Here is a short reading list that will help:

• https://en.wikipedia.org/wiki/Coriolis_acceleration
• https://en.wikipedia.org/wiki/Rossby_wave
• http://science.sciencemag.org/content/247/4946/1061

But it is a difficult concept to get one's head around. Explanations like : "We propose that Saturn’s long-lived polygonal structures correspond to wavemodes caused by the nonlinear equilibration of barotropically unstable zonal jets." require a lot of unpacking (and critical thinking)

• http://people.seas.harvard.edu/~rwordsworth/aguiar2009.pdf

They report : "the phenomenon is as yet poorly understood."

• http://ciclops.org/media/sp/2011/6793_16343_0.pdf
• https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2013GL059078

As already discussed it is not well understood but there a number of papers that have been published that make some attempt at explaining.

Morales-Juberías, R., Sayanagi, K.M., Simon, A.A., Fletcher, L.N. and Cosentino, R.G., 2015. MEANDERING SHALLOW ATMOSPHERIC JET AS A MODEL OF SATURNʼS NORTH-POLAR HEXAGON. The Astrophysical Journal Letters, 806(1), p.L18.

Rostami, M., Zeitlin, V. and Spiga, A., 2017. On the dynamical nature of Saturn’s North Polar hexagon. Icarus, 297, pp.59-70.

Cosgrove, J.M. and Forbes, L.K., 2017. A δ-plane simulation of anticyclones perturbing circumpolar flows to form a transient north polar hexagon. Monthly Notices of the Royal Astronomical Society, 469(4), pp.4133-4147.

Also this experiment (from the paper posted above by Ian Kennedy) shows how it is possible to create a stable hexagon like structure in a rotating fluid. Although it's not quite the same and Saturn's hexagon should be thought of as more 3D than a single layer.

https://www.youtube.com/watch?v=iheK6JX3rLg

In addition, there has also been seasonal changes in the colour of the Hexagon during the time Cassini was in orbit around Saturn.

Recent research about Saturn's North Polar Hexagon done at LMD:

Article On the dynamical nature of Saturn’s North Polar hexagon

explains the dynamical nature of hexagon and its dependancy to Saturn's North Polar Cyclone.