Harmonic Quadrangles & The 13 Configuration 
Harmonic Quadrangles & The 13 Configuration
by Richard Southwell
Video Lecture 8 of 16
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Views: 1,498
Date Added: March 5, 2015

Lecture Description

We introduce harmonic ranges, harmonic conjugates and harmonic pencils. We discuss motivating examples from Apollonias and angle bisection. We also describe how harmonic ranges get induced by quadrangles, and how harmonic pencils get invoked by quadrilaterals. We also discuss the famous 13 configuration.

This videos uses some material from Wildberger's early lectures on universal hyperbolic geometry. The crucial point of our lectures is that harmonic conjugates are preserved by projection.

The cross ratio


is a more general projective invariant which takes a value of minus 1 when harmonic ranges occur.

Course Index

Course Description

Protective geometry is deeper and more fundamental than standard euclidean geometry, and has many applications in fundamental physics, biology and perspective drawing. We shall introduce it visually, without relying upon equations. The hope is make this beautiful subject accessible to anybody, without requiring prior knowledge of mathematics. At the same time, there are some very deep, rarely discussed ideas in this subject which could also benefit experts.


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