In grade school, students learn a standard set of Euclidean triangles. Among this set, the usual 45–45–90 and 30–60–90 triangles are the only right triangles with rational angles and side lengths each containing at most one square root. Are there any other such right triangles? We answer this question completely and present an elegant complement called Ailles’ rectangle that deserves to be in e...

Abstract. Monotone triangles are certain triangular arrays of integers, which correspond to n × n alternating sign matrices when prescribing (1, 2, . . . , n) as bottom row of the monotone triangle. In this article we define halved monotone triangles, a specialization of which correspond to vertically symmetric alternating sign matrices. We derive an operator formula for the number of halved mo...

It is shown that Bernoulli numbers and tangent numbers (the derivatives of the tangent funcion at 0) can be obtained by means of easily defined triangles of numbers in several ways, some of them very similar to the Catalan triangle and a Motzkin-like triangle. Our starting point in order to show this is a new expression of ζ(2n) involving Motzkin paths.

Extending a similar result about triangles, we show that each Heronian tetrahedron may be positioned with integer coordinates. More generally, we show the following: if an integral distance set in R can be positioned with rational coordinates, then it can in fact be positioned with integer coordinates. The proof, which uses the arithmetic of quaternions, is tantamount to an algorithm.

We study a family of substitution tilings with similar right triangles two sizes which is obtained using the rule introduced in Danzer and van Ophuysen (Res. Bull. Panjab Univ. Sci. 50(1–4), 137–175 (2000)). In that paper, it proved this can be from local decorated tiles. That is, sofic. present we provide an alternative proof fact. use more tiles than (22 place 10). However, our decoration sup...