The three-fold symmetry of chromogeometry | Rational Geometry Math Foundations 141 | NJ Wildberger

There are three planar metrical geometries that fit together beautifully; the usual Euclidean geometry (which we call blue), and two relativistic geometries (red and green). Most of the fundamental theorems of the subject hold for all three. But even more remarkably, when we transcend Klein's Erlangen program and consider all three geometries simultaneously, many completely new and unexpected relations and patterns appear.

This is a key door into 21st century geometry. Hundreds of fascinating discoveries await those who venture forth and learn how to think about geometry in this bigger, richer way. It also provides a lot of evidence to support the purely algebraic, logical point of view that results from letting go of the `real number' dreaming!

Video Content:
00:00 Intro to chromogeometry!
2:23 The green relativistic structure
6:28 Projective line
8:43 Exercises
11:10 Blue, red and green quadrances
12:45 Coloured quadrances theorem
15:31 Perpendicularity
17:46 Relation between three perpendiculars

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Here are the Insights into Mathematics Playlists:

[postbodzapism]

Professor, I was thinking like perhaps the three geometries actually fit into one rank-3 tensor g_{abc} where a, b, c run from 1 to 3. This acts as a metric and the quadrance would become a "cubance", and one works in the projective plane, therefore giving rise to 3 coordinates.
Working in Riemannian geometry, the problem is how would curvature be defined from this rank-3 "metric"? And is there such thing as projective Riemannian geometry?

[txikitofandango]

Just had a prediction: if the triple quad formula holds for all three colors, then there should be some nice correspondences between points (lines through the origin) in different colors.