Multidimensional projections on 2d plane

For 3 dimensions you can imagine that A,B,C are your 3 dimensional axes turned in such a way that looking at them they appear like this:

Define a point P on the plane to be P=a*A+b*B+c*C, with a,b,c values from the sliders...

Then we can look at all the ways to fix 2 of the sliders at 1 or 0 and move the 3rd, and tracing the paths they make gives:

Which is the "shadow" or projection of the 3d cube on the plane...

Now what if we have four orthogonal axes, so that their projection on the 2d plane looked like this:

We can define P to be P=a*A+b*B+c*C+d*D

There are a lot of ways to hold all but one of the sliders fixed at 1 or 0 and move the other one, doing so traces this onto the plane:

This is the projection on the plane of a 4 dimensional cube!