4-D Graphing

[jordan]

Can someone show me how to graph a point in four dimensions? Particularly with cartesian coordinates. And then as a follow up, are there equivalents of spherical and cylindrical coordinates in 4-D? And there should be another new system or coordinates also, perhaps? Thanks.

[the tree]

Well there's no way in hell your gonna manage to draw one meaningfully. The co-ordinates work exactly the same: (x,y,z,t).

[Klaynos]

And there is pretty much an infinite number of coordinate systems,,,

 

Drawing 4D on a 2D piece of paper is veyr very difficult!

[BhavinB]

Ternary phase diagrams use a 2D method to graph four dimensions....sort of.

 

http://www.sv.vt.edu/classes/MSE2094_NoteBook/96ClassProj/experimental/ternary2.html

[jordan]

There's one thing I can't get past with 4-D graphing. I don't understand where the fourth dimension is. I still don't have a logical answer for where is exists that couldn't be described with 3 dimensions. I realize everything i've seen is a 2-D projection of a 4-D object but it still seems like all you'd have to do to get to the fourth dimension in these pictures is to extend the third dimension out a ways.

 

And yeah, Klaynos, I realize there are a ton of ways to make coordinate systems but are there direct translations of spherical and/or cylindrical coordinates in 4-D?

[CanadaAotS]

well you could have 4 dimensions if you added color to a 3 dimensional graph and let hot / cold colors represent different numbers.

[Connor]

I think we're looking for a forth spacial dimension though. Very hard to visualize.

 

I imagine there would be cylindrical (hyper-cylindrical?), spherical (cylindro-spherical?), and hyper-spherical coordinate systems possible in a four dimensional space

[CanadaAotS]

well to display 4 Spatial dimensions on a 2D surface is definitely impossible... possibly if it was some kind of 3D medium being drawn upon...

[Connor]

yeah, I think any 2D representation would be too jumbled to make sense. I'm not even sure we could comprehend a 3D version, because our brains have just never experienced 4D vision.

[Dave]

A much easier way (on the eye) to "plot" 4-dimensional graphs is to take the level sets. For example, if you have some function [imath]f(x,y,z)[/imath], then its graph is a subset of [imath]\mathbb{R}^4[/imath]. However, if you consider the set:

 

[imath]f^{-1}© = \{ (x,y,z) \in \mathbb{R}^3 \ | \ f(x,y,z) = c \}[/imath]

 

then it's easy to plot this since it lies in [imath]\mathbb{R}^3[/imath]. One can then plot the set for various values of c and get a general idea of the behaviour of the graph.

[Dave]

After posting this, I realised that it might not be clear what I mean by "level sets", so here's a follow-up. Consider the function [imath]f:\mathbb{R}^2 \to \mathbb{R}[/imath] defined by [imath]f(x,y) = x^2 + y^2[/imath]. (This is called a paraboloid; it looks like the surface generated by y = x² rotated around the z-axis).

 

Now, whilst its possible to graph this function, it's handy to use it as an example to show what level sets are. The idea is to look at the plane parallel to the xy-plane, and adjust its height to build up an image of what the graph does. This is particularly easy for this function.

 

Solving [imath]f(x,y) = c[/imath] we get the equation [imath]x^2 + y^2 = c[/imath]. This clearly defines a circle with radius [imath]\sqrt{c}[/imath] if c > 0, and if c < 0 then we get no solutions (i.e. the graph doesn't exist). So basically, as we move the "height-gauge" up and down the z-axis, we build up an image of what the graph should look like.

 

You can extend this idea by another dimension, but instead of curves, you're going to get some kind of surface. This is pretty much the only way I know of getting a good idea of what the graph actually does.

[Connor]

It's kind of like trying to graph complex function, I guess... you can take paths on a 2-D graph and map them onto another 2-D graph.

 

when I try to visualize a fourth dimension, I think of looking on our 3-space from a viewpoint outside of it... so you could see everypoint in our 3-space, inside and out. I don't know how our eyes would handle that, being three dimensional and seeing only two dimensions. It would be kind of like trying to visiualize a 4-space on a 2-space.