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) ************ Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Wild Egg Maths YT channel: https://www.youtube.com/channel/UCriF... Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos)    • Elementary Mathematics Explained (K-6)   (40 videos)    • Ancient Mathematics and insights of Howard...   (7 videos)    • Wild West Banking: A mathematician goes We...   (7 videos)    • Playing Go: the ancient oriental board game   (19 videos)    • Maths and Music   (21 videos)    • Year 9 Mathematics (review fractions, deci...   (10 videos)    • Wild Trig: An introduction to Rational Tri...   (94 videos) Wild Egg Maths Playlists:    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)