What if we define 1/0 = ∞? | Möbius transformations visualized

Head to https://brilliant.org/Mathemaniac/ to get started for free with Brilliant's interactive lessons. The first 200 listeners will also get 20% off an annual membership. 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 channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject. This video was sponsored by Brilliant. Video chapters: 00:00 Intro 02:38 Chapter 1: The 2D perspective 08:43 Chapter 2: More about inversion 14:33 Chapter 3: The 3D perspective (1/z) 19:38 Chapter 4: The 3D perspective (general) --------------------------------------------------- SOURCES: [That 2012 paper] Rigid motion 1-1 Möbius map: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?article=1218&context=rhumj Möbius transformations revealed: https://www.youtube.com/watch?v=0z1fIsUNhO4 Accompanying paper: https://www-users.cse.umn.edu/~arnold//papers/moebius.pdf Unitary iff rotation: https://users.math.msu.edu/users/shapiro/pubvit/Downloads/RS_Rotation/rotation.pdf Möbius iff sphere: https://home.iitm.ac.in/jaikrishnan/MA5360/files/mobius.pdf Rotation of Riemann sphere: https://people.reed.edu/~jerry/311/rotate.pdf Circle-preserving implies Möbius: https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.19670330506 Problem of Apollonius video: https://www.youtube.com/watch?v=Z6GG8zsMWH8 Power of a point: https://www.nagwa.com/en/explainers/798164323509 -------------------------------------------------------- MORE CONNECTIONS OF MÖBIUS MAPS: Sir Roger Penrose lecture on the book with Rindler (Spinors and space-time): https://www.youtube.com/watch?v=fzYV6VrsHyQ The book: https://www.cambridge.org/core/books/spinors-and-spacetime/B66766D4755F13B98F95D0EB6DF26526 Hyperbolic geometry: https://assets.cambridge.org/97811071/16740/excerpt/9781107116740_excerpt.pdf Conformal mapping (fluid mechanics): https://math.berkeley.edu/~iliopoum/Topics_121A/Conformal%20mapping%20in%20fluid%20mechanics.pdf -------------------------------------------------------- Music used: Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White River Asher Fulero - The Closing of Summer -------------------------------------------------------- Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: https://forms.gle/QJ29hocF9uQAyZyH6 If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe! Social media: Facebook: https://www.facebook.com/mathemaniacyt Instagram: https://www.instagram.com/_mathemaniac_/ Twitter: https://twitter.com/mathemaniacyt Patreon: https://www.patreon.com/mathemaniac (support if you want to and can afford to!) Merch: https://mathemaniac.myspreadshop.co.uk For my contact email, check my About page on a PC. See you next time!