Some new parameterizations for the Diophantine bi-orthogonal monoclinic piped

TL;DR

This paper introduces new parameterizations for the Diophantine bi-orthogonal monoclinic parallelepiped, providing formulas that encompass most known solutions and approaching the perfect cuboid problem.

Contribution

It presents novel parameterizations for the monoclinic piped, covering 99.5% of solutions and advancing understanding of the Diophantine parallelepiped structure.

Findings

Two new parameterizations for the monoclinic piped are provided.

A parameterization covers 99.5% of solutions found by computer searches.

Sequences approaching the perfect cuboid are listed.

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.