First steps in hyperbolic geometry | Universal Hyperbolic Geometry 4 | NJ Wildberger

This video outlines the basic framework of universal hyperbolic geometry---as the projective study of a circle, or later on the projective study of relativistic geometry. Perpendicularity is defined in terms of duality, the pole-polar correspondence introduced by Apollonius, and we explain that the three altitudes of a triangle meet in a point- the orthocenter H. The basic measurements of quadrance and spread in this geometry arise from the cross ratio of suitable points and lines. We state the main formulas: Pythagoras' theorem, the Triple quad formula, Pythagoras' dual theorem, the Triple spread formula, the Spread law and the Cross law and its dual. These are closely related to, but different from the corresponding laws in Rational Trigonometry. CONTENT SUMMARY: notion of perpendicularity @04:48 Perpendicularity via duality @05:33 Do the altitudes of a triangle meet in a point? @10:54 Quadrance: m'ment between points @15:14 exercise @18:41 remark on Beltrami-Klein model @19:11 Pythagoras' theorem and Triple quad formula @20:30 Spread: m'ment between lines and quadrance spread duality theorem @23:29 Remark on Beltrami-Klein model @26:45 Pythagoras' dual theorem @28:43 Main formulas for triangles that involve both quadrances and preads @31:13 (THANKS to EmptySpaceEnterprise) Video Content: 00:00 Introduction 5:27 Perpendicularity via duality 10:54 Quadrance: measurement between points 15:07 Quadrance: measurement between lines 18:41 remark on Beltrami-Klein model 20:30 Spread: measurement between lines 28:38 Pythagoras' dual theorem 31:11 Spread law ************ 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)