# Visualising finite fields

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?

Here's an interesting take on the problem of visualizing finite fields: Thomas Alden Gassert, a graduate student, studies finite fields generated by the solutions of iterated polynomials, yielding a visualization of finite fields via cycles. See the pictures on his website and especially in his papers: https://www.math.umass.edu/~gassert/

• The link seems no longer to work. Perhaps this is one of the papers you meant? If you know of other links that could be added, that would be very helpful. – Will Orrick Jun 15 '15 at 15:12
• Yes, that's the paper. Gassert is currently at the University of Colorado at Boulder as a postdoc. Here's the full list of his arXiv papers: arxiv.org/find/math/1/au:+Gassert_T/0/1/0/all/0/1 – Jeff Jun 16 '15 at 1:30

While not giving visualizations of finite fields you might find the book: A Geometrical Picture Book by Burkard Polster, Springer, NY, 1998 of interest. It has ways of visualizing various combinatorial objects such as finite affine planes and finite projective planes (some of which can be co-ordinatized using finite fields), as well as means for drawing other finite combinatorial objects (Steiner triple systems, generalize quadrangles, etc.).

John Conway has constructed the finite field $\mathbf F_{2^n}$ using the game of Nim. It is quite a surprising construction, without a known counterpart for odd primes (as far as I know). You can see the Wikipedia entry and the references at the bottom.

• +1: But unless you go to infinite ordinals, you only get the fields $\mathbf{F}_{2^{2^n}}$ from the construction. – Jyrki Lahtonen Nov 1 '13 at 18:36

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).