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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)
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SOURCES:
[That 2012 paper] Rigid motion 1-1 Möbius map: scholar.rose-hulman.edu/cgi/viewcontent.cgi?articl…
Möbius transformations revealed: • Möbius Transformations Revealed [HD]
Accompanying paper: www-users.cse.umn.edu/~arnold//papers/moebius.pdf
Unitary iff rotation: users.math.msu.edu/users/shapiro/pubvit/Downloads/…
Möbius iff sphere: home.iitm.ac.in/jaikrishnan/MA5360/files/mobius.pd…
Rotation of Riemann sphere: people.reed.edu/~jerry/311/rotate.pdf
Circle-preserving implies Möbius: onlinelibrary.wiley.com/doi/epdf/10.1002/mana.1967…
Problem of Apollonius video: • Problem of Apollonius - what does it ...
Power of a point: www.nagwa.com/en/explainers/798164323509
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MORE CONNECTIONS OF MÖBIUS MAPS:
Sir Roger Penrose lecture on the book with Rindler (Spinors and space-time): • Sir Roger Penrose on collaborating wi...
The book: www.cambridge.org/core/books/spinors-and-spacetime…
Hyperbolic geometry: assets.cambridge.org/97811071/16740/excerpt/978110…
Conformal mapping (fluid mechanics): math.berkeley.edu/~iliopoum/Topics_121A/Conformal%…
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Music used:
Aakash Gandhi - Heavenly / Kiss the Sky / Lifting Dreams / White River
Asher Fulero - The Closing of Summer
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Mathemaniac