How Chromogeometry Transcends Klein's Erlangen Program for Planar Geometries - N J Wildberger

Explore a groundbreaking lecture on chromogeometry, a revolutionary approach that unifies Euclidean and relativistic geometries to transform our understanding of planar geometry. Delve into the foundations of Rational Trigonometry and discover three fundamental planar metrical geometries. Examine the generalization of concepts like Euler lines, orthocenters, and circumcenters, culminating in the innovative Omega triangle. Investigate a novel perspective on conic sections, including the introduction of ellipse diagonals and chromogeometric aspects of ellipses and parabolas. Gain insights into the power of rational thinking in mathematics, challenging traditional notions of infinite sums and real numbers. Follow along as the lecture progresses through topics such as quadrance, spread, Pythagoras theorems in different geometries, colored altitudes, Euler lines, and chromatic parabolas. Witness how this new approach to geometry transcends Klein's Erlangen Program and opens doors to exciting mathematical discoveries.

Note: Around the diagram for the green geometry has one of the "green squares" incorrectly drawnthe one with area 18 --it too should be a parallelogram.
Intro to three-fold symmetry in planar geometry
Quadrance between points
Pythagoras and Triple quad formula
Spread between lines
Spread as a normalized squared determinant
Laws of affine rational trigonometry
Thales' theorem
Proof of Cross law
Blue Pythagoras
Red Pythagoras
Green Pythagoras
Coloured altitudes
Blue Euler line
Red Euler line
Green Euler line
Omega triangle
Omega triangle and more
Blue ellipse and its diagonals
Red ellipse
Green ellipse and diagonals
Three sets of foci and directrices
A chromatic parabola
References