Here is another interesting phenomenon.
youtu.be/MFzDaBzBlL0
You can't ride this bike because a mechanic changed an important thing...
youtu.be/MFzDaBzBlL0
You can't ride this bike because a mechanic changed an important thing...
@gambit_Pnav said in #3:
> Concave mirrors, like spoons, would produce upside-down images.
Real and Inverted
> Concave mirrors, like spoons, would produce upside-down images.
Real and Inverted
@e4e5f4exf4nicht
That was interesting. I wonder what would happen if you also rigged the pedals so you had to pedal backwards to go forward.
That was interesting. I wonder what would happen if you also rigged the pedals so you had to pedal backwards to go forward.
@PxJ said in #14:
> It's because left-right is not a property of the space itself, but of the observer.
i think euclidian space is separable if you put an origing somewhere and take a subspace to create a partition.
the normal to that subspace would then become an orientation basis.
I think you meant, that one needs a referential like i did. which could be an observer location. i guess it depend on what one is willing to put on in the definition of space... is that going in circles...
replace mirror with a space transformation like a point based reflection (maybe i am not using the right english math terms, i translate from my french way back geometry notions). one could have refection based on a subspace. in 3D that could be a point, an axis or even a plane.
all of these can be looked at, keeping the same original referential orienting structure. and describe a vector image by such transformation, and ask about its orientation.
so not sure how the mirror thing translates in to this.. or if i made some fallacies..
> It's because left-right is not a property of the space itself, but of the observer.
i think euclidian space is separable if you put an origing somewhere and take a subspace to create a partition.
the normal to that subspace would then become an orientation basis.
I think you meant, that one needs a referential like i did. which could be an observer location. i guess it depend on what one is willing to put on in the definition of space... is that going in circles...
replace mirror with a space transformation like a point based reflection (maybe i am not using the right english math terms, i translate from my french way back geometry notions). one could have refection based on a subspace. in 3D that could be a point, an axis or even a plane.
all of these can be looked at, keeping the same original referential orienting structure. and describe a vector image by such transformation, and ask about its orientation.
so not sure how the mirror thing translates in to this.. or if i made some fallacies..
@dboing said in #24:
> so not sure how the mirror thing translates in to this.. or if i made some fallacies..
Well a mirror is exactly a reflection with respect to a plane.
Also "separable" usually means that it has a countable dense subset.
> so not sure how the mirror thing translates in to this.. or if i made some fallacies..
Well a mirror is exactly a reflection with respect to a plane.
Also "separable" usually means that it has a countable dense subset.
Now once you've chosen an orientation of the 2D plane. If you take an arbitrary vector u you can find a second vector v such that (u, v) has positive orientation. You could say that "v is on the left of u".
But if you consider a point in space, you can't "choose a vector that has positive orientation" because the notion of orientation applies to a pair of vectors, not to a single vector.
The vector you pick in the beginning of my explanation is the direction in which the observer is looking at. The point in the second paragraph is the position where they stand.
A reflection with respect to a line in the 2D space reverts the orientation whereas a point based reflection does not.
But if you consider a point in space, you can't "choose a vector that has positive orientation" because the notion of orientation applies to a pair of vectors, not to a single vector.
The vector you pick in the beginning of my explanation is the direction in which the observer is looking at. The point in the second paragraph is the position where they stand.
A reflection with respect to a line in the 2D space reverts the orientation whereas a point based reflection does not.
@PxJ said in #25:
> Well a mirror is exactly a reflection with respect to a plane.
> Also "separable" usually means that it has a countable dense subset.
dense and countable. sounds like number theory.. could we stay in plain intuitive geometry... i mean a 2D plane will separate the 3D euclidian space into 2 half open set each of 3D ball coverings. ok that was compromise into topology... boundary stuff.
i forgot want dense meant.. is it about containing all its sequences and their limit. or being closed in itself.. i don't recall why this is related to separable.. but it must be. sequence and enumeration, is one way to look at topology.
but human intuition of the space they could swim in if it were liquid, is pretty good at not making logical errors about topological notions.. and putting a giant plane in a pool, is going to separate the pool into 2 parts... the point of the separatrix (here a plane) in the normal direction to that plane could serve as an origin on the line colinear to that normal.. or the plane partitioning the pool into pool 1 and pool 2.
maybe you are right about the need for 2 vectors... as putting a referential like i mention does install an orientated set of unit vectors that can completely describe a vector of interest, and its reflection with respect to some line or point or plane.. i feel like there is a circle somewhere. will review tomorrow.. or you can correct me meanwhile. i like that, memory might accumulate small errors, would be nice to clean that up, been relying on internal version of that past knowledge....
> Well a mirror is exactly a reflection with respect to a plane.
> Also "separable" usually means that it has a countable dense subset.
dense and countable. sounds like number theory.. could we stay in plain intuitive geometry... i mean a 2D plane will separate the 3D euclidian space into 2 half open set each of 3D ball coverings. ok that was compromise into topology... boundary stuff.
i forgot want dense meant.. is it about containing all its sequences and their limit. or being closed in itself.. i don't recall why this is related to separable.. but it must be. sequence and enumeration, is one way to look at topology.
but human intuition of the space they could swim in if it were liquid, is pretty good at not making logical errors about topological notions.. and putting a giant plane in a pool, is going to separate the pool into 2 parts... the point of the separatrix (here a plane) in the normal direction to that plane could serve as an origin on the line colinear to that normal.. or the plane partitioning the pool into pool 1 and pool 2.
maybe you are right about the need for 2 vectors... as putting a referential like i mention does install an orientated set of unit vectors that can completely describe a vector of interest, and its reflection with respect to some line or point or plane.. i feel like there is a circle somewhere. will review tomorrow.. or you can correct me meanwhile. i like that, memory might accumulate small errors, would be nice to clean that up, been relying on internal version of that past knowledge....
@dboing said in #27:
> dense and countable. sounds like number theory..
Nah it's topology.
> i mean a 2D plane will separate the 3D euclidian space into 2 half open set each of 3D ball coverings.
Yep but that corresponds to choosing a direction east-west, not an orientation left-right.
> i forgot want dense meant.. is it about containing all its sequences and their limit. or being closed in itself..
It means its closure is the whole space (ie every open neighbourhood of every point in your space contains a point from your dens subset). Doesn't matter for the matter at hand anyway. Was just nitpicking on terminology.
> maybe you are right about the need for 2 vectors...
Of course I am right. If you look at your friend in the eyes, your left will be his right and vice-versa. Because the notion of left and right is defined with respect to the direction (vector) in which you are looking. It is exactly the same as the notion of orientation in the 2D space.
> dense and countable. sounds like number theory..
Nah it's topology.
> i mean a 2D plane will separate the 3D euclidian space into 2 half open set each of 3D ball coverings.
Yep but that corresponds to choosing a direction east-west, not an orientation left-right.
> i forgot want dense meant.. is it about containing all its sequences and their limit. or being closed in itself..
It means its closure is the whole space (ie every open neighbourhood of every point in your space contains a point from your dens subset). Doesn't matter for the matter at hand anyway. Was just nitpicking on terminology.
> maybe you are right about the need for 2 vectors...
Of course I am right. If you look at your friend in the eyes, your left will be his right and vice-versa. Because the notion of left and right is defined with respect to the direction (vector) in which you are looking. It is exactly the same as the notion of orientation in the 2D space.
@PxJ said in #28:
> Of course I am right. If you look at your friend in the eyes, your left will be his right and vice-versa. Because the notion of left and right is defined with respect to the direction (vector) in which you are looking. It is exactly the same as the notion of orientation in the 2D space.
putting people in and the possibility of who's looking at who is disorienting for me. Seems supplementary degree of freedom to store in short term memory for further reasoning.
just one person holding an arrow so the only flip is if the arrow was pointing toward the mirror. I would hold the arrow going away from me, but in the refection is would be pointing toward me (not my reflection). i think we need to keep the referential.
but all of that seems a distraction from the op (or how i viewed the gist of it):
I think it was more a psychological question, about why the writing serifs or discriminating geometrical gizmos in alphabet are more robust to a flip in one direction or another. maybe the mirror or reflection is not needed. just go for the flip question which we can all understand, however one needs to set a referential to make it a proper transformation.
as a favor to a fading mathematician, could you remind me about the separable definition.. about what kind of set, does it have to be the whole space... can you use only countable basis of open neigborhoods to get to connected set, simply connected, convex sets, and interior of a subset.
see for me I don't need the dense notion, (does not mean it can't be define or even used) to define a separating curve in a 2D real euclidian space. Think of the jordan lemna i think it is.. a basic 2D topological result that any closed curve in R2, separates it into 2 distinct parts. ok these parts would be open sets. The notion of inside and outside cound then be defined.. holes can be defined. for me that is useful topology.. but i may have forgotten the exact plumbing, so it would be nice if you would refresh my memory... aboout separable something.
> Of course I am right. If you look at your friend in the eyes, your left will be his right and vice-versa. Because the notion of left and right is defined with respect to the direction (vector) in which you are looking. It is exactly the same as the notion of orientation in the 2D space.
putting people in and the possibility of who's looking at who is disorienting for me. Seems supplementary degree of freedom to store in short term memory for further reasoning.
just one person holding an arrow so the only flip is if the arrow was pointing toward the mirror. I would hold the arrow going away from me, but in the refection is would be pointing toward me (not my reflection). i think we need to keep the referential.
but all of that seems a distraction from the op (or how i viewed the gist of it):
I think it was more a psychological question, about why the writing serifs or discriminating geometrical gizmos in alphabet are more robust to a flip in one direction or another. maybe the mirror or reflection is not needed. just go for the flip question which we can all understand, however one needs to set a referential to make it a proper transformation.
as a favor to a fading mathematician, could you remind me about the separable definition.. about what kind of set, does it have to be the whole space... can you use only countable basis of open neigborhoods to get to connected set, simply connected, convex sets, and interior of a subset.
see for me I don't need the dense notion, (does not mean it can't be define or even used) to define a separating curve in a 2D real euclidian space. Think of the jordan lemna i think it is.. a basic 2D topological result that any closed curve in R2, separates it into 2 distinct parts. ok these parts would be open sets. The notion of inside and outside cound then be defined.. holes can be defined. for me that is useful topology.. but i may have forgotten the exact plumbing, so it would be nice if you would refresh my memory... aboout separable something.
@dboing I will assume you know the definition of a topological space.
A subset D of a topological space E is dense if for every element x of E and every open neighbourhood U of x, U also contains an element of D.
D is countable if there is a bijection between D and the natural numbers.
Now if E is a topological space which has a dense, countable subset, then we say E is separable.
Exemple: the real numbers, with the topology from the order, are separable because the subset of rationals is open and dense.
Counter-example: an uncountable totally disconnected topological space is not separable, because every element is an open neighbourhood of itself, so a dense subset has to be the whole space.
A subset D of a topological space E is dense if for every element x of E and every open neighbourhood U of x, U also contains an element of D.
D is countable if there is a bijection between D and the natural numbers.
Now if E is a topological space which has a dense, countable subset, then we say E is separable.
Exemple: the real numbers, with the topology from the order, are separable because the subset of rationals is open and dense.
Counter-example: an uncountable totally disconnected topological space is not separable, because every element is an open neighbourhood of itself, so a dense subset has to be the whole space.
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