This course will explore how one of our senses -- our sense of sight -- relies on our brain's seemingly hardwired understanding of a fascinating geometric space. What's more, this space -- known as the two dimensional Projective Plane -- is fantastically unlike the Euclidean space in which we live and work.By examining some basic questions, like why parallel lines appear to converge in our vision, we can identify and analyze the defining properties of this space — properties which are well known to our visual sense, but openly defy our logical intuition!As the course progresses, we'll situate perspective drawing in the framework of projective geometry, and get familiar with the algebra of homogeneous coordinates, which will allow us to translate our sensory intuition into precise information we can easily communicate. We'll also get to know the topological space known as the real projective plane, as well the Lie group that governs its transformations.Being a four-week long elective, this course will nicely complement standard courses on manifolds, geometry, lie groups, and topology, by giving hands on experience with several important mathematical objects you’ll meet in those classes. And you’ll also get a free art lesson to boot!As part of the series 'Our Mathematical Senses', we plan to offer follow-up courses in subsequent semesters, exploring our sense of spatial orientation (the Topology of Movement) and our sense of hearing (the Algebra of Sound).INTENDED AUDIENCE :Any Interested LearnersPRE-REQUISITES :Linear Algebra is the only prerequisite
WEEK 1: Perspective Drawing from First Principles
Lecture 1: Why do families of parallel lines appear to converge in our vision?
Lecture 2: Can we determine the position of the observer in a perspective drawing?
WEEK 2: An Invitation to Projective Geometry
Lecture 3: A geometry without parallel lines?
Lecture 4: Does projective geometry tell us anything about nonlinear curves?
WEEK 3: The Cross Ratio: A Projective Invariant
Lecture 5: What is preserved under perspective transformations?
Lecture 6: What can the cross ratio tell us?
WEEK 4: Automorphisms of the Projective Plane
Lecture 7: Homogeneous coordinates and the topology of the real projective plane
Lecture 8: Automorphisms of the real projective plane