Why does 3-D math work the way it does?

[questionposter]

So if I graph 3-D patterns like a ripple in a pond or sound waves, instead of it involving something like sin(x^3) and sin(z^3) its sin(x^2+y^3). But how could actual movement 3-D movement in 4 dimensional space be built out of the warping of 2 dimensional things? How could a 2-D thing actually move in a 3 dimensional way? Does x^2 not actually have anything to do with 2 dimensions, is it just coincidence that that's how you find a square or is x^3 coincidentally a unit for a cube? What about x^4units^4? Shouldn't that be a 4th dimensional object? Is space somehow a 2-D plane in every direction? And how is that different than just a plain 3-D object?

Edited November 8, 2011 by questionposter

[Schrödinger's hat]

So if I graph 3-D patterns like a ripple in a pond or sound waves, instead of it involving something like sin(x^3) and sin(z^3) its sin(x^2+y^3). But how could actual movement 3-D movement in 4 dimensional space be built out of the warping of 2 dimensional things? How could a 2-D thing actually move in a 3 dimensional way?

Does x^2 not actually have anything to do with 2 dimensions,

It does...sort of.

You're conflating dimension and power of x.

 

If x were in distance then [math]x^2[/math] would have units of [math]m^2[/math], which would represent some kind of area-like or two dimensional thing.

 

But that doesn't mean everything with [math]x^2[/math] in it will be area-like.

 

Take [math]F = \frac{GMm}{r^2}[/math] for example. There's an [math]r^2[/math], but when you take the whole thing into consideration it has units of force.

 

[math]\sin{(x^2 + y^3)}[/math] isn't the kind of formula that would come up in physics. For one we can't really take the sin of something with units. Secondly the x^2 and the y^3 have different units.

But if you are only considering x and y to be numbers (no units, so x^2 is just a number, it doesn't have dimension or units) then this is perfectly valid.

Think of it as just a relationship between three numbers.

 

is it just coincidence that that's how you find a square or is x^3 coincidentally a unit for a cube? What about x^4units^4? Shouldn't that be a 4th dimensional object? Is space somehow a 2-D plane in every direction? And how is that different than just a plain 3-D object?

[DrRocket]

The short answer is that polynomial exponents have nothing whatever to do with dimension.