Multidimensional integer trigonometry

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.

Figures (4)

• Figure 1: An example of the sail of the angle formed by the rays $\left\{\left(x,\frac{8}{5}x\right)|x\geq{0}\right\}$ and $\left\{\left(x,0\right)|x\geq{0}\right\}$. See Example \ref{['LLS']}.
• Figure 2: Examples of transpose and adjacent angles.
• Figure 3: An example of angle summation and the effect this has on the LLS sequences, see Example \ref{['anglesum']}.
• Figure 4: An example of the conditions imposed on the LLS sequences of the angles inside an integer triangle, see Example \ref{['triangle']}.

Theorems & Definitions (2)

• proof
• proof