This graph represents the last integers leading to 1 when subjected to the Collatz function, which, for any positive integer n, returns n/2 if n is even, or 3n+1 if n is odd. The Collatz conjecture states that the repeated connection of even numbers n to n/2, and odd numbers n to 3n+1, links all natural numbers to a single tree having 1 as a root. Once the value 1 has been reached, the cycle (1 4 2) repeats itself endlessly. Here, the numbers highlighted in orange are powers of 2, while the ones in purple are odd.
Two representations for the abelian sand pile model, constructed in the following way: assume that a point in Z^2 contains an integer number (weight) of chips, such that if this number is greater or equal to 4, the position will topple over, equally distributing over the four neighbours of the point. Continuing this way, every position will eventually contain either 0, 1, 2 or 3 chips.
The representation is once realized for an initial weight of 500000 chips placed at the origin of Z^2, and for two disjoint weights placed at distinct positions.
The first picture from the left represents an 80x80 grid with open (blue) and closed (black) sites. We are interested in the length of the shortest path (if it exists) of blue sites that goes from top to bottom. You can see the shortest path in red and its length is 97. While this can be computed directly for a predetermined system of open and closed sites, it is an interesting research problem to study the behavior of the shortest path where sites are opened at random with some probability. To the right you can see the probability that there is a path of open sites from top to bottom as we increase the probability that a site is open.
The first video represents the following situation: By changing the value r of the equation with initial values (10,10,10), we see that there is a range where the convergence to the positive or negative convergence points stays stable. It is quite surprising to have this form of stability in a complete chaos theoretic system of equations, and one is free to look for themselves where this might come from.
The second video corresponds to the following: By changing the value f of the periodic function, the stability of convergence is also modified. One can clearly see that the system switches from "nearly-lorenz" to "tornado-like" behaviour, which results in a very beautiful video.
The first picture is a realization of a 1D symmetric random walk up to the first time it is at distance 5000 from the origin (it takes more than 35 million steps!). Then you can see 1000 symmetric random walks run in parallel (picture is rotated 90 degrees). The last picture presents a realization of a symmetric random walk in 2D with a color gradient corresponding to time. For more graphs, check out the project report.
The first picture shows a minimal sphere packing that is more efficient than the linear 3-dimensional sphere packing. Download from here the project report on convex hulls of finite packs of spheres
to read why this is of interest, and for a description of a record-breaking 4-dimensional analogue.
The following five pictures are sphere packings obtained from the five platonic solids.