Perfect Parallelepipeds Exist

TL;DR

This paper demonstrates the existence of perfect parallelepipeds with all edges, face diagonals, and body diagonals as positive integers, providing explicit examples and computational searches for such shapes.

Contribution

It proves the existence of perfect parallelepipeds and presents explicit examples, including those with rectangular faces, through computational brute force searches.

Findings

Existence of perfect parallelepipeds with all integer edges and diagonals

Explicit example with specified edge and diagonal lengths

Dozens of primitive perfect parallelepipeds found via brute force

Abstract

There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths all positive integers. In particular, there is a parallelepiped with edge lengths 271, 106, 103, minor face diagonal lengths 101, 266, 255, major face diagonal lengths 183, 312, 323, and body diagonal lengths 374, 300, 278, 272. Focused brute force searches give dozens of primitive perfect parallelepipeds. Examples include parallellepipeds with up to two rectangular faces.