The Algebraic Topology: A Beginner's Course
The Algebraic Topology: A Beginner's Course (UNSW). Taught by N. J. Wildberger, this course provides an introduction to algebraic topology, with emphasis on visualization, geometric intuition and simplified computations. Algebraic topology is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This course introduces a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. And it also has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical.
| Lecture 12 - Rational Curvature, Winding and Turning |
This lecture introduces an important re-scaling of curvature, using the natural geometric unit rather than radians or degrees. We call this the turn-angle, or tangle, and use it to describe polygons, convex and otherwise. We also introduce winding numbers and the turning number of a planar curve.
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