A Logical Concern about Dimensional Spaces

[ryan]

So in physics and mathematics, superpositions are allowed. For example, if I apply 4 Newtons of force on an object in the positive x direction and Bob pushes with 3 Newtons in the positive y direction, then that is equivalent to 5 Newtons of force about 38 degrees in the positive y direction from the positive x direction.

But dimensions are only perpendicular to each other. So on a 2 dimensional plane, there should only be up, down, left and right directions. What would it mean to say that there is a diagonal direction on a 2 dimensional plane?

 

[Juma]

A diagonal is a line between to nonadjacent corners in a polygon. I have no clue what you want it to mean here...

 

[ryan]

no idea?

 

[J842P]

You seem to be conflating several related concepts. Mathematically, a dimension is a property of a space. In Newtonian mechanics, this is generally a vector space like\(\mathbb{R}\)³. So, dimensions aren't perpendicular to each other, that's a misuse of the concept. People pick perpendicular basis vectors for a coordinate system because it's convenient, but any 3 linearly independent vectors in \(\mathbb{R}\)³ will form a basis. Furthermore, just because you pick a basis doesn't mean that those are the only directions, they are merely three directions that you use to speak about the space, that is, you can define a linear combination of those three vectors to define any other vector, and the constants of that linear combination are the coordinates!

 

[ryan]

You seem to be conflating several related concepts. Mathematically, a dimension is a property of a space. In Newtonian mechanics, this is generally a vector space like\(\mathbb{R}\)³. So, dimensions aren't perpendicular to each other, that's a misuse of the concept. People pick perpendicular basis vectors for a coordinate system because it's convenient, but any 3 linearly independent vectors in \(\mathbb{R}\)³ will form a basis. Furthermore, just because you pick a basis doesn't mean that those are the only directions, they are merely three directions that you use to speak about the space, that is, you can define a linear combination of those three vectors to define any other vector, and the constants of that linear combination are the coordinates!

Two linearly independent vectors with a basis in R2, for example, are really only independent if one is orthogonal (perpendicular) component relative to the other.

A 3 dimensional space only needs the three vectors that are perpendicular to each other.

 

[beero1000]

You seem to be conflating several related concepts. Mathematically, a dimension is a property of a space. In Newtonian mechanics, this is generally a vector space like\(\mathbb{R}\)³. So, dimensions aren't perpendicular to each other, that's a misuse of the concept. People pick perpendicular basis vectors for a coordinate system because it's convenient, but any 3 linearly independent vectors in \(\mathbb{R}\)³ will form a basis. Furthermore, just because you pick a basis doesn't mean that those are the only directions, they are merely three directions that you use to speak about the space, that is, you can define a linear combination of those three vectors to define any other vector, and the constants of that linear combination are the coordinates!

Two linearly independent vectors with a basis in R2, for example, are really only independent if one is orthogonal (perpendicular) component relative to the other.

A 3 dimensional space only needs the three vectors that are perpendicular to each other.

That is not true. Take some time and look up the definitions for  linear independence,  orthogonality,  basis (linear algebra), and  dimension (vector space).

 

[ryan]

Two linearly independent vectors with a basis in R2, for example, are really only independent if one is orthogonal (perpendicular) component relative to the other.

A 3 dimensional space only needs the three vectors that are perpendicular to each other.

That is not true. Take some time and look up the definitions for  linear independence,  orthogonality,  basis (linear algebra), and  dimension (vector space).

It's been a while since I took linear algebra, but what the hell else is needed to create an R3 vector space? What about (1,0,0),
(0,1,0),
(0,0,1)?

 

[beero1000]

That is not true. Take some time and look up the definitions for  linear independence,  orthogonality,  basis (linear algebra), and  dimension (vector space).

It's been a while since I took linear algebra, but what the hell else is needed to create an R3 vector space? What about (1,0,0),
(0,1,0),
(0,0,1)?

What does that have to do with the quoted posts?

If you want anyone to understand what you are trying to get at, you will need to think hard and post what you mean using precise language. Right now, all I can tell is that you are confused about basic concepts of linear algebra.

 

[ryan]

Let's recap:

ryan: A 3 dimensional space only needs the three vectors that are perpendicular to each other.

beero: that's not true

ryan: (1,0,0),
(0,1,0),
(0,0,1)

 

[beero1000]

Let's recap:

ryan: A 3 dimensional space only needs the three vectors that are perpendicular to each other.

beero: that's not true

ryan: (1,0,0),
(0,1,0),
(0,0,1)

I understand that your statements make sense to you. However, they are not leaving your head as sensible statements to anyone else.

Think CAREFULLY, and say exactly what you mean - using the language of linear algebra, not informal language like "needs" or "has", etc.

 

[ryan]

Let's recap:

ryan: A 3 dimensional space only needs the three vectors that are perpendicular to each other.

beero: that's not true

ryan: (1,0,0),
(0,1,0),
(0,0,1)

I understand that your statements make sense to you. However, they are not leaving your head as sensible statements to anyone else.

Think CAREFULLY, and say exactly what you mean - using the language of linear algebra, not informal language like "needs" or "has", etc.

Can you agree that walking on a 1 dimensional line requires you only to able to be on that line? If you do, then imagine a perpendicular line crossing the one you are on. You can now walk either line but only by going through the intersection first. Now imagine a "completely dense" number of lines intersecting with a completely dense set of lines perpendicular to the others making up a field.

At any point on the field, you will be at a coordinate (a,b); a, b are elements of R. Now you can only follow north/south "lines" and west/east "lines". At no point do you follow any other direction other than the lines that you can travel on.

 

[beero1000]

I understand that your statements make sense to you. However, they are not leaving your head as sensible statements to anyone else.

Think CAREFULLY, and say exactly what you mean - using the language of linear algebra, not informal language like "needs" or "has", etc.

Can you agree that walking on a 1 dimensional line requires you only to able to be on that line? If you do, then imagine a perpendicular line crossing the one you are on. You can now walk either line but only by going through the intersection first. Now imagine a "completely dense" number of lines intersecting with a completely dense set of lines perpendicular to the others making up a field.

At any point on the field, you will be at a coordinate (a,b); a, b are elements of R. Now you can only follow north/south "lines" and west/east "lines". At no point do you follow any other direction other than the lines that you can travel on.

 

[Juma]

I understand that your statements make sense to you. However, they are not leaving your head as sensible statements to anyone else.

Think CAREFULLY, and say exactly what you mean - using the language of linear algebra, not informal language like "needs" or "has", etc.

Can you agree that walking on a 1 dimensional line requires you only to able to be on that line? If you do, then imagine a perpendicular line crossing the one you are on. You can now walk either line but only by going through the intersection first. Now imagine a "completely dense" number of lines intersecting with a completely dense set of lines perpendicular to the others making up a field.

At any point on the field, you will be at a coordinate (a,b); a, b are elements of R. Now you can only follow north/south "lines" and west/east "lines". At no point do you follow any other direction other than the lines that you can travel on.

What is your purpose of this text?

I thought at first that you wanted to describe a coordinate system but the reason seems rather of some sort of metaphysical speculation of how a coordinate system maps a 2 dim surface.

But then it completely avoids me why you must follow a specific line when you are on a plane?

And you use linear (in)dependence wrong: linear dependences is a property versus a set of vectors. Any vector in a set of two vectors are linear independent unless the vectors have the same direction. They do not need to be orthogonal.

 

[ryan]

Sorry Juma, but beero1000 wasted all of the time I had for this tonight.

 

[Juma]

Sorry Juma, but beero1000 wasted all of the time I had for this tonight.

Dont blame the truthsayer.

 

[ryan]

Can you agree that walking on a 1 dimensional line requires you only to able to be on that line? If you do, then imagine a perpendicular line crossing the one you are on. You can now walk either line but only by going through the intersection first. Now imagine a "completely dense" number of lines intersecting with a completely dense set of lines perpendicular to the others making up a field.

At any point on the field, you will be at a coordinate (a,b); a, b are elements of R. Now you can only follow north/south "lines" and west/east "lines". At no point do you follow any other direction other than the lines that you can travel on.

What is your purpose of this text?

I thought at first that you wanted to describe a coordinate system but the reason seems rather of some sort of metaphysical speculation of how a coordinate system maps a 2 dim surface.

And you use linear (in)dependence wrong: linear dependences is a property versus a set of vectors. Any vector in a set of two vectors are linear independent unless the vectors have the same direction. They do not need to be orthogonal.

I meant "has an orthogonal component".

But then it completely avoids me why you must follow a specific line when you are on a plane?

Any line, you must follow the lines because that is all that is there.

 

[Bomb#20]

So in physics and mathematics, superpositions are allowed. For example, if I apply 4 Newtons of force on an object in the positive x direction and Bob pushes with 3 Newtons in the positive y direction, then that is equivalent to 5 Newtons of force about 38 degrees in the positive y direction from the positive x direction.

But dimensions are only perpendicular to each other. So on a 2 dimensional plane, there should only be up, down, left and right directions. What would it mean to say that there is a diagonal direction on a 2 dimensional plane?

It sounds to me like what you're trying to get at is the idea of a Normed vector space. Logically, the space of points in a universe is a separate concept from the space of distances among those points; in principle you could have two universes with the same points but where the distance from A to B is different in universe 1 from what it is in universe 2, and the shortest path from A to B takes you through point C in one universe but not in the other. So when you wake up and find yourself in an N-dimensional universe, you still have to make observations in order to figure out what geometry you're in.

To say "there should only be up, down, left and right directions" is to say that intuitively you expect to find yourself in a universe with a "p-norm" where p=1. What it means to say that there is a diagonal direction is simply to say that this intuition is wrong for the universe in question -- that in fact the local p is some other value, typically 2 -- you're in a universe where the Pythagorean Theorem applies: distance is (up² + left²)^1/2. You could have found yourself in a universe with a different p. I spend my working life in the universe you expect, where p=1 and distance is (up¹ + left¹)^1/1, i.e., up plus left. There are no diagonal directions -- to get a wire from transistor A to transistor B you have to go only up, down, left and right, because the routing software won't allow you to do otherwise. (It's written that way because to try to go diagonally will probably block a hundred other wires you haven't laid out yet.)

The point is, to say "So in physics and mathematics, superpositions are allowed." isn't quite right. In mathematics, whether superpositions are allowed is up to you. You can allow them or not, and then study the implications of your choice. In physics, whether they're allowed is up to the world you find yourself in. You can't rely on intuition; you have to observe the world and see what its p is.

 

[ryan]

So in physics and mathematics, superpositions are allowed. For example, if I apply 4 Newtons of force on an object in the positive x direction and Bob pushes with 3 Newtons in the positive y direction, then that is equivalent to 5 Newtons of force about 38 degrees in the positive y direction from the positive x direction.

But dimensions are only perpendicular to each other. So on a 2 dimensional plane, there should only be up, down, left and right directions. What would it mean to say that there is a diagonal direction on a 2 dimensional plane?

It sounds to me like what you're trying to get at is the idea of a Normed vector space. Logically, the space of points in a universe is a separate concept from the space of distances among those points; in principle you could have two universes with the same points but where the distance from A to B is different in universe 1 from what it is in universe 2, and the shortest path from A to B takes you through point C in one universe but not in the other. So when you wake up and find yourself in an N-dimensional universe, you still have to make observations in order to figure out what geometry you're in.

To say "there should only be up, down, left and right directions" is to say that intuitively you expect to find yourself in a universe with a "p-norm" where p=1. What it means to say that there is a diagonal direction is simply to say that this intuition is wrong for the universe in question -- that in fact the local p is some other value, typically 2 -- you're in a universe where the Pythagorean Theorem applies: distance is (up² + left²)^1/2. You could have found yourself in a universe with a different p. I spend my working life in the universe you expect, where p=1 and distance is (up¹ + left¹)^1/1, i.e., up plus left. There are no diagonal directions -- to get a wire from transistor A to transistor B you have to go only up, down, left and right, because the routing software won't allow you to do otherwise. (It's written that way because to try to go diagonally will probably block a hundred other wires you haven't laid out yet.)

The point is, to say "So in physics and mathematics, superpositions are allowed." isn't quite right. In mathematics, whether superpositions are allowed is up to you. You can allow them or not, and then study the implications of your choice. In physics, whether they're allowed is up to the world you find yourself in. You can't rely on intuition; you have to observe the world and see what its p is.

I guess this all comes down to the strangeness of infinity. I was thinking about a completely dense plane equivalent to R2. So if you were to travel diagonally, you would never come to a gap in the fabric of the continuum of the plane, and I realized that there is no shape that allows this and still fills in the entire plain.

Now I just realised that they are dimensionless points, and somehow if you put enough of them together, specifically with the density of the real number line, the points become an area.

I just have to deal with how amazing and strange infinity is.

 

[Bomb#20]

I guess this all comes down to the strangeness of infinity. I was thinking about a completely dense plane equivalent to R2. So if you were to travel diagonally, you would never come to a gap in the fabric of the continuum of the plane, and I realized that there is no shape that allows this and still fills in the entire plain.

Well, you can fill the plane with hexagons or equilateral triangles and then travel vertex-to-vertex in six directions instead of four. Curiously enough, one of my former bosses spent his last few years before retirement trying to write a wire router that could take advantage of this without causing a horrendous traffic jam. It's harder than it looks.

I just have to deal with how amazing and strange infinity is.

Don't we all?

 

[J842P]

That is not true. Take some time and look up the definitions for  linear independence,  orthogonality,  basis (linear algebra), and  dimension (vector space).

It's been a while since I took linear algebra, but what the hell else is needed to create an R3 vector space? What about (1,0,0),
(0,1,0),
(0,0,1)?

Those are fine, but to be a basis they merely have to span the space and be linearly independent. While orthogonality implies linear independence, not all linear independent vectors are orthogonal. You can have a basis of non-orthogonal yet linearly independent vectors.