Apollonius and harmonic conjugates | Universal Hyperbolic Geometry 2 | NJ Wildberger
28.4 هزار بار بازدید -
13 سال پیش
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Apollonius introduced the important idea
Apollonius introduced the important idea of harmonic conjugates, concerning four points on a line. He showed that the pole polar duality associated with a circle produces a family of such harmonic ranges, one for every line through the pole of a line. Harmonic ranges also occur in the context of vertex bisectors, as combinations of vectors, and associated with the sides of a quadrangle.
CONTENT SUMMARY: polarity holds for general conics @03:58 geometers sketchpad in use @05:02 How to find the polar of a null point @05:50 Harmonic conjugates @08:58 discussion of various types of geometry @12:17 More on harmonic conjugates @16:37 examples of harmonic ranges and harmonic ranges theorem @24:00 Harmonic pencils and Harmonic bisectors theorem @28:34 Harmonic vector combinations theorem @32:37 Harmonic quadrangle theorem @34:34
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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
CONTENT SUMMARY: polarity holds for general conics @03:58 geometers sketchpad in use @05:02 How to find the polar of a null point @05:50 Harmonic conjugates @08:58 discussion of various types of geometry @12:17 More on harmonic conjugates @16:37 examples of harmonic ranges and harmonic ranges theorem @24:00 Harmonic pencils and Harmonic bisectors theorem @28:34 Harmonic vector combinations theorem @32:37 Harmonic quadrangle theorem @34:34
************************
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
13 سال پیش
در تاریخ 1390/01/29 منتشر شده
است.
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