The Torus of Triangles

We show that when parameterized by triples of angles, the set of similarity classes of labeled, oriented, possibly degenerate triangles has the natural structure of the Clifford torus $\sf{T}$, a compact abelian Lie group. On this torus the main triangle types form distinguished algebraic structures: subgroups and cosets. The construction relies on a natural definition of similarity for degenerate triangles. We analyze the set of (unrestricted) similarity classes using a uniform probability measure on $\sf{T}$ and compute the relative measures of the different triangle types. Our computations are compatible with the spherically symmetric probability distribution analyzed in [Por94] and [ES15], which are based on vertices/side lengths instead of angles

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v