter how fast he goes, the squirrel moves as fast in the opposite direction, and always keeps the tree between himself and the man, so that never a glimpse of him is caught. The resultant metaphysical problem now is this: Does the man go round the squirrel or not? He goes round the tree, sure enough, and the squirrel is on the tree; but does he go round the squirrel? In the unlimited leisure of the wilderness, discussion had been worn threadbare. Everyone had taken sides, and was obstinate; and the numbers on both sides were even. Each side, when I appeared, therefore appealed to me to make it a majority. Mindful of the scholastic adage that whenever you meet a contradiction you must make a distinction, I immediately sought and found one, as follows: “Which party is right,” I said, “depends on what you practically mean by ’going round’ the squirrel. If you mean passing from the north of him to the east, then to the south, then to the west, and then to the north of him again, obviously the man does go round him, for he occupies these successive positions. But if on the contrary you mean being first in front of him, then on the right of him, then behind him, then on his left, and finally in front again, it is quite as obvious that the man fails to go round him, for by the compensating movements the squirrel makes, he keeps his belly turned towards the man all the time, and his back turned away. Make the distinction, and there is no occasion for any farther dispute. You are both right and both wrong according as you conceive the verb ’to go round’ in one practical fashion or the other.”
"Mathematics can be fun" is an awesome book
A group of people decided to seat together and talk about some casual math problems
The one problem I keep remembering is a bet about 1000 men walking by on the street in a row. Random chance is not guaranteed - especially when it's suddenly a parade :)
This tickled my brain in a nice way.
I'm probably butchering this, but in my mind it is something like:
1. From the squirrels frame of reference and local coordinate system, the man has remained "in front" of the squirrel. The squirrel is orienting and rotating in sync with the man and therefore has not observed that the man has "gone round" it.
2. From our perspective (and on reflection from the man), the man has circled the squirrel in the global coordinate system of the scene.
As the reader we assume that our perspective is the authoritative one, but I am sure the squirrel disagrees.
One could argue that the moon is orbiting the sun. The fact that it's orbit is a little wobbly because of interferes from the earth is a rounding detail, no?
> An enduring myth about the Moon is that it doesn't rotate. While it's true that the Moon keeps the same face to us, this only happens because the Moon rotates at the same rate as its orbital motion, a special case of tidal locking called synchronous rotation.
https://science.nasa.gov/resource/the-moons-rotation/
My colleagues once spent a good hour trying to explain this fact to me and I still really struggle to accept it. I can see that the moon is rotating on its own axis from the point of view of a space that is external to the system it forms with the earth. But then isn’t everything on earth rotating about its own axis with respect to that external space? It seems arbitrary to isolate the moon from all this other stuff and make a special case of it…
1. Unlike position and velocity, which are relative (there is no given "origin" for them, no way to say where a thing is or how fast it's moving except relative to other things), rotation is absolute. A thing is either rotating or not, regardless of its relation to other things. Objects that rotate "experience (centrifugal) forces as a result" or "require (centripetal) forces to hold them together" depending on how you choose to describe it. This is detectable: hook two weights together with a newton-meter in space and the newton-meter will read non-zero when the assemblage is rotating, zero when not. The reading tells you how fast it is rotating regardless of any external reference point. (An equivalent device to detect position or velocity is not possible, but it is for acceleration.)
2. Yes, everything "at rest" on earth is in fact rotating at the rate the earth rotates. If you stand on the equator at midday and do not rotate you will be standing on your head at midnight.
>no way to say where a thing is except relative to other things
This is always true. The origin is just a thing that other things are relative to. It's just as possible to define an origin in the real world as it is on a piece of graph paper.
Thanks for this explanation. If I understand correctly then, the moon requires some centripetal force in order not to dissipate due to its rotation whereas e.g. my head or the Eiffel Tower do not because they are not subject to absolute rotation.
They're rotating too.
If you rotate as part of some larger rotating thing then you still rotate. (You also move around.) It's all absolute.
Indeed. The Eiffel tower and your head do both have some (extremely small) centripetal force compensating for their rotation along with the earth.
(You can break that down in different ways, i.e. use various choices of generalised coordinates to describe it, so exactly what constitutes "centripetal", "centrifugal", "gravitational", "tidal", etc. forces depends on that. I'm being pretty vague in how I decribe it. Regardless, rotation is absolute, or in other words the equations of physics take a different form in a rotating frame of reference than in a non-rotating one.)
Why is the Eiffel Tower not rotating?
It is.
> But then isn’t everything on earth rotating about its own axis with respect to that external space?
From the point of view of the moon, for the purposes of action due to gravity, anything on Earth is essentially part of the Earth, not an entity that is massive enough to be considered separately. The aggregate centre of mass is what counts. Similar for the Sun looking at the Earth/Moon system: from that PoV Earth+Moon is it a single mass with a centre somewhere between the two major masses that form it.
If the Moon where sufficiently consistent in its shape and density, it could rotate freely in any direction while orbiting the Earth, that fact that it is more dense on one side means that it is more energy efficient for it to spin in step with its orbit such that the dense side keeps facing us. If something massive hit the moon (let's assume this somehow happens without significantly affecting its orbit or causing significant problems for Earth too!) it might push the rotation off for a bit, but it would slowly be pulled back into sync. If something sufficiently massive simply landed on the moon, that would affect the mass distribution and the exact face that points at us would slowly change to reach a new equilibrium.
Pick the sun as reference: the moon rotates. Pick the earth as reference: the moon rotates. Stand on the moon and pick any star as reference: the moon rotates.
From which reference frame would it not rotate?
Yes but from the point of view of the earth the moon does not appear to be rotating around its own axis since it is tidally locked. In that sense it’s confusing to me to distinguish it from everything else on earth but the comment above about centripetal force clarifies this for me I think
We can see the stars moving behind the moon, thus it must be rotating with us.
But is a geostationary satellite going around the Earth?
Due to the Earth's rotation, the man could be circling the squirrel even when standing still.
I remember this from a Martin Gardner “aha!” book (or one in that series) which my parents gave to me somewhere around the age of 10. Those books had a profound effect on me.
(Of course, the text in the linked article predates Gardner’s work.)
I remember this being mentioned by Charles Peirce as an argument for pragmatism (the philosophical kind): It's a nonsensical question unless you can phrase it in terms where the answer has some practical consequence.
That's true. The man did not go around the squirrel. They both were orbiting some point near the center of the tree trunk. Otherwise, one could say that the farther point on the Moon's surface is going around the point that is facing Earth.
From the title I thought this was going to be a variation on the bear and swimmer puzzle.
https://www.quantamagazine.org/can-math-help-you-escape-a-hu...
I take it that the squirrel didn't circle the man? Two squirrels running around the same tree, are they circling each other? Or is it that when two bodies are orbiting the same center, then the body with the larger orbit is circling the one with the smaller? What is the definition of "circling"?
Similarly, a rolling wheel (without slipping and on flat ground) does a pure rotation around the touch point and not its center.
It took me longer than it should to get this!
On reflection, does the squirrel consider itself to have been "gone round"? I don't think so.
Say instead of just walking, the man was laying down a net/barricade around the tree. As soon as the man completes the circumference, the squirrel must admit that it has been gone around.
Now let us suppose the squirrel is at the same distance as the man.
Has the man have gone around the squirrel and the squirrel around the man?
If it's only radii less than the other, where is the limit?
To get it I think I have to re-frame it like this:
If you hold out an object toward the centre, you clearly go around it when completing an orbit.
If you keep extending that to the origin but then go beyond, so your arm is longer than the radius, then you still go around it, until your arm reaches twice the radius.
The article says the man was trying to see the squirrel's back implying a larger radius.
But yeah if your circuit completely fits inside the other person's circuit, then you've been gone around, no matter how slow or fast you both are.
To get what?
That the man technically went around the squirrel without ever having caught up to it.
It is still not clear to me. The periodicity of their orbit around the tree is the same. I think this is an instance of us meaning different things by “go around”
The original:
> Some years ago, being with a camping party in the mountains, I returned from a solitary ramble to find everyone engaged in a ferocious metaphysical dispute. The corpus of the dispute was a squirrel—a live squirrel supposed to be clinging to one side of a tree-trunk; while over against the tree’s opposite side a human being was imagined to stand. This human witness tries to get sight of the squirrel by moving rapidly round the tree, but no matter how fast he goes, the squirrel moves as fast in the opposite direction, and always keeps the tree between himself and the man, so that never a glimpse of him is caught. The resultant metaphysical problem now is this: Does the man go round the squirrel or not? He goes round the tree, sure enough, and the squirrel is on the tree; but does he go round the squirrel? In the unlimited leisure of the wilderness, discussion had been worn threadbare. Everyone had taken sides, and was obstinate; and the numbers on both sides were even. Each side, when I appeared, therefore appealed to me to make it a majority. Mindful of the scholastic adage that whenever you meet a contradiction you must make a distinction, I immediately sought and found one, as follows: “Which party is right,” I said, “depends on what you practically mean by ’going round’ the squirrel. If you mean passing from the north of him to the east, then to the south, then to the west, and then to the north of him again, obviously the man does go round him, for he occupies these successive positions. But if on the contrary you mean being first in front of him, then on the right of him, then behind him, then on his left, and finally in front again, it is quite as obvious that the man fails to go round him, for by the compensating movements the squirrel makes, he keeps his belly turned towards the man all the time, and his back turned away. Make the distinction, and there is no occasion for any farther dispute. You are both right and both wrong according as you conceive the verb ’to go round’ in one practical fashion or the other.”
— William James, Pragmatism, 1907