Chinese interest in 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...

Pushing fillings in right-angled Artin groups

We define a family of quasi-isometry invariants of groups called higher divergence functions, which measure isoperimetric properties “at infinity.” We give sharp upper and lower bounds on the divergence functions for right-angled Artin groups, using different pushing maps on the associated cube complexes. In the process, we define a class of RAAGs we call orthoplex groups, which have the proper...

Surface Subgroups of Right-Angled Artin Groups

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group A(K) has such a subgroup if its defining graph K contains an n-hole (i.e. an induced cycle of length n) with n ≥ 5. We construct another eight “forbidden” graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or co...