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?