A Logical Concern about Dimensional Spaces

[Juma]

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.

You seem to investigate"how objects have position in space", "how points are connected" and "which moves are allowed".

Neither of which has any obvious connection to the naive notion of orthogonal coordinate systems.

Instead there should be a lot of insight by studying quantum relativity physics (and maybe string theory).

 

[J842P]

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.

You can decompose a vector into non-orthogonal components too. For a space with dimension N, in fact, you can decompose a vector into components in terms of any N linearly independent vectors. Orthogonality is merely a choice that helps for solving physics problems in straightforward ways. There can be useful cases, I suppose, where a non-orthogonal basis might make the calculations more straightforward.

 

[beero1000]

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.

You can decompose a vector into non-orthogonal components too. For a space with dimension N, in fact, you can decompose a vector into components in terms of any N linearly independent vectors. Orthogonality is merely a choice that helps for solving physics problems in straightforward ways. There can be useful cases, I suppose, where a non-orthogonal basis might make the calculations more straightforward.

An example from elementary linear algebra - matrix diagonalization usually involves a non-orthogonal eigenbasis.

 

[J842P]

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.

You can decompose a vector into non-orthogonal components too. For a space with dimension N, in fact, you can decompose a vector into components in terms of any N linearly independent vectors. Orthogonality is merely a choice that helps for solving physics problems in straightforward ways. There can be useful cases, I suppose, where a non-orthogonal basis might make the calculations more straightforward.

An example from elementary linear algebra - matrix diagonalization usually involves a non-orthogonal eigenbasis.

Yeah, in mathematics a non-orthogonal basis is hardly exotic.

 

[beero1000]

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.

You can decompose a vector into non-orthogonal components too. For a space with dimension N, in fact, you can decompose a vector into components in terms of any N linearly independent vectors. Orthogonality is merely a choice that helps for solving physics problems in straightforward ways. There can be useful cases, I suppose, where a non-orthogonal basis might make the calculations more straightforward.

An example from elementary linear algebra - matrix diagonalization usually involves a non-orthogonal eigenbasis.

Yeah, in mathematics a non-orthogonal basis is hardly exotic.

Yup, and diagonalization is extremely useful in economics, physics, engineering, etc. The entire idea is that there are bases that are more conducive to calculation than the standard basis. It's only a rare case when these bases happen to be orthonormal.

 

[Malintent]

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.

It's stranger still... If you have two fixed points with distance 1 (meter), you can traverse the distance in a set amount of time (T=1). However, before you traverse that complete distance, you first must make it half way there, in half the time. (T=1/2). Now you are halfway there.. .however, before you make it the rest of the way, you must first traverse half the remaining distance (T=1/2 + 1/4)....

Given the infinite set {1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64....},

Eventually, you will make it all the way to distance = 1. Therefore,

{1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64....} = 1.

An infinite set can sum to a finite value!

There's your "diagonal" direction... the infinitesimal steps near the far "end" of that set.

In math, resolving infinite geometry is called "super solids". The steps you take to make it halfway across the remaining distances are "tasks".

To walk in a diagonal direction, you are "super tasking"

 

[EricK]

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?

Imagine a plane with an origin and a normal x and y axis, perpendicular to each other. Now imagine another random point in the plane. Construct a rectangle with the origin and this new point as opposite corners and with portions of the x any axes as two of the sides. It is clear that there is a unique way to do this. The lengths of the sides of the rectangle are the coordinates of the point.

Now imagine a plane with an origin and two news axes, which are not at right angles to each other. Call them the a and b axes. If we take a new point in the plane, we can do something similar to the above, but instead of creating a rectangle, we create a parallelogram (note that a rectangle is a special type of parallelogram, so we're not doing anything radically new). Again, there is a unique way to create that parallelogram. Now, just as the lengths of the rectangle's sides were the coordinates in the xy plane, the lengths of the parallelogram's sides are the coordinates in the ab plane.

This shows that the two directions for axes do not need to be perpendicular in order to act as dimensions (in fact, they just need to be linearly independent). I'd any 2 non-parallel lines can act as a basis for the plane.

Note, also, that these constructions of parallelograms are how forces etc combine as in your OP. So a force of 3N along the x axis combined with a force of 4N along the t axis gives a force of 5N, which is the length of the rectangle's diagonal, in the direction of the far corner of the rectangle. Similarly, a force of 3N along the a axis combined with a force of 4N along the b axis will give a force equal to the length of the parallelogram's diagonal, in the direction towards the far corner of the parallelogram.

 

[Cheerful Charlie]

Relativity. Your dimensions depend on your speed. Just to throw some kinks into the discussion of 3 dimensional space.

 

[fromderinside]

Relativity. Your dimensions depend on your speed. Just to throw some kinks into the discussion of 3 dimensional space.

Want to get really kinky. Remove time.