多维整数三角学
Multidimensional integer trigonometry
AI总结：
本文介绍多维整数三角学，将二维整数三角函数推广至任意维度，研究其基本性质，探讨与Euclidean算法和连分数的关系，并引入单纯锥的最佳逼近概念。
摘要：
本文致力于介绍多维整数三角学。我们从2008年引入的二维整数三角学阐述开始，并以此将整数三角函数推广到任意维度。接着我们研究整数三角函数的基本性质。我们找到了转置与相邻单纯锥，以及生成相同单纯形的锥的整数三角关系。此外，我们讨论了整数三角学、Euclidean算法与连分数之间的关系。最后，我们利用相邻与转置锥引入单纯锥的最佳逼近概念。在二维情形下，该最佳逼近概念与实数最佳逼近的经典概念一致。
英文摘要：
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.
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