Isometry groups of the projective line III | Rational Geometry Math Foundations 140 | NJ Wildberger

We extend our discussion of elementary metrical projective geometry in one dimension to incorporate Einstein's special theory of relativity. This remarkable new understanding of Einstein transformed much of 20th century physics, but its effect on pure mathematics has been surprisingly modest.

In this video we see that even in the one dimensional setting, relativistic geometry, also associated with the names of Lorentz and Minkowski, is a rich alternative framework. This is a useful introduction for physicists to a geometry that figures very prominently in modern physics. One of the key new features is the existence of null points, which are analogous to the situation that we met over F_5 in our previous lecture on the Euclidean setting, and which correspond to the space-time trajectories of photons.

This video is an introduction to a fascinating new world of geometry, much bigger than the one we usually thing about. A key feature is that we are required to think more algebraically, and let go of that real number dreaming that currently distorts our geometrical understanding.

************************

Here are the Insights into Mathematics Playlists:

[txikitofandango]

Ex. 140.5: Let x1^2 be less than y1^2. Now, without loss of generality, there are two cases. Case 1: x2^2 < y2^2 and Case 2: x2^2 > y2^2 (they can't be equal). For Case 1, in the quadrance formula without the "1 minus, " the numerator is always negative, both factors of the denominator are always negative, and so the whole fraction is negative. For Case 2, use the other quadrance formula (1 minus the big fraction). The numerator is always negative, but this time the factors of the denominator have opposite signs, and so that fraction is overall negative, and so the quadrance is 1 minus a negative number, and hence greater than 1.

This probably could've been solved without this two-case setup by forming some quadratic equation.

[rewtnode]

@25:15 is this relativistic P1 representable by split complex numbers? With z = a + j b and j^2=1. Assuming one can make that work over rational numbers.