Getting used to manifolds

It’s confusing to think about 2-d spaces, the necessity for us to look at them in 3-d, and the added requirement that 2-d manifolds require a fourth dimension to intersect and transform. This is a good introductory video that sorts out some of the terms and transformations that allow Lacan to use topology to think about the subject of psychoanalysis that lives up to Freud’s saying, “Pysche is extended; knows nothing of it!” Drew’s Campfire, “Paradox of the Möbius Strip and Klein Bottle – A 4D Visualization.” Embark on a mind-bending journey into the 4th dimension as we explore the fascinating geometry of the Möbius Strip and Klein Bottle. This video will take you on a whirlwind tour of time travel, geometric paradoxes, 4D visualization, and sentient primitive shapes.

A Miraculous Proof (Ptolemy’s Theorem) – Numberphile

Wikipedia: Zvezdelina Stankova began attending the Ruse math circle as a fifth grader in Bulgaria, the same year she learned to solve the Rubik’s Cube and began winning regional mathematics competitions. She worked at the University of California, Berkeley as Morrey Assistant Professor of Mathematics before joining the Mills College faculty in 1999, and continues to teach one course per year as a visiting professor at Berkeley. This video explains the rules of the inversion circle and how to draw lines, circles, and other shapes as they are projected from inside to outside and back again.

Dirac’s belt trick, topology, and 1/2-spin particles

Noah Miller explains the method invented by the physicist Paul Dirac to explain the mysterious 1/2 spin of the electron. In the course of this exposition, we are introduced to rotation space, vector spin, and the idea of a circuit as a 4π or double rotation returning to the origin.

What if we define 1/0 = ∞?

From the YouTube site: “Defining 1/0 = ∞ isn’t actually that bad, and actually the natural definition if you are on the Riemann sphere -∞ is just an ordinary point on the sphere! Here is the exposition on Möbius maps, which will explain why 1/0 = ∞ isn’t actually something crazy. And this video will also briefly mention the applications of the Möbius map.” As is the case for all videos in the series, this is from Tristan Needham’s book “Visual Complex Analysis.” There will also be things like circular and spherical inversion, which are really neat tools in Euclidean geometry to help us establish lots of interesting results, this one included. This video sponsored by “Mathemaniac,” Brilliant.org. synthesizes a number of issues surrounding inversive geometry, rotational space, cover spaces, and projectivity.