(1) Your Reference  is misleading. O'Neill (programmer) and Laubscher (analyst) were not the first ones to come up with the Quadrilateralized Spherical Cube (QLSC). The original work was done by Chan (analyst) and O'Neill (programmer) in 1973 and published as a Navy report in 1975. Go to Google and Wikipedia for "Quadrilateralized Spherical Cube".
(2) The work of O'Neill and Laubscher did not truly "quadrilateralize" a sphere but actually "triangularized" it. As a result, there are mathematical singuarities along the diagonals of the cube and, hence, also at the poles. This is evident from the sharp breaks in the latitudes. Even their longitudes are evidently far from being straight. On the other hand, the earlier work of Chan and O'Neill did not have such singularities and the cube is truly quadrilateralized. This is evident in the smoothness of the latitudes and the straightness of the longitudes. See diagrams in following link: http://www.progonos.com/furuti/MapProj/Dither/ProjPoly/projPoly2.html
(3) There is no exact closed form solution for the QLSC mapping. The original work of Chan and O'Neill expressed the mapping as a truncated convergent infinite series (which is the norm when dealing with infinite series). Because of this, Laubscher erroneously referred to it as "approximate" and claimed his own work as "exact" even though it was not a QLSC. This point was buried in the fuss raised by Laubscher and escaped the notice of the general public.
(4) The original QLSC was adopted for use by the NASA Cosmic Background Explorer (COBE). Results from COBE demonstrated the anisotropy of the cosmic microwave background radiation only moments after the Big Bang. A brief description of the QLSC is given in the link: http://adsabs.harvard.edu/full/1992ASPC...25..379W
(5) Additional history of the QLSC may be found in the "Quadrilateralized Spherical Cube Forum". Look up Google with these keywords.
(6) John Snyder's equal area projection onto a spherical cube is not quadrilateralized and also has mathematical singularities evidently along the diagonals of the cube and the poles. Go to Google and look up "John Snyder Equal Area Polyhedra".