The rotation of the hypercube, whose vertices have coordinates each ±1, is the composite of rotations in the

*w-x*,

*w-y*, and

*w-z* planes, in that order. Combined with

*Mathematica*'s built-in rotation of 3D graphics, this gives all six degrees of freedom of four-dimensional rotations. The bounding cubes are drawn in the order from least

*w* coordinate to greatest. If all the opacities are set to one, the display shows the cubes "visible" looking from out on the positive

*w* axis. The projection is accomplished by dropping the

*w* coordinate; therefore it is a parallel projection, although the projection onto the screen has the default perspective of

*Mathematica *3D graphics. The positive

*x*,

*y*,

*z*,

*w* axes of the hypercube may be shown; these rotate with the hypercube, which helps illustrate how the rotations work.