Tractatus Logico-Philosophicus A Book by Ludwig Wittgenstein www.gutenberg.org The totality of factsEverything that can be saidI am my worldEthics and aesthetics are oneWhereof one cannot speak
What we have known since long A Quote by Ludwig Wittgenstein The problems are solved, not by giving new information, but by arranging what we have known since long. notetakingunderstandingproblemsinformation
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