I've enjoyed playing around with projective geometry recently: that's even more basic, since it doesn't involve a compass, only a straightedge! The fundamental

I've enjoyed playing around with projective geometry recently: that's even more basic, since it doesn't involve a compass, only a straightedge! The fundamental objects in projective geometry are points and lines: - Given two distinct points, a unique line joins them - Given two distinct lines, they meet at a unique point We can't do anything with just a single line/point. With a pair we can find their meet/join, but that's it. With three we can find all the meets and joins (forming a triangle). It's only once we have four objects that things get interesting, and we can start joining points, then meeting those lines, then joining those meets, and so on. Note that lines don't always meet in Euclidean geometry, since parallel lines never meet. Projective geometry avoids this by including "points at infinity". Modelling that with normal 2D diagrams is quite mysterious, especially since opposite directions approach the same point at infinity. Yet it becomes very simple if we switch to 3D: - We can draw our points on a hemisphere instead of a plane, with great-circles for lines: distinct great-circles will always meet, and the "points at infinity" are simply those on the equator. - Instead, we can choose an "origin" sitting above our plane, and connect it to our points (forming lines) and lines (forming planes). In that case, the "points at infinity" are just the lines through the origin which are parallel to the plane (thus never meeting it). There's lots of fascinating results in this framework, from the Greeks to modern times: - Here's a nice video with physical 3D models https://www.youtube.com/watch?v=dBH-Id8VC3U https://www.youtube.com/watch?v=dBH-Id8VC3U - Here's a lecture on its history https://www.youtube.com/watch?v=NYK0GBQVngs https://www.youtube.com/watch?v=NYK0GBQVngs In fact the latter channel has loads of videos on the subject https://www.youtube.com/results?search_query=insights+into+mathematics+projective https://www.youtube.com/results?search_query=insights+into+m...