The Cube Unfolded


The surface of a cube has no intrinsic curvature, except at the eight vertices, 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 vertices, 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.


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