A fast algorithm to find reduced hyperplane unit cells and solve N‐dimensional Bézout's identities

• Published in: Acta crystallographica. Section A, Foundations and advances Vol. 77; no. 5; pp. 453 - 459
• Main Author: Cayron, Cyril
• Format: Journal Article
• Language: English
• Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.09.2021
• Subjects: hyperplane unit cell; integer relation; N‐dimensional Bézout's identity; Research Papers; twinning
• ISSN: 2053-2733; 2053-2733
• DOI: 10.1107/S2053273321006835

Deformation twinning on a plane is a simple shear that transforms a unit cell attached to the plane into another unit cell equivalent by mirror symmetry or 180° rotation. Thus, crystallographic models of twinning require the determination of the short unit cells attached to the planes, or hyperplanes for dimensions higher than 3. Here, a method is presented to find them. Equivalently, it gives the solutions of the N‐dimensional Bézout's identity associated with the Miller indices of the hyperplane. The paper describes a method to determine a short unit cell attached to any hyperplane given by its integer vector p. Equivalently, it gives all the solutions of the N‐dimensional Bézout's identity associated with the coordinates of p.