# Classifying Diophantine parallelepipeds

## TL;DR

This paper classifies Diophantine parallelepipeds into five classes based on surface angles, analyzes their rational components, and uses computer searches to discover notable examples including perfect parallelepipeds and rectangular cuboids.

## Contribution

It introduces a new classification scheme for Diophantine parallelepipeds based on angles and rationality checks, and identifies numerous examples through extensive computational search.

## Key findings

- Identified five unique classes of Diophantine parallelepipeds.
- Reduced rationality checks from 83 to 27 for analysis.
- Discovered notable examples including the perfect parallelepiped and rectangular cuboid.

## Abstract

By examining the 3 surface angles which exist at any of the 8 vertices of a Diophantine parallelepiped, and classifying them by the appearance of a right angle, it is discovered that 5 unique classes of Diophantine parallelepipeds exist. It is proposed to name these classes: acute (triclinic), obtuse (triclinic), 1-ortho (biclinic), 2-ortho (monoclinic), and rectangular, according to the count of rights angles which may exist. A Diophantine analysis of the 83 possible rational components of the piped reveals that only 27 rationality checks need to be made when examining for rationality; such as skew triangles, body or face parallelograms, face diagonals, body diagonals, and volume. A computer search of 1,981,336,681 tetrahedrons with 6 rational face diagonals uncovers interesting examples of pipeds, including the perfect parallelepiped of Sawyer-Reiter (and 5 others), and the rectangular integer cuboid. Other interesting pipeds were also discovered in the 115 unique categories which the computer searches revealed. Some questions, conjectures and possible studies are provided at the conclusion.