What could we reasonably call “a geometry”? What sort of structures emerge if we try to do geometry on different surfaces, with different axioms, with lengths and distances different from what we are used to? What other interesting classes of shapes besides polygons are there, and what can we say about them?
Geometry is not just about measuring and drawing figures on a flat plane (although we’ll do some of that, too). We will introduce aspects of geometry that answer the questions above and some others, including non-Euclidean and projective geometry, metric spaces, topology, graph theory, and convexity. Each day of the semilab will we will present the main concepts and examples of a new topic, work on exercises and puzzles related to the topic, and look for connections between the day's theme and other geometry ideas.
Prerequisites: middle-school algebra and keen interest in math; some high-school geometry is expected, but not strictly necessary.