Each angle formed by two rays with integer slopes has two basic integer invariants: its height and its width. An angle is called Vitruvian (after a Roman architect Vitruvius advocating proportions between height and width) if its height divides its length. A Vitruvian polygon is a polygon, such that all of its angles are Vitruvian. Vitruvian polygons form a distinguished class of polygons in Tropical Planimetry.

After a breakthrough idea of Galkin and Usnich from 2010, Vitruvian triangles (studied, under a different guise, by Hacking and Prokhorov, buiding up on an earlier work of Manetti to obtain the complete classification of toric degenerations of the plane) started to play a prominent role also in Symplectic Geometry. In the talk, I review some of these applications, as well as a new symplectic application, involving use of Vitruvian quadrilaterals (work in progress, joint with Richard Hind and Felix Schlenk).