In this piece, there is a grid of pixels in three colors: one representing rock, one representing paper, and one representing scissors. When pixels are mostly surrounded by pixels which “beat” them, they switch (so rock mostly surrounded by paper becomes paper, for example). The system evolves but never settles, as waves of rock, paper, and scissors cross the screen. With the right color choices, the result is striking.
Chip-firing is a model in dynamical combinatorics in which chips are passed back and forth along the edges of a graph. In this model, one can construct the ‘‘sandpile group’’ of a graph; this group’s identities have a beautiful fractal-esque structure which can be visualized for certain types of graphs.
This post discusses the use of probability theory in combinatorics, specifically to search for objects with particular properties (or prove no such objects exist). This is aimed at anyone interested in combinatorics who has basic exposure with probability theory (indicator variables, expected values).
This post is of the notes from a talk I gave on an application of the probabilistic method to problems in algebra: the problem of being given a subset S of a finite abelian group A, and trying to find the largest subset T of S which is “sum-free”. Here, we give a bound on the size of T in terms of S, and show that it is tight.
This image depicts the Julia set (that is, the set of initial values which remain bounded under iteration) of a holomorphic extension of the Collatz function (which is initially defined on any positive integers n to divide n by 2 if n is even and multiply n by 3 and add 1 otherwise). This results in a fractal representing the complexity of the Collatz Conjecture.