The Cube Unfolded

The surface of a cube has no intrinsic curvature, except at that 
eight verticies, where the curvature is singular.  This implies that 
straight lines can be drawn unambiguously on the surface, from one
face to another, as long as the line doesn't pass precisely through 
a vertex.  Beginning from any rational point on the surface of a 
cube, the set of rays emanating outward with irrational slopes on 
the surface from that point can be extended indefinitely without
striking a vertex.  This enables us to map the surface of the cube 
to a plane by essentially "unfolding it" along each of these rays.
This mapping completely "tiles" the infinite plane, although of
each individual point on the cube is mapped to infinitely many
different points on the plane, corresponding to the different ways
in which it is possible to proceed from the origin to that point
along a straight line on the surface.

If we color each of the six faces of the cube with a different
color, using red, green, and blue for the three pairs of opposite
faces, with light coloring for one and dark for the other, and if
we place the origin at the center of the dark blue face, then the
cube unfolded along the rays emanating from that point is as shown
below:



Interestingly, this represents an application of what are called
Riemann coordinates in differential geometry, and it gives a nice
illustration of the non-commutativeness of parallel transport on
curved surfaces.  The uniqueness of this situation is that all the
intrinsic curvature of the surface is contained in the singular
verticies, breaking up the geodesic rays into discrete patches.
Similar maps can be generated for the surfaces of other polyhedrons,
such as the Platonic and Archmedian solids.