Hexagrammum Mysticum 3 | Affine and projective geometry and a proof of Pappus' theorem | Wild Egg

We want to explore one of the most remarkable developments of 19th century geometry -- the Hexagrammum Mysticum arising from Pascal's theorem viewed more symmetrically. A good place to start is with the corresponding situation for the precursor to Pascal's theorem, namely Pappus' theorem. We will in fact uncover a wide range of new previously undiscovered phenomenon here, with plenty of scope for amateur investigations! Here we develop some preliminary material for this. We look at some basic facts about affine and projective geometries in the 2D situation, for example the definitions of points and lines, and the usefulness of projective coordinates even in the affine case. We discuss also symmetries of the affine plane. And then we give a simple proof of Pappus' theorem, using the analytic freedom that symmetries provide, allowing us to choose the two initial lines in a particularly simple and pleasant fashion. The more complete playlist is available to Members at Hexagrammum Mysticum :    • Playlist   Video Contents: 0:00 – Introduction: Pappus' Theorem & Projective Geometry 4:25 – Incidence in Projective Geometry 7:23 – The Power of Projective Coordinates for Computation 12:36 – Duality: Join of Points and Meet of Lines 17:25 – Translations as Simple Parallelism-Preserving Transformations 23:09 – Linear Transformation to Align Lines with Coordinate Axes 26:25 – Checking Collinearity of Points c1, c2, c3 Using a 3x3 Determinant 27:17 – Efficient Computations Using Coordinate Transformations and Computers ************************ My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated.