Some new parameterizations for the Diophantine bi-orthogonal monoclinic piped

Abstract

The bi-orthogonal monoclinic Diophantine parallelepiped is introduced, then the s-parameters and their governing equation for the bi-orthogonal monoclinic Diophantine parallelepiped are discussed. Previous discoveries and parameterizations of the monoclinic piped are noted. Then two parameterizations P[1/2,s_2,s_3,s_4], s_i in Q,Z are given for a specific type of Diophantine bi-orthogonal monoclinic parallelepiped. Next, a parameterization P[s_1,s_2,s_3,s_4], s_i in Q,Z is presented which covers 99.5% of solutions found by raw computer searches. Several asymptotic sequences approaching the perfect cuboid are listed, and some final comments made.

4 figures, 2 tables