---
title: 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