Additional “Note on Right-Angled Triangles.”—

Classroom note: Almost-isosceles right-angled triangles

We provide an elementary method to show that there exist infinitely many right-angled triangles with integral sides in which the lengths of the two non-hypotenuse sides differ by 1. The method also enables us to construct all such right-angled triangles recursively. 1. Introduction There does not exist any isoceles right-angled triangle with integral sides. Does there exist a right-angled trian...

A Ramsey-type problem on right-angled triangles in space

An Extension of the Fundamental Theorem on Right-angled Triangles

{3, 4, 5} is perhaps the most famous Pythagorean Triple with interest in such triples dating back many thousands of years to the ancient people of Mesopotamia. In this article, we shall consider such triples, with the restriction that the elements of these triples must not have any common factors they are Primitive Pythagorean Triples (PPTs). In particular, we shall consider the question of how...

Lattices Acting on Right-angled Hyperbolic Buildings

Let X be a right-angled hyperbolic building. We show that the lattices in Aut(X) share many properties with tree lattices. For example, we characterise the set of covolumes of uniform and of nonuniform lattices in Aut(X), and show that the group Aut(X) admits an infinite ascending tower of uniform and of nonuniform lattices. These results are proved by constructing a functor from graphs of grou...

Lattices acting on right-angled buildings

Let X be a right-angled building. We show that the lattices in Aut(X) share many properties with tree lattices. For example, we characterise the set of covolumes of uniform and of nonuniform lattices in Aut(X), and show that the group Aut(X) admits an infinite ascending tower of uniform and of nonuniform lattices. These results are proved by constructing a functor from graphs of groups to compl...