Published On Apr 20, 2011
Pappus' theorem is the first and foremost result in projective geometry. Another of his significant contributions was the notion of cross ratio of four points on a line, or of four lines through a point.
We discuss various important results: such as the Cross ratio theorem, asserting the invariance of the cross ratio under a projection, and Chasles theorem for four points on a conic. We show that the notion of cross ratio also works for four concurrent lines.
CONTENT SUMMARY: Pappus' theorem @00:52 cross ratio @02:46 cross ratio transformation theorem @11:08 cross ratio theorem @13:54 Chasles theorem @16:19 The cross ratio is the most important invariant in projective geometry 9:09
Video Contents:
00:00 Hyperbolic geometry as based on projective geometry
00:30 Pappus of Alexandria
02:45 Exercise - Verify Pappus' theorem for special cases
03:55 Cross ratio
08:41 Experience with cross ratio
11:03 Cross ratio transformation theorem
13:49 Cross ratio theorem
16:13 Chasles' theorem
19:07 Projective geometry and Desargues
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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.
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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
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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
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