Experimental Mathematics Lab

Image Gallery

Knights and Queens

The gif is an animation of two movements of a knight on a chess board. The picture represents the number of jumps needed for the knight to reach every square on a torus-shaped chess board

Magic Square of Squares (Group A)

The following visualisations are the result of some generalization of the concept of magic squares: Magic tetrahedron of sum 576. Magic star of sum 630. Magic Sierpiński tetrahedron of sum 1292.

Music in Fibonacci numbers

The first audio file represents the Fibonacci sequence (mod 5) in the E minor Pentatonic scale. The second one corresponds to the A Minor Blues Scale where we take the sequence mod 12.

Machine Learning for Trading

Price prediction using hidden Markov models and LSTM neural networks:

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.

Perturbations of the Lorenz system

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.

Magic Square of Squares

Below you can see a few generalization of the concept of a Magic square.

Runs in random sequences

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.

Non-standard fair dice

3D-printed non-standard fair dice, as described in the project report

Visualization of Bianchi Fundamental Polyhedra

Visualization of Bianchi Fundamental Polyhedra, carried out by Kelly Jost in her Master thesis.

Convex hulls of finite packs of spheres

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.