Regular 2-Dimensional Polygons

3D Polyhedra

In this course, Professor Satyan Devadoss (University of San Diego) explores 3-dimensional polyhedra and objects in higher dimensions. In the first mini-lecture, we introduce 2-dimensional polygons, in particular regular polygons. In the second mini-lecture, we think about the construction of regular 3-dimensional polyhedra out of polygons, and consider the constraint that must be placed on the corner angles. In the third mini-lecture, we use our constraint equation from the previous mini-lecture to discover that there are only five regular polyhedra, known as the Platonic solids. In the fourth mini-lecture, we address how to express 4D objects in our 3D world by considering how to express 3D objects in a 2D world using the Shlegel diagram. Finally, in the fifth mini-lecture, we explore regular polytopes in 4D, 5D, and beyond.

Regular 2-Dimensional Polygons

In this mini-lecture, we explore the motivations behind studying 3D polyhedra and think about 2D polygons. In particular, we think about: (i) how 2D polygons make up the sides of 3D polyhedra; (ii) finding beauty in mathematics such as numbers, patterns, shapes, ideas, and symmetry; (iii) regular polygons. i.e. polygons where the sides are all the same length and the angles are all the same — examples including the triangle, square, pentagon, and hexagon, where we see that there are an infinite number of regular polygons; (iv) and how to find the corner angles of a regular n-sided polygon using triangulation.