Image Gallery
The Collatz conjecture (Group B)
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.
Illustrations of the abelian sand pile
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.
Solving polynomial equations
Newton-Raphson method
Newton-Raphson method (revisited)
Random matrices
Marchenko-Pastur: Consider a matrix Y=XX^T, where X is a random, real matrix of size N x M with i.i.d. entries having finite variance. If M and N are great, then the histogram of Y’s eigenvalues corresponds roughly to the curve of a function that only depends on M/N, N and the variance of X’s entries. For the example, we have chosen N=1500, M/N=5, and X’s entries to be normally distributed with mean 50 and variance 1.
Circular law: Consider a random, complex, square matrix A of large size with i.i.d. entries having finite variance. Then A's eigenvalues appear to be roughly uniformly distributed in a complex disc centred at 0, whose radius only depends on A’s size and the variance of A’s entries. The example shows the eigenvalues of a matrix of size 1500 whose entries are normally distributed with mean 100 and variance 5.
Wigner: Consider a random, real, symmetric matrix A whose entries all have the same finite variance. Under the right assumptions of independence and distribution of A’s entries, the histogram of the matrix’ eigenvalues corresponds roughly to a half-ellipse whose axes only depend on A’s size and its entries’ variance. The example shows the histogram corresponding to a matrix A of size 1500, whose entries are Poisson distributed with parameter equal to 6.
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.
Schottky groups
Machine Learning for Trading
Price prediction using hidden Markov models and LSTM neural networks:
Percolation
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.
Your browser does not support the video tag.
Your browser does not support the video tag.
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.
Punctured torus group actions
Convex hulls of hyperbolic limits sets, various parameters, and an anti-de Sitter limit set.
Visualisation of the p-adic numbers
7-adic integers (pdf, precision 2)
11-adic integers (precision 3)
Visualisation of the distribution of prime numbers
Ulam spiral size 25
Ulam spiral with primes produced by a quadratic polynomial
Visualization of flexible polyhedra
(G. Palmirotta, July 2015)
The Steffen flexible polyhedron (view 1)
The Steffen flexible polyhedron (view 2)
The Steffen flexible polyhedron (view 3)