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Why are these three equivalence relations special? [closed]
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Consider the set of all possible bivectors $mathfrak{B}$ in $mathbb{R}^3$. Then there are three possible equivalence relations.
Equipollence: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever a has the same normal vector as b.
Orientation: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever the boundary of a rotates in the same way as b.
Size: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever Area(a) = Area(b).
Now is there anything else that singles out these three equivalence relations as special? There are many other equivalence relations we can put on $mathfrak{B}$, like equivalence depending on whether one of the edges is the same. But they all seem stupid. These three equivalence relations seem special compared to all of the others. Why?
differential-forms equivalence-relations
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
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closed as off-topic by José Carlos Santos, abc..., Robert Z, Jyrki Lahtonen, Xander Henderson Feb 10 at 14:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – abc..., Robert Z, Jyrki Lahtonen, Xander Henderson
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Consider the set of all possible bivectors $mathfrak{B}$ in $mathbb{R}^3$. Then there are three possible equivalence relations.
Equipollence: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever a has the same normal vector as b.
Orientation: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever the boundary of a rotates in the same way as b.
Size: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever Area(a) = Area(b).
Now is there anything else that singles out these three equivalence relations as special? There are many other equivalence relations we can put on $mathfrak{B}$, like equivalence depending on whether one of the edges is the same. But they all seem stupid. These three equivalence relations seem special compared to all of the others. Why?
differential-forms equivalence-relations
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
$endgroup$
closed as off-topic by José Carlos Santos, abc..., Robert Z, Jyrki Lahtonen, Xander Henderson Feb 10 at 14:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – abc..., Robert Z, Jyrki Lahtonen, Xander Henderson
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Consider the set of all possible bivectors $mathfrak{B}$ in $mathbb{R}^3$. Then there are three possible equivalence relations.
Equipollence: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever a has the same normal vector as b.
Orientation: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever the boundary of a rotates in the same way as b.
Size: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever Area(a) = Area(b).
Now is there anything else that singles out these three equivalence relations as special? There are many other equivalence relations we can put on $mathfrak{B}$, like equivalence depending on whether one of the edges is the same. But they all seem stupid. These three equivalence relations seem special compared to all of the others. Why?
differential-forms equivalence-relations
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
$endgroup$
Consider the set of all possible bivectors $mathfrak{B}$ in $mathbb{R}^3$. Then there are three possible equivalence relations.
Equipollence: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever a has the same normal vector as b.
Orientation: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever the boundary of a rotates in the same way as b.
Size: The equivalence relation $(mathfrak{B}, sim)$ such that a ~ b whenever Area(a) = Area(b).
Now is there anything else that singles out these three equivalence relations as special? There are many other equivalence relations we can put on $mathfrak{B}$, like equivalence depending on whether one of the edges is the same. But they all seem stupid. These three equivalence relations seem special compared to all of the others. Why?
differential-forms equivalence-relations
differential-forms equivalence-relations
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
edited Mar 9 at 9:42
Lord_Farin
15.7k636110
15.7k636110
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
asked Jan 28 at 13:49
ErotemeObelusErotemeObelus
775718
775718
closed as off-topic by José Carlos Santos, abc..., Robert Z, Jyrki Lahtonen, Xander Henderson Feb 10 at 14:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – abc..., Robert Z, Jyrki Lahtonen, Xander Henderson
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by José Carlos Santos, abc..., Robert Z, Jyrki Lahtonen, Xander Henderson Feb 10 at 14:33
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – abc..., Robert Z, Jyrki Lahtonen, Xander Henderson
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
add a comment |
1 Answer
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$begingroup$
$mathbb{R}^3$ is special in the sense that the dimension of the bivector space is the same as the original space (which is usually not the case).
Under the usual identification of bivectors with vectors via the usual cross product we get that equipollence is the same es representatives being in the same linear subspace and for size we get that representatives have the same magnitude.
I'm not sure what you mean by 'boundary rotates in the same way'. Orientation is a good equivalence class for n-vectors in n-dimensional space.
Similarly 'compare one of the "edges" ' is not really an equivalence relation as $ a wedge b = - b wedge a $ so we'd have a problem with reflexivity
The other two equivalence relations are quite special as you can see they are invariant under action of $ SO(3) $ (the usual rotations) on your vector space. Any symmetric partition of $ mathbb{R} $ can be extended into an equivalence relation of a similar kind: e.g. all bivectors with size < 1 are equivalent and all others are equivalent only if they are proportional. I guess there is some additional level of 'specialness' in the choice of equivalence classes on $ mathbb{R} $ in the examples you've chosen but I'll leave it at that...
answered Feb 5 at 0:52
RadostRadost
65512
$endgroup$
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
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– ErotemeObelus
Feb 5 at 17:16
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$mathbb{R}^3$ is special in the sense that the dimension of the bivector space is the same as the original space (which is usually not the case).
Under the usual identification of bivectors with vectors via the usual cross product we get that equipollence is the same es representatives being in the same linear subspace and for size we get that representatives have the same magnitude.
I'm not sure what you mean by 'boundary rotates in the same way'. Orientation is a good equivalence class for n-vectors in n-dimensional space.
Similarly 'compare one of the "edges" ' is not really an equivalence relation as $ a wedge b = - b wedge a $ so we'd have a problem with reflexivity
The other two equivalence relations are quite special as you can see they are invariant under action of $ SO(3) $ (the usual rotations) on your vector space. Any symmetric partition of $ mathbb{R} $ can be extended into an equivalence relation of a similar kind: e.g. all bivectors with size < 1 are equivalent and all others are equivalent only if they are proportional. I guess there is some additional level of 'specialness' in the choice of equivalence classes on $ mathbb{R} $ in the examples you've chosen but I'll leave it at that...
answered Feb 5 at 0:52
RadostRadost
65512
$endgroup$
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
$endgroup$
– ErotemeObelus
Feb 5 at 17:16
add a comment |
$begingroup$
$mathbb{R}^3$ is special in the sense that the dimension of the bivector space is the same as the original space (which is usually not the case).
Under the usual identification of bivectors with vectors via the usual cross product we get that equipollence is the same es representatives being in the same linear subspace and for size we get that representatives have the same magnitude.
I'm not sure what you mean by 'boundary rotates in the same way'. Orientation is a good equivalence class for n-vectors in n-dimensional space.
Similarly 'compare one of the "edges" ' is not really an equivalence relation as $ a wedge b = - b wedge a $ so we'd have a problem with reflexivity
The other two equivalence relations are quite special as you can see they are invariant under action of $ SO(3) $ (the usual rotations) on your vector space. Any symmetric partition of $ mathbb{R} $ can be extended into an equivalence relation of a similar kind: e.g. all bivectors with size < 1 are equivalent and all others are equivalent only if they are proportional. I guess there is some additional level of 'specialness' in the choice of equivalence classes on $ mathbb{R} $ in the examples you've chosen but I'll leave it at that...
answered Feb 5 at 0:52
RadostRadost
65512
$endgroup$
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
$endgroup$
– ErotemeObelus
Feb 5 at 17:16
add a comment |
$begingroup$
$mathbb{R}^3$ is special in the sense that the dimension of the bivector space is the same as the original space (which is usually not the case).
Under the usual identification of bivectors with vectors via the usual cross product we get that equipollence is the same es representatives being in the same linear subspace and for size we get that representatives have the same magnitude.
I'm not sure what you mean by 'boundary rotates in the same way'. Orientation is a good equivalence class for n-vectors in n-dimensional space.
Similarly 'compare one of the "edges" ' is not really an equivalence relation as $ a wedge b = - b wedge a $ so we'd have a problem with reflexivity
The other two equivalence relations are quite special as you can see they are invariant under action of $ SO(3) $ (the usual rotations) on your vector space. Any symmetric partition of $ mathbb{R} $ can be extended into an equivalence relation of a similar kind: e.g. all bivectors with size < 1 are equivalent and all others are equivalent only if they are proportional. I guess there is some additional level of 'specialness' in the choice of equivalence classes on $ mathbb{R} $ in the examples you've chosen but I'll leave it at that...
answered Feb 5 at 0:52
RadostRadost
65512
$endgroup$
$mathbb{R}^3$ is special in the sense that the dimension of the bivector space is the same as the original space (which is usually not the case).
Under the usual identification of bivectors with vectors via the usual cross product we get that equipollence is the same es representatives being in the same linear subspace and for size we get that representatives have the same magnitude.
I'm not sure what you mean by 'boundary rotates in the same way'. Orientation is a good equivalence class for n-vectors in n-dimensional space.
Similarly 'compare one of the "edges" ' is not really an equivalence relation as $ a wedge b = - b wedge a $ so we'd have a problem with reflexivity
The other two equivalence relations are quite special as you can see they are invariant under action of $ SO(3) $ (the usual rotations) on your vector space. Any symmetric partition of $ mathbb{R} $ can be extended into an equivalence relation of a similar kind: e.g. all bivectors with size < 1 are equivalent and all others are equivalent only if they are proportional. I guess there is some additional level of 'specialness' in the choice of equivalence classes on $ mathbb{R} $ in the examples you've chosen but I'll leave it at that...
answered Feb 5 at 0:52
RadostRadost
65512
answered Feb 5 at 0:52
RadostRadost
65512
answered Feb 5 at 0:52
RadostRadost
65512
answered Feb 5 at 0:52
RadostRadost
65512
65512
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
$endgroup$
– ErotemeObelus
Feb 5 at 17:16
add a comment |
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
$endgroup$
– ErotemeObelus
Feb 5 at 17:16
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
$endgroup$
– ErotemeObelus
Feb 5 at 17:16
$begingroup$
Here is my problem. Points can have an orientation of "things coming in" (internal) or "things going out" (external). I'm trying to see if there are any MORE properties one can assign to points, even in spaces other than 3D. Does that make sense? I will award you the points but I cannot give you the green checkmark because I need you to either find generalizations of orientation or explain why what I'm asking for is impossible.
$endgroup$
– ErotemeObelus
Feb 5 at 17:16
add a comment |
