Abstract: In 1751, Euler wrote a letter to Goldbach in which he conjectured a formula for the number of triangulations of a regular polygon with n sides. It turns out that the triangulations of a regular polygon are in bijection with many other geometric and combinatorial sets of objects. There is a mythical polytope, called the associahedron, whose vertices correspond to the triangulations of a regular polygon. The associahedron itself has a long mathematical history, starting with work of Tamari and Stasheff. The associahedron today has connections to many diverse areas of mathematics, including moduli spaces and topology, quiver representation theory, cluster algebras and toric varieties. We will discuss the beginnings of this story, starting with cutting small polygons into triangles using non-crossing diagonals.