4-dimensions on a 2 dimensional plane

[Chowmein]

A co-worker and I are working on representing 4 dimensions in a 2 dimensional space. This might not make a lot of sense, but I’ll try to explain it as best I can. For this we are using the assumption that the first 3 dimensions represent a specific location and the 4th dimension represents that location in time.

Start with a 2 dimensional graph drawn on paper. You can plot coordinates X,Y. This is basically a 2 dimensional graph drawn on a 2 dimensional plane. Draw a slanted line thru the center of the graph and you can plot coordinates X,Y,Z. This will give a 3 dimensional graph on a 3 dimensional plane.

Here’s the kicker, we want to plot time. We can plot point 5,5,5. We can assume this is at Time = 1. How would we plot the same point at Time = 2? So far, we have come up with the idea of using multiple 3-dimentional graphs each one labeled with a Time = Whatever. This is not the exact solution we want. We want to be able to plot the same point at different times on the same graph. Is this even possible? Does this even make sense?

[ultrafilter]

It's tough. The standard solution is to use the line y = x as the z-axis and the line y = -x as the w-axis. But for any kind of complicated graph, it's going to be unpleasant to look at, and hard to get information from.

The best would be an animation that shows the system evolving over time. That can be tough, though.

[Orbifold]

Colour may also be of some use (i.e. use the colour of a point to indicated its position in one or two of the dimensions), but as ultrafilter says it's just plain difficult to show 4-D data in 2-D in an accessible way.

[Grey]

I agree that an animation would make the most visible sense. Off the top of my head you could :

1. Stack graphs as timeslices to show change.

2. Colour code the date based on rate of change of the value. I.e. Red points indicate lots of d/dt that would show interesting regions of the dimensional data.

[CalMeacham]

You could represent your data as a vector, rather than as a point. The head of the vector is at (x1,y1) = (x,y) and the tail is at (x2,y2) = (z,t). That way your vector uniquely encodes four pieces of data, or four dimensions.

[Grey]

I like that. Any examples you can link to?

Sorry Chowmein, I've kicked this around too and never liked my solutions much.

[knock knock]

You might try something like CalMeacham's suggestion, except make the point a 3-dimensional point (x,y,z) on your paper & make the length of the arrow indicate the time. I'm not sure what you would do about the direction of the arrow - maybe make them all go in the same direction?