3 dimensions and 90 degree angles

[DaveC426913]

I am not sure how to form a coherent question about this; I can't figure out how to phrase it:

Why are our three dimensions all at 90 degrees to each other?

Drat, I can't even form the question. I've been reading about extra dimensions, such as the further 6 postulated in string theory, and how they overlap our current three, so this is not a rudimentary question looking for a rudimentary answer.

But I still can't seem to ask the question I want. Every explanation I try to form ends up making the initial assumoption that there are 3 dimensions we have to play with in the first place.

I'm going to leave this open for feedback, and clarify my question as I go.

 

[Integral]

The phrase you are looking for is orthogonal. Orthogonality has some nice properties, but is not necessary. The Cartesian Coordinate system is orthogonal, but cylindrical and spherical coordinates are not. Orthogonality is nice but not necessary, the only necessary condition for a coordinate system is that you have a linear independent basis which spans the space.

 

[chroot]

The dimensions themselves are not "90 degrees to each other," i.e. orthogonal. Orthogonality is a property of a simple coordinate system than human beings arbitrarily impose on a vector space. There are many other acceptable coordinate systems (cylindrical, spherical, "squashed" Cartesian, etc.) which do not have the property of having orthogonal basis vectors.

The reason there are three dimension is that it seems to require three numbers to uniquely label any given point in our space. That is just our human experience; it is possible that there are additional dimensions which are so small in extent that we don't even notice them.

- Warren

edit: Damn you Integral!

 

[dextercioby]

The phrase you are looking for is orthogonal. Orthogonality has some nice properties, but is not necessary. The Cartesian Coordinate system is orthogonal, but cylindrical and spherical coordinates are not. Orthogonality is nice but not necessary, the only necessary condition for a coordinate system is that you have a linear independent basis which spans the space.

I RESENT THAT! Curvilinear coordinates can be orthonormal...(look for Lamé coefficients).

Daniel.

 

[dextercioby]

The dimensions themselves are not "90 degrees to each other," i.e. orthogonal. Orthogonality is a property of a simple coordinate system than human beings arbitrarily impose on a vector space. There are many other acceptable coordinate systems (cylindrical, spherical, "squashed" Cartesian, etc.) which do not have the property of having orthogonal basis vectors.

The reason there are three dimension is that it seems to require three numbers to uniquely label any given point in our space. That is just our human experience; it is possible that there are additional dimensions which are so small in extent that we don't even notice them.

- Warren

I resent that! (no caps lock) Do some reading. Look for Lamé coefficients...

Daniel.

 

[chroot]

dexter,

Do elaborate. I'm actually mildly confused, even though I typed virtually the same response as Integral. Given any point in a space described with a spherical coordinate system, the tangent planes of the coordinate surfaces are actually orthogonal, so now I'm second-guessing myself. It just happens that every point in the space has a different set of these orthogonal basis vectors.

The only example I can think of where the basis is indeed not orthogonal is the "squashed" Cartesian coordinate system, which might have a better name than "squashed" Cartesian.

- Warren

 

[dextercioby]

The basis vectors are tangent to the coordinate curves...What are those curves for spherical coordinates...?

Daniel.

 

[chroot]

I usually just imagine Earth when thinking of spherical coordinates. I suppose it makes quite good sense that the basis vectors are the tangents of the coordinate curves at all points in any space, with any kind of coordinate system imposed on it.

The coordinate curves for spherical coordinates are radius, theta (i.e. longitude on Earth) and phi (i.e. latitude on Earth). On the Earth's surface, the three basis vector r, theta, and phi are always orthogonal at any point.

- Warren

 

[dextercioby]

Voilà.It happens the same with plane polar and their generalization,the cylidrical...

Daniel.

 

[chroot]

So, okay. The spherical coordinate system is really "orthogonal" then, in that sense. The cylindrical coordinate system does not seem so, however...

- Warren

 

[DaveC426913]

Yah, in doodling while refining my question, I came up with some alternate coord systems that aren't 90 degrees, such as polar. But do they all require exactly *3* coords? (3 must be the minimum, or they'd've been replaced with the simpler one, wouldn't they? And it'd be easy to create systems with *more* coords, just by limiting the freedom in the existing three, it's just that 3 is the minimum, right?)

Ultimately I am trying to get at this concept of these extra dimensions, and the idea of whether I'm really passing my arm through them when I wave it.

 

[chroot]

Yah, in doodling while refining my question, I came up with some alternate coord systems that aren't 90 degrees, such as polar. But do they all require exactly *3* coords? (3 must be the minimum, or they'd've been replaced with the simpler one, wouldn't they? And it'd be easy to create systems with *more* coords, just by limiting the freedom in the existing three, it's just that 3 is the minimum, right?)

You could invent a coordinate system with 20 coordinates, if you'd like, but the basis vectors would not all be linearly independent, so some of them would be redundant. You cannot, however, use fewer than three.

Ultimately I am trying to get at this concept of these extra dimensions, and the idea of whether I'm really passing my arm through them when I wave it.

The coordinates of your arm in other ("higher") dimensions would certainly be changing, so you can say your arm was moving through them, sure.

- Warren

 

[jcsd]

The phrase you are looking for is orthogonal. Orthogonality has some nice properties, but is not necessary. The Cartesian Coordinate system is orthogonal, but cylindrical and spherical coordinates are not. Orthogonality is nice but not necessary, the only necessary condition for a coordinate system is that you have a linear independent basis which spans the space.

Actually both the cylidrical and spherical coordinate systems are orthogonal by the normal meaning of orthogonal coordiantes systems, as an orthongogonal coordinate system is a coordinate system for which at every point each basis vector is orthogonal to each other basis vector (one way to tell an orthogonal coordinate system is if the matrix represneting the metric tensor has no non-zero off-diagonal entires), what differentiates rectilinear coordinate systsems (The Cartesian coordinate is the case of an orthonormal rectilinear coordinate system) from other orthogonal curvilinear coordinates systems is that the basis vector fields which define the coordinate system are all constant.

 

[HallsofIvy]

"Three dimensions" only means that we need 3 numbers to designate a single point. It does not follow that "3 dimensions are all at 90 degrees to each other". It is not the "dimensions" that are orthogonal but the coordinate axes- and those are our choice.

It is convenient to set up axes that way! For one thing the "metric tensor" is diagonal and so very simple. A result of that is that the axes to do not "interact"- we can add vectors (and do other things) on individual coordinates. If we use non-orthogonal axes (which we certainly can do) computations become much more complicated.

 

[DaveC426913]

The coordinates of your arm in other ("higher") dimensions would certainly be changing, so you can say your arm was moving through them, sure.

I guess the operative question though, is whether it is possible to move it through those other dimensions *independently* of our mundane three.

I think the answer is that the strings-that-make-up-the-particles that-make-up-the-atoms that-make-up-my-arm *are* moving throuigh those extra dimensions independently, but that doesn't mean I can swing my arm through them independently.

 

[jcsd]

As Halls of Ivy poimts out the concept of diemsion is more primitive than the conncept of orthognality, i.e. we needn't even define a way (i.e define an inner product) of telling if two vectors are orthogonal to find the dimension of a vector space.

Interestingly we can parametreize, for example, a 3-D real space with single real numbers (this is just a statement of the fact that R and R³ are bijective) the 3 numbers are needed to parametrize it continiously.

 

[Gokul43201]

The cylindrical coordinate system does not seem so, however...

Why does it not ?

Edit : nevermind