How Chromogeometry transcends Klein's Erlangen Program for Planar Geometries| N J Wildberger

Описание: We give a pictorial introduction to chromogeometry, a remarkable new three-fold symmetry which brings together Euclidean geometry and two relativistic geometries to transform our understanding of planar geometry. And it also urges us to re-evaluate Felix Klein's Erlangen Program---which viewed geometry as a group acting transitively on a space. There is actually quite a lot more to it than that.
We give a quick introduction to Rational Trigonometry, introduce three fundamental planar metrical geometries, and then go on to generalize Euler lines, orthocenters, circumcentres, and put them together as the Omega triangle of a triangle. We also have a novel look at the conic sections, introduce the diagonals of an ellipse, and then illustrate chromogeometric aspects of ellipses and parabolas.
There is a lot of new mathematics here. It really shows the power of thinking rationally, and putting aside all those dreamy arithmetical fantasies about "infinite sums" and "real numbers". When we start doing correct mathematics, a lot of doors open for us!

Note: Around 15:50 the diagram for the green geometry has one of the "green squares" incorrectly drawn(the one with area 18) --it too should be a parallelogram.

Thanks to Joshua Ho for videoing.

Video Content:
00:00 Intro to three-fold symmetry in planar geometry
4:33 Quadrance between points
6:14 Pythagoras and Triple quad formula
7:41 Spread between lines
8:53 Spread as a normalized squared determinant
9:23 Laws of affine rational trigonometry
10:38 Thales' theorem
11:19 Proof of Cross law
12:33 Blue Pythagoras
14:02 Red Pythagoras
15:49 Green Pythagoras
17:01 Coloured altitudes
19:11 Blue Euler line
19:50 Red Euler line
21:50 Green Euler line
22:26 Omega triangle
23:57 Omega triangle and more
25:13 Blue ellipse and its diagonals
27:21 Red ellipse
27:48 Green ellipse and diagonals
27:55 Three sets of foci and directrices
33:36 A chromatic parabola
39:01 References

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