# Perfect parallelepipeds exist

@article{Sawyer2009PerfectPE,
title={Perfect parallelepipeds exist},
author={Jorge F. Sawyer and Clifford A. Reiter},
journal={Math. Comput.},
year={2009},
volume={80},
pages={1037-1040}
}
• Published 1 July 2009
• Physics
• Math. Comput.
There are parallelepipeds with edge lengths, face diagonal lengths and body diagonal lengths that are 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.
• Mathematics
Math. Comput.
• 2014
It is proved the existence of an infinite family of dissimilar perfect parallelepipeds with two nonparallel rectangular faces that can be obtained of this form with the angle of the nonrectangular face arbitrarily close to 90◦.
• Mathematics
• 2013
A perfect parallelepiped has edges, face diagonals, and body diagonals all of integer length. We prove the existence of an infinite family of dissimilar perfect parallelepipeds with two nonparallel
A rectangular parallelepiped is called a cuboid (standing box). It is called perfect if its edges, face diagonals and body diagonal all have integer length. Euler gave an example where only the body
• R. Sharipov
• Mathematics
Journal of Mathematical Sciences
• 2020
A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet
A perfect cuboid is a rectangular parallelepiped in which the lengths of all edges, the lengths of all face diagonals, and also the lengths of spatial diagonals are integers. No such cuboid has yet
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
We discuss generating parallelepipeds, with 4 rectangular faces, which have rational lengths and all face and space diagonals also rational.
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.
Four integer parametrizations for the bi-orthogonal monoclinic Diophantine parallelepiped are given.
A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called quadratic if the degrees of its