Apollonius and polarity | Universal Hyperbolic Geometry 1 | NJ Wildberger

This is the start of a new course on hyperbolic geometry that features a revolutionary simplifed approach to the subject, framing it in terms of classical projective geometry and the study of a distinguished circle. This subject will be called Universal Hyperbolic Geometry, as it extends the subject to arbitrary fields, as well as to the outside of the light cone/null circle.

We begin by going back to Apollonius of Perga (in present day Turkey, not Italy!) and his understanding of the crucial role of polarity in studying conics, in particular the circle. Given a fixed circle, to each point in the plane we associate a line called the polar, and conversely to a line we associated a point called its pole. This duality is all important for hyperbolic geometry

CONTENT SUMMARY: conics @00:00 the circle @5:00 Thales thm @05:20 Greek m'ment @06:20 polarity, pole of A and polar of a @09:06 def' of polarity (projective def') @14:40 Polar independence Theorem @19:13 projective def' of polarity @23:34 polar of a point inside the circle @25:10 3-way symetry @26:40 hands_on experience @27:52 pole, polar starting with four points on circle @29:12 Quadrangle and quadralateral @29:26 Polar duality thm @31:30 The distinguished circle @37:40 Pole of a line thm @38:00 (THANKS to EmptySpaceEnterprise)

************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.

My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...

My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.

Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!

If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon: njwildberger Your support would be much appreciated.

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: playlist
list=PL8403C2F0C89B1333
Year 9 Maths:   Year9Maths
Ancient Mathematics: Ancient Mathematics
Wild West Banking: Wild West Banking
Sociology and Pure Mathematics: Sociology and Pure Mathematics
Old Babylonian Mathematics (with Daniel Mansfield): playlist
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: MathHistory: A course in the History ...
Wild Trig: Intro to Rational Trigonometry: WildTrig: Intro to Rational Trigonometry
MathFoundations: Math Foundations
Wild Linear Algebra: Wild Linear Algebra
Famous Math Problems: Famous Math Problems
Probability and Statistics: An Introduction: Probability and Statistics: an introd...
Boole's Logic and Circuit Analysis: Boole's Logic and Circuit Analysis
Universal Hyperbolic Geometry: Universal Hyperbolic Geometry
Differential Geometry: Differential Geometry
Algebraic Topology: Algebraic Topology
Math Seminars: MathSeminars
************************

And here are the Wild Egg Maths Playlists:

Triangle Centres: ENCYCLOPEDIA OF TRIANGLE CENTERS
Six: An elementary course in pure mathematics: Six: An elementary course in Pure Mat...
Algebraic Calculus One: Algebraic Calculus One
Algebraic Calculus Two: Algebraic Calculus Two
м

• #education
• #univhypgeom1
• #hyperbolic
• #geometry
• #universal
• #wildberger
• #course
• #education
• #apollonius
• #circle
• #polarity
• #pole
• #polar