Isometry groups of the projective line (II) | Rational Geometry Math Foundations 139 | NJ Wildberger

In this video we show that the algebraic approach to the metrical structure of the projective line, including the group of isometries including rotations and reflections, can all be defined and studied over a finite field. This is quite a remarkable fact. It leads us to think that perhaps much of Euclidean geometry has a valid interpretation in such a finite setting---and in fact this turns out to be the case. This new meeting ground of traditional geometry and a concrete combinatorial domain will inevitably lead to many new avenues of investigation: in particular we can now count things, since there always a finite number of objects around!

Here we look at the field F3 with three elements and the field F5 with five elements. It turns out that there are significant differences between these two cases, and in fact we discover aspects of Euclidean geometry that are invisible over the much more familiar setting of the rational numbers, and that connect naturally with the relativistic geometry of Lorentz, Einstein and Minkowski.

In fact this finite Euclidean geometry is a good introduction to relativistic thinking!

Video Content:
00:00 Intro to projective metrical structure
1:45 Notation for projective points
6:39 Exercise
10:18 The projective line
14:45 Rotations and reflections
19:22 A group multiplication table
23:36 The projective line p_1 over F5
29:23 Formulas for projective rotations and reflections

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[norbertfrohlich]

In exercise 139.3 with [1 : -2] rho[1 : 2] I get a proportion [0 : 0], which is invalid, not [1 : 2]