I wrote a paper on this, see http://rangevoting.org/WarrenSmithPages/homepage/works.html paper #41. This at the time (still?) was the best known lower bound on simplexity of the N-cube, it was simply hyperbolic volume ratio. Another good lower bound method called "linear programming method" arises by noting that each face of the N-cube needs to be dissected into simplices also (and so on, for lower dimensional flats than "faces" too) so you can by combining all these constraints write a linear program whose solution provides a lower bound on simplexity. Even better lower bounds presumably would result from combining both hyperbolic volume and linear programming ideas, but last time I looked nobody had ever worked out the result of that. Meanwhile for upper bounds, it is a matter of finding good triangulations. Sallee, Cottle, and Haiman have got various ways to produce triangulations better than the naive-recursive triangulation involving N! simplices. Last I looked the best lower bounds had behavior (N/2)! * C^N for some C>1, and the best upper bounds had the form N! * K^N for some constant K with 0<K<1. Note the large gap. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)