Standard 2D Crystalline Patterns and Rational Points in Complex Quadrics

TL;DR

This paper explores the connection between rational points on complex quadrics and standard realizations of 2D crystalline patterns, linking Diophantine problems with crystallography through graph theory.

Contribution

It introduces a novel framework connecting rational points on complex quadrics to 2D crystal structures via standard realizations, expanding the mathematical understanding of crystallography.

Findings

Rational points on complex quadrics relate to 2D crystal realizations.

A finite graph is associated with a complex projective quadric over Q.

The approach links Diophantine problems with crystallography.

Abstract

A certain Diophantine problem and 2D crystallography are linked through the notion of standard realizations which was introduced originally in the study of random walks. In the discussion, a complex projective quadric defined over Q is associated with a finite graph. "Rational points" on this quadric turns out to be related to standard realizations of 2D crystal structures.