Isometry groups of the projective line (I) | Rational Geometry Math Foundations 138 | NJ Wildberger

The projective line can be given a Euclidean structure, just as the affine line can, but it is a bit more complicated. The algebraic structure of this projective line supports some symmetries. Symmetry in mathematics is often most efficiently encoded with the idea of a group--a technical term referring to the way symmetries can be composed to generate new symmetries, satisfying certain properties.

Here we look at two kinds of symmetries called isometries which preserve the Euclidean structure: rotations and reflections (it turns out these are the only types of isometries in this context). We derive algebraic relationships between these isometries. This is a nice introduction to Group Theory, rather explicit and concrete--which of course is the best way to learn about abstract objects!

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

Here are the Insights into Mathematics Playlists:

[txikitofandango]

For Exercise 138.1, I just threw this into Wolfram Alpha "simplify and it works! All the a's and b's vanish! I don't know if subscripts are possible in Wolfram Alpha, so I used [u:v] for [x_2, y_2].