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:
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!