TITLE: Visualising finite fields
QUESTION [23 upvotes]: I'm interested in finding visual and/or physical approaches to understanding finite fields. I know of a few: V. I. Arnold has a few pictures of 'finite circles' and 'finite tori' in his book Dynamics, Statistics and Projective Geometry of Galois Fields. Also, N. Carter displays what you might call 'double Cayley diagrams' of the fields of order $4=2^2$ and $8=2^3$ in his book Visual Group Theory, which I reproduce here:

The solid lines are the graph for addition and the dotted lines are the graph for multiplication. I like how you can see the structure of the additive group as a product of cyclic groups with the order of the characteristic, and if you look closer you can also see how the multiplicative group is cyclic.
Are there any other interesting visual/physical ways of understanding finite fields?

REPLY [3 votes]: You can geometrically construct the addition and multiplication operations of a given finite field $\mathbb{F}_n$ on a finite projective plane of order $n$, which is built from the vector space $\mathbb{F}_n^3$ (the points of the projective plane are all the lines through the origin in the vector space, which you can visualise as an infinite repeating 3d lattice, or a discrete 3d finite torus). There's a nice exposition of the constructions in §6.4 of Stillwell's Four Pillars of Geometry for the real projective plane which you can adapt to a finite projective plane.
Here's an example I drew for $1+1=0$ in $\mathbb{F}_2$:

The line at infinity is arbitrarily chosen, $\mathscr{L}$ is a line 'parallel' to the $x$-axis (i.e. meeting it at the line at infinity), and the construction lines are drawn thicker with the final line pointing at the result (0).