Multidimensional integer trigonometry

TL;DR

This paper introduces and generalizes multidimensional integer trigonometry, exploring its properties, relations, and applications to approximations, extending concepts from two-dimensional integer trigonometry to higher dimensions.

Contribution

It extends integer trigonometry from two dimensions to arbitrary dimensions and investigates its properties, relations, and approximation methods.

Findings

Established integer trigonometric relations for cones

Connected integer trigonometry with Euclidean algorithm and continued fractions

Proposed a notion of best approximations for simplicial cones

Abstract

This paper is dedicated to providing an introduction into multidimensional integer trigonometry. We start with an exposition of integer trigonometry in two dimensions, which was introduced in 2008, and use this to generalise these integer trigonometric functions to arbitrary dimension. We then move on to study the basic properties of integer trigonometric functions. We find integer trigonometric relations for transpose and adjacent simplicial cones, and for the cones which generate the same simplices. Additionally, we discuss the relationship between integer trigonometry, the Euclidean algorithm, and continued fractions. Finally, we use adjacent and transpose cones to introduce a notion of best approximations of simplicial cones. In two dimensions, this notion of best approximation coincides with the classical notion of the best approximations of real numbers.