# The Unit Plane Graph

Suppose we were to take the plane, $\mathbb{R}^2$, and turn it into a graph – not the function plotting kind, the edge and vertex kind.

We let every point in the plane be a vertex, and we draw an edge between two vertices if they are at a distance of exactly $1$ from each other (Euclidean metric. We could take other metrics or other base spaces, but that is perhaps a topic for a further post).

We’ve now got a pretty formidable object! Uncountably many vertices and uncountably many edges at each vertex! But this graph is amenable enough after a little thought. Three quick ideas: