What Do Equations Expressing Higher Dimensions Look Like?

[schlynn]

Ok, I often hear that things become more easily understood in higher dimensions. I also hear that it is easier to express them in math than with words. But what does equations that express higher dimensions look like? Is it something as simple as (x,y,z,w)? I might be over thinking this far more than I should, but what are some examples of equations that take advantage of higher dimensions? For example something like x²+y²+z²=w². Is something like that a example?

 

[Mentallic]

But what does equations that express higher dimensions look like?

I'm assuming you're speaking of higher spatial dimensions, and ignoring time.
Should anyone truly know the answer to this and have been exposed to it visually, they are either high or godlike

And yes just throw in an equation with 4 variables, and you'll have just that, a 4-d graph.

Maybe if you think of it like this, things will be cleared up a bit.
On a 2-d coordinate system, the equation x=0 will represent a line.
With a 3-d coordinate system, the equation x=0 will represent a 2-d plane.
On this theoretical 4-d system, the equation x=0 is 3-d space as we know it.

But looking at the first example, the line is only one 'segment' of the plane itself, there are an infinite number of lines that can fit on this cartesian plane.
The same thing goes for the third example. Even though all known space is consumed by the equation x=0, there are still an infinite number of these spaces that can fit into the 4-d system.