Make cube rotate around its edge without having to keyframe the anchor point?

This is an interesting and tricky problem. I've been tinkering with it for a couple of days and here's what I've got. If the layer is a solid with the anchor point centered, you can use a position expression like this to "roll" the square in accordance with the rotation property:

w = width;
r = Math.sqrt(2*w*w)/2;
a = rotation%90;
a += a < 0 ? 90 : 0;
n = Math.floor(rotation/90);
a0 = Math.PI*5/4;
a1 = Math.PI*7/4;
a2 = linear(a,0,90,a0,a1);
c0 = Math.cos(a0);
x = n*w + r*(Math.cos(a2) - c0);
s0 = Math.sin(a0);
y = r*(Math.sin(a2) - s0);
value + [x,y]

If you're using a square shape layer (with anchor point centered), you need to change the first line to this:

w = content("Rectangle Path 1").size[0];