Why 2-Dimensional Polygons?

2D Polygons

In this course, Professor Satyan Devadoss (University of San Diego) explores the importance of 2-dimensional polygons, their properties, and their uses. In the first mini-lecture, we define 2D polygons, discuss what motivates us to study them, and differentiate between something that is continuous and something that is discrete. In the second mini-lecture, we explore how a process called diagonalization allows us to break polygons down into triangles until the polygon is composed solely of triangles (triangulation), making triangles the fundamental building blocks of polygons. In the third mini-lecture, we use our understanding of polygons from the previous mini-lectures to help us think about sensors and security systems, looking in particular at the Art Gallery theorem. In the fourth mini-lecture, we discuss polygon congruency and similarity before delving deeper into a new concept called scissors congruency and the 1833 Wallace–Bolyai–Gerwien theorem. In the fifth mini-lecture, we seek to extend our understanding of 2D polygons to 3D polyhedra, where we see that many of the concepts in the world of 2D polygons fail when applied to 3D polyhedra.

Why 2-Dimensional Polygons?

In this mini-lecture we think about why 2-dimensional polygons are worth studying and give the definition of a polygon. In particular, we consider: (i) what motivates our study of 2D polygons; (ii) the difference between something that is continuous and something that is discrete; (iii) how anything continuous can be approximated with near-perfect accuracy by something discrete; (iv) how computer think digitally, in discrete packets; (v) the need for calculus to understand the continuous; (vi) the definition of a polygon: a closed region of the plane, surrounded by a finite collection of line segments that do no intersect; (v) the importance of objects represented by finite data, which permits computers to store and optimise this data; (vi) examples of polygons; (vii) and the definitions of vertices (singular: vertex) and edges (singular: edge).