In brief. In the window below, unit cubes in dimensions 3, 4, and 5 rotate freely in their respective dimension; their projection onto three dimensions is displayed. The projected cubes may be rotated in 3 dimensions by dragging the mouse over the applet. Click on the 3D/4D/5D cube button to toggle between the various dimensional cubes.
In detail. The rotation on/off button turns on and off the free rotation of the cube. Regardless of the mode, you can rotate the projected cube in three dimensions by dragging the mouse over the applet window.
The wire/solid frame button controls how the projected cube is displayed in three dimensions: either as a simple perspective line drawing (wire frame), or with edges thickened (solid frame).
Remark. The projections of the unit 4-cube (hypercube) and unit 5-cube (hyper-hypercube) used in the above applet are orthogonal. Specifically, points (w,x,y,z) on the 4-cube are mapped to (x,y,z) in 3 dimensions; similarly, points (v,w,x,y,z) on the 5-cube are mapped to (x,y,z) in 3 dimensions. While there are an infinite number of other (orthogonal) projections to choose from, the fact that the unit cube is rotating freely in its own dimension makes the use of any other orthogonal projection redundant: the cube will eventually rotate through all possible orthogonal projections.