# 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