In regular spaces with a notion of distance, geodesics are locally distance minimizing paths. In smooth spaces, they satisfy a differential equation. One can define geodesics in spacetimes using an analogous differential equation, but they no longer have an interpretation in terms of distance. We create computer visualizations of geodesics on surfaces of revolution embedded in 3D Minkowski space and use this to study how curvature bounds on the surface affect the behavior of geodesics.
An Apollonian Sphere Packing is a collection of spheres. To construct a packing, we first consider a collection of four mutually tangent spheres. There are always two spheres that are tangent to all four original spheres, a small one nested in the gap between the spheres and a large one which wraps around the outside. We add these two to our collection. Then we pick another four mutually tangent spheres from our collection of six and repeat. Iterating this procedure over and over, we create an Apollonian Sphere Packing. This semester, we focused on the 3D Farey-Ford Sphere Packing, where two of the "spheres" are the planes $z=0$ and $z=1$. We use a generalized Möbius transform to construct this packing in Mathematica (as in the picture).
Suppose we cut through this packing with a horizontal plane at height $h$ above the $z=0$ plane and color the intersection as in the picture. If we measure the ratio between the area of the intersection and the appropriate part of the intersecting plane, then, as $h$ approaches 0, this area ratio should approach some limit. We estimate the limit to be $\approx 0.855$.
A lithium-ion battery charges as lithium ions flowing from the anode to the cathode of the battery bind to buffers in the electrode. Once there is a high enough concentration of lithium ions bound to the tin cells, the battery is fully charged.
We are interested in finding an optimal distribution of buffers to minimize the time it takes to charge the battery. To do this, we created a model of the flow of lithium-ions through a cross-section of the cathode of the battery. This cross-section is represented as a grid and it's dual graph where each section of the grid is a node in the graph.
We perform iterations of lithium flow (in which 1 unit of lithium moves one unit of the grid) through the battery until the cumulative charge of the buffers is 95% of the total capacity of buffers in the battery. In order to minimize the charge time of the battery, we need to minimize the number of iterations.
Our research this semester has focused on testing all possible configurations of buffers in the battery. This is a brute force strategy and we have utilized the Campus Cluster in order to cover all the configurations. Below we have two sample configurations, one a spiral and the other a random distribution of buffers. Both configurations have a 50% density of buffers. The images show the initial configuration of the buffers, and also the concentration of lithium throughout the battery at 95% charged.
Many people believe that the shortest path between two points is smooth, yet this may not always be the case. Our project focuses on finding some hints illustrating this idea. Using a set of differential equations, we are exploring a variety of solutions in order to find the shortest, non-straight and non-curved path between two points. In other words, we are looking for a path which is non-smooth, such as a corner of a square which consists of two lines meeting at a point. These equations are as follows: \begin{eqnarray*} x'(t) = v(t),& &y'(t) = u(t)\\ v'(t) = F(x, y)u(t),& &u'(t) = -F(x,y)v(t)\\ F(x,y) &=& y(y-x\epsilon) \end{eqnarray*}
By construction we have a non-smooth solution called the baseline trajectory of length two. By drawing a line of length one from the origin to the starting point and another line of length one from the origin to the ending point, we have a non-smooth path of length two for comparison. If we can produce a solution to the given set of equations with a length greater than that of the baseline trajectory, then the optimal path would be the baseline trajectory itself. We plan on doing this through an exploration of varying values of epsilon. As a result we will have found one example of a non-smooth optimal path between two points, which we refer to as a non-smooth extremal.
When we talk about knots, we mean a closed curve in three dimensions that doesn't intersect itself. More simply, a knot can be imagined as what one would get by taking a piece of string and intertwining it around itself and finally taping the ends together. Now, imagine laying this knot flat and looking at it head on. We would see places where the string goes over or under itself. These are called crossings and the picture of the knot in this view is the knot diagram. If we try untangling this knot without cutting the string, we might end up with a different knot diagram (perhaps with more or less crossings), but since we left the string intact, we still have the same knot we started with. The crossing number of the knot is the smallest number of crossings a knot diagram of this knot can have. There may be many different knots with the same crossing number and in fact, the number of different knots of a given crossing number grows so rapidly that for only 17 crossings, there are millions of different knots. For this reason, generating different knots for higher crossing numbers explicitly becomes computationally infeasible. Our project explored ways of generating high crossing knots randomly and studying various properties these knots have using the software called SnapPy.
The rigidity of structures is a problem that concerns architects and engineers. Our project is concerned with the rigidity of 3D quasicrystal framings. Tony Robbin is a mathematical artist that has constructed 3D quasicrystals framings. He has made these framings rigid by placing plates in the faces of the rhombohedra in the quasicrystal. The goal of our project is to find the minimum number of plates needed to make a quasicrystal structure rigid and where to put them.
While we have yet to answer this problem in the 3D case, we have studied the solution for the 2D case given by engineer Ture Wester. In this case we look at a Penrose framework that comes from a quasicrystal. Wester's solution allows us to find the minimum number of rhombi to make the 2D framework rigid. We believe that a modified version of his theorem may solve our 3D problem.
Another part of our project has been making our research accessible and interactive. While working on Wester's theorem and the 3D problem, we have found it very beneficial to construct and play with 3D and 2D models. Playing with the 2D model of a Penrose tiling has been so much fun we developed a corresponding game. We have also developed 2 computer programs. The first displays a rotating 3D quasicrystal framework. The second program allows interactive manipulations of the 2D Penrose framework so users can further understand the proof behind Wester's theorem.
Many properties of the natural numbers can be encoded as sequences of $1$'s and $-1$'s. On the surface, such sequences often show no obvious pattern and indeed seem to behave much like sequences generated by true random experiments such as coin tosses. In this project we seek to obtain a deeper understanding of the behavior of such sequences via certain ``random walks'' in the plane formed with these sequences. These random walks provide a natural way to visualize the degree of randomness inherent in a sequence and to detect, and possibly explain, hidden patterns, but they can also open up new mysteries that defy explanation.
In Spring 2013, we focused on a particular class of such random walks, defined in terms of the expansions of integers in a given base. These random walks exhibit particularly interesting, and intriguing, features. To quantify the degree of randomness in such random walks, we measured their rates of growth and compared them to that of a true random walk. Through this, we found a significant disparity that demonstrates a not-so-random behavior in our random walks.
Our team presented their work at a conference at the Rose-Hulman Institute of Technology and at the University of Illinois Undergraduate Research Symposium.
This project is part of a broader program to seek out and explore interesting new or little known applications of $n$-dimensional integrals. This year we focused on two such applications, each motivated by a well-known classical problem. The first of these problems is the volume of the ``Steinmetz solid'', the region of intersection of three pairwise perpendicular cylinders of unit radius depicted in the figure below. We introduced appropriate notions of Steinmetz solids in higher dimensions and obtained exact formulas for the volume of higher-dimensional Steinmetz solids with up to five pairwise perpendicular cylinders.
The second problem is the "Broken Stick Problem", which asks for the probability that the three pieces obtained by breaking up a stick at two randomly chosen points can form a triangle. In our project we explored analogous questions for broken sticks with an arbitrary number of pieces.
Members of our team have given talks on this work at conferences at the University of Texas at Austin and the Rose-Hulman Institute of Technology, and the entire team participated at the University of Illinois Undergraduate Research Symposium with two poster presentations. We also engaged in a variety of outreach activities, including visits to local middle and high schools, and participation in the University of Illinois Public Engagement Symposium.
The goal of our project is to use a genetic algorithm to produce first examples of $SL(3,\mathbb{C})$ character equivalent words in $F_2$. The idea is to generate thousands of very long words and compute their characters and keep those ones which are close to being character equivalent and then produce a better generation by combining and mutating the words obtained in the first generation. This requires a great amount of processing power which can be solved by parallel programming and using the campus cluster. An element of a free group $F_2$ of rank 2 is a word $w$ in $\{a, b, a^{-1},b^{-1}\}$ where no two consecutive letters in $w$ are inverses of each other. A representation of $F_2$ on $SL(3, \mathbb{C})$ is a homomorphism from $F_2$ to $SL(2,\mathbb{C})$ which amounts to assigning a matrix $A$ to letter $a$, and another matrix $B$ to letter $b$. For example, the image of $w = abba^{-1}b^{-1}a$ is the product matrix $W = ABBA^{-1}B^{-1}A$. A character of a word $w$ is the \emph{trace} of the corresponding matrix $W$. Two words $w_1, w_2$ are called character equivalent if for every pair of matrices $A$ and $B$ we have $tr(W_1) = tr(W_2)$.
Let $P$ be a $n$-vertex closed polygon with the usual counterclockwise orientation. Erect squares on each edge and connect the nearby corners of the outer edges of the squares to create a new polygon $T(p)$ with $2n$ vertices and $2n$ edges; half the edges are the edges of the original polygon $p$.
If $v$ and $w$ are consecutive edges in a polygon $p$, then $v$, $i(v-w)$, $w$ are consecutive edges in the new polygon $T(p)$. Thus the way to think about this is to write the pair $(v,w)$ as a column vector, and multiply it by the matrices $\begin{bmatrix} 1 & 0 \\ i & -i \end{bmatrix}$ and $\begin{bmatrix} i & -i \\ 0 & 1 \end{bmatrix}$. This procedure makes it possible to continue iterating the construction, even when it becomes non-convex (as it must no later than the third stage). It also happens that the group generated by these two matrices has order $96$. Thus the dimensions of the $n$th iterate grow at most linearly in $n$, whereas the number of sides grows exponentially.
We investigate the relationship among the perimeters of the iterates of a fixed polygon $p$. We show that the perimeters satisfy a certain recurrence, and moreover, we can generalize to other functions on the edges of $p$. In particular, given a polygon with three points, we determine what the growth rate of the perimeter is depending on the shape of the polygon.
Given a triangulation $\tau$ of $T$ into $N$ triangle, the trapezoid approximation of $$\int\int_Tf(x,y)dx\,dy$$ is $$\sum_{\triangle_{ABC}\in\tau}\frac{f(A) + f(B) + f(C)}{3} \text{area}(\triangle_{ABC}).$$ We studied ways of finding the triangulation that minimizes the error between the approximation and the actual integral. The best we found so far is to start with a regular triangulation defined in some canonical way and then to evolve the triangulation by descending along the gradient of the error function.
This method is better than the greedy recursive division algorithm which we initially tried. This method for the example below, $f=x^4 + x^2y^2 + y^3$, produced $16\%$ better errors than simple regular triangulation. Presumably, for functions that have larger second order derivatives the error improvement should be much better.
In the spring semester we have continued to identify as many mathematical models as possible from the University of Illinois collection. Our database now consists of 355 images of plaster, paper, metal, and wood models, 201 of which have been identified using known catalogs and other collections. Over half of these were manufactured and sold by Martin Schilling (Leipzig and Halle, Germany) in the late 19th and early 20th centuries. We know we have a number of models designed, and possibly made, by Ludwig Brill (Munich, Germany), as well as those made by Professor Arnold Emch, a UI faculty member starting in 1911. Documentation of these models will continue, with a plan to eventually label, organize, and better display the artifacts. In addition, we will make the database of models available online, and have missing or damaged one reconstructed, where possible.