Since the title is almost longer than the talk, I will probably not completely cover everything promised. But this is a talk embedded in a series so things can always be pushed forward. The origin of this subject for me is my fascination with square ice. This is a 2-d model like everything in the talk in which every edge connecting 2-lattice points that have a distance 1 from each other has an arrow either up or down, or right or left. The configuration is complete and satisfies the condition that exactly 2 arrow heads touch a lattic point. This means there are '4 choose 2' = 6 local configurations possible at each point. This is known to be equivalent to the 3 color model in which the plane is tiled by 1 by 1 squares each colored one of Red, Blue, Green with the condition that no adjacent tiles have the same color. (There is also an equivalent dimer (i.e. domino) model. We show other equivalences. This model is connected with the modular group as can be seen by its connections with the Farey tree. Applying the formalism of the Farey tree to symbols instead of numbers gives a family of minimal Sturmian transformations and these code to create the matrices A_n, n = 1, 2, . . . so that (Trace((A_n)^n))^(1/n) converges to the entropy of these models. We can also look at the graph represented by these mostly 0-1 matrices and see that it is a kind of mediant triangulation of the hypercube. The normalized spectral radius of these graphs also give the entropy and other invariants of the square ice process. With a little squinting one can see the formalism of quantum mechanics within these models. Abstracting these models to Temperley-Lieb algebras leaves the beauty of these entropy relationships bare and connects them to the objects in the title and many more, including the quantum teaser.