# Geometry

## Introduction

Probably the simplest, most-tactile form of geometry is projective geometry, which only requires a straight-edge (no compass, no ruler, no protractor, etc.).

With such a basic setup, we’re limited to just two sorts of object: points and lines. Two distinct points are joined by a line. For example, these two points labelled `A` and `B` are joined by the line shown:

``````    A      B
────⋆──────⋆────
AB``````

We can write one point after another to indicate the line joining them; in this case `AB` is the join of `A` and `B`. Note that we can swap the order without changing the line:

``XY = YX for any points X and Y``

Similarly, any two distinct lines will meet at a point. For example, these two lines labelled `l` and `m` meet at the point labelled `lm` (again, just writing one after the other):

``````╲
╲lm
──⋆──l
╲
╲m``````

Just like joins, meets also don’t change if we swap their order:

``xy = yx for any lines x and y``

## Parallelism

Projective geometry doesn’t have the concept of parallelism: any distinct lines will meet at a point; even those which might be considered “parallel” by other geometries.

There are a few ways to interpret this: as great-circles meeting on the equator of a sphere; as points located “at infinity”; as a surface with its edges “glued together backwards” (like a Mobius band); as plnes through a point in 3D; etc. I’ll remain agnostic about these interpretations, go with whichever you find most helpful!

lines w two lines meet, even those which You might be wondering what happens for two parallel lines, since they don’t meet. they don’t meet

## Constructing a Number Line

Projective geometry lets us contruct an interesting form of number line. We start with two points