Welcome to class An Essay by Bill Tozier vaguery.com I differ from almost all your previous instructors in three ways: First, I acknowledge that this is true, whereas they have for the most part lied to you (and themselves) and declared you competent, even though they’ve had to re-train you from scratch in every damned class. Second, unlike them I intend to do something about it. And, third, in order to do something about it, I will let you—no, make you—cheat. teaching
Wang tiles Wang tiles (Hao Wang, 1961) are a class of formal systems. They are modelled visually by square tiles with a color on each side. A set of such tiles is selected, and copies of the tiles are arranged side by side with matching colors, without rotating or reflecting them. The basic question about a set of Wang tiles is whether it can tile the plane or not, i.e., whether an entire infinite plane can be filled this way. The next question is whether this can be done in a periodic pattern. In 1966, Wang's student Robert Berger solved the problem in the negative. He proved that no algorithm for the problem can exist, by showing how to translate any Turing machine into a set of Wang tiles that tiles the plane if and only if the Turing machine does not halt. The undecidability of the halting problem then implies the undecidability of Wang's tiling problem. Wikipedia en.wikipedia.org Truchet TilesThe Tiling Patterns of Sebastien Truchet and the Topology of Structural Hierarchy mathalgorithms