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In the Alexander group, we worked on visualizations of the ``unit sphere" in flat space-time. In space-time, or Minkowski space, the norm of a vector $(x_1,x_2,x_3)$ is $\sqrt{(x_1^2+x_2^2-x_3^2)}$, and the unit sphere consists of vectors of square norm $1$ and $-1$. This gives the two parts of the sphere in space-time, the deSitter space, and two copies of hyperbolic space. This semester we worked primarily on projections of the sphere: we constructed central projections of the sphere into spacelike and timelike planes, and investigated the shapes of geodesics under these maps. Pictured are some projections of geodesics.

An Apollonian Circle Packing is a collection of disjoint circles in which each set of three mutually tangent circles are all mutually tangent to two more circles. Instead of looking at an arbitrary packing, we like to study the Farey-Ford packing, which consists of circles with centers at $(\frac{p}{q},\frac{1}{2q^2})$ and radii $\frac{1}{2q^2}$, which are all tangent to the $x$-axis. (The $x$-axis can be thought of as a circle with infinitely large radius.)

Suppose we cut through the Farey-Ford packing with a horizontal line and color the intersections with the circles red (as in the picture). Let $F(h)$ measure the length of the red segments at height $h$ on the interval $[0,1]$. As $h$ approaches $0$, $F(h)$ approaches $3/\pi$. We studied the error in this limit as $h$ approaches $0$: the second picture shows how the plot of $|F(1/x^2)-3/\pi|$ is well-approximated by $1/x^{3/2}$.

We have begun working on how this problem is different in other Apollonian Circle Packings and how it might be generalized to a three-dimensional sphere packing.

Project Report Software Package

The Lithium-Ion battery is charged as tin cells within the electrode absorb the Li-ions. The charge time of the battery depends on the distributions of the tin cells. We are interested in finding the best distributions of these tin cells within the electrode of the battery to minimize the charge time. To do this, we have divided the surface into a grid, where each square in the grid is represented by a node in a graph. Each node has a concentration, is of type electrolyte or buffer, and is connected to its neighbors in the grid via edges in the graph. After a certain number of iterations modeling the flow of Li-ions through the grid, each node will have a certain concentration. After all the buffer nodes are charged to a maximum concentration, the graph tells us the total time for all the nodes to become charged. Belwo we show two different distributions of the tin cells: a random distribution, and a spiral distribution, both of which have 50% density. The images show the initial distribution of tin cells, and then the final concentrations of the tin cells and electrolyte once the total concentration has reached 95%.

Project Report Software Package

A holograph is a recording of the interference patterns formed between two beams of coherent light coming from a laser on a light-sensitive media such as photographic film. The light beam coming from a laser is broken up into two beams by a beamsplitter. One beam is directed onto a 3D object, and the other beam goes to the photographic plate. Two sets of waves, from the object and the laser, form an interference pattern on the plate, and form a hologram. Since these images are not flat and contain highly oscillating patterns, the JPEG algorithm is highly inefficient when trying to compress holographic images.

Our project was to develop methods to compress holographic images and to determine which of these methods are most efficient. Given an image, we can extract the image data consisting of ordered triples $\{R,G,B\}$ representing the red, green, and blue color channels and put the data into a matrix. Since it is hard to generate a true holographic image, we take the Fast Fourier transform (FFT) of our $N \times N$ to act as our approximate hologram:

Next, we split our approximate hologram into square submatrices of some arbitrary size and take the Discrete Cosine transform (DCT) of each submatrix, a process borrowed from the JPEG compression algorithm.
Now we approximate the compression of the image by zeroing out certain elements in the submatrices using some condition. This corresponds to storing fewer bits in our image. We then take the inverse DCT, combine the submatrices and take the inverse FFT to retrieve a compressed version of the original image.

Our goal was to make interactive demonstrations for Professor Bradlow's three-week course on 'Curves' at the African Institute of Mathematical Sciences (AIMS). One demonstration was a physical set of conic sections. In order to print these with the IGL 3D printer, each piece of the figure was described analytically, typed into Mathematica, and then imported to the IGL 3D printer to print. Another model involved describing the focal points of an ellipse by where the spheres tangent to both the cone and ellipse (called Dandelin spheres) touched the ellipse. As before, each separate part of this figure was described and then drawn by the 3D printer to create a physical representation of this concept.

Another project involved constructing an applet in Sage (a python-like programming language) which allowed students to interact with the equation for the conic sections. By computing how to represent the conic section as the solution to a matrix equation, how to find the eigenvectors, and how to draw the axes for the curve, we were able to program this into Sage. With the resulting applet, students can explore how changing coefficients in the equation affected the conic section's type and appearance. Students are also able to use this applet to explore the connection between the axes of the conic section and the eigenvectors of the matrix. By giving each group of students their own copy of the conic sections and Sage applet, they can interact with the demonstrations by themselves.

Please click here for the description of the course at AIMS.

The Mathematics Department at Illinois holds one of the world's largest collections of mathematical models dating to the late 19th and early 20th century. These date to a time when models were the best way to illustrate mathematical formulae and concepts, given that powerful computers and software would not exist for another 70-90 years. Although many of the models at Illinois had been photographed before the start of this project, they were not displayed and organized in a manner useful to mathematicians.

With this project, our goal is to document the models for current mathematicians' use. As of the end of Fall 2012, the full collection of 358 plaster cast, string, metal, glass, plastic, and paper models have been photographed; the identification process has started. Already, we have found that Ludwig Brill (Mathematical Institute of the Royal Polytechnicum, Munich), Martin Schilling (Halle and Leipzig), and Arnold Emch (University of Illinois) published most of our models.

Later in the project, we anticipate the development of a website which would include images of the models, their history, and their significance, along with virtual versions created using the 3D printer available to the Illinois Geometry Lab.

The sequence of quadratic residues modulo a prime $p$ is a sequence of $p-1$ symbols $1$ or $-1$ that encodes the solubility of quadratic congruences modulo $p$ and that behaves in many respects like a random binary sequence of length $p-1$. In this project we studied a certain ``random walk'' in the plane formed with this sequence, the ``Quadratic Residue Random Walk'' (QRRW). The QRRW modulo $p$ is a finite walk in the plane consisting of $p-1$ steps of unit length and starting at the origin.

A famous result of Gauss predicts the \emph{end point} of a QRRW, but what happens along the way is rather mysterious and has not been unexplored in the literature. A cursory examination of QRRW graphs shows random-like features such as sudden turns, sharp cusps, and ragged edges, but also some unexpected symmetries, though no obvious patterns.

The goal of this project was to unravel some of these mysteries. We identified
six distinct shapes for a QRRW, and we correlated these shapes with
congruence classes of $p$ modulo
small primes. We developed efficient algorithms and C code to facilitate large
scale computations of QRRWs, and we used the campus computing cluster to carry
out these computations. We computed a variety of quantities associated with a
QRRW, such as the maximal distance to the origin and the amount of time spent in
each quadrant. We also created animations showing the evolution of a QRRW.

This project is part of a broader program to seek out and explore interesting new or little known applications of $n$-dimensional integrals. In Fall 2012 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 an appropriate notion of an $n$-dimensional Steinmetz solid, and we obtained exact formulas for its volume for dimensions up to $5$, and numerical values for dimensions up to ~$9$.

The second problem is the ``Broken Stick Problem'', which first appeared in a 19th century examination at Cambridge University. The problem 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 considered analogous questions for broken sticks with $n$ pieces, and we obtained both theoretical and numerical results on these questions.

Our team also created animations and interactive modules to help visualize the geometric aspects of these problems, and we gave a presentation on this work to the Math Club Team at Urbana High School.

A word $u \in F_2$ can be represented as a string of letters from the set $\left\{a, b, a^{-1}, b^{-1}\right\}$ such that no letter is followed by its inverse. The inverse of a word is the word written backwards with each letter written as its inverse. Two words $u,v \in F_2$ are conjugate if there exists a word $w \in F_2$ such that $u = w^{-1}vw$. A matrix representation is a homomorphism $\phi : F_2 \to SL(3,\mathbb{C})$ which can be thought of as mapping each letter of a word $u \in F_2$ to corresponding matrices $A, B, A^{-1}, B^{-1} \in SL(3,\mathbb{C})$. Two words $u, v \in F_2$ are character equivalent if $trace(\phi(u)) = trace(\phi(v))$ for all $\phi$.

It has been shown that by taking a word $u$ and its reverse $u^R$ ($u$ written backwards), one can generate a pair of words which are not conjugate in $F_2$ but which are character equivalent in $SL(2,\mathbb{C})$ (Horowitz, 1972). However, no non-trivial examples of $SL(3,\mathbb{C})$ words are known to exist. The goal of this project was to find a pair of such words by creating a genetic algorithm in Sage to generate these words based on pairing a word with its reverse. We were able to finish the genetic algorithm, but as yet have not run it enough or tested extensively enough to generate any suitable words. We hope to continue the project next semester and finish the algorithm, and then hope to generate a formal proof that such pairs exist. We also hope to explore representations to $SL(2,\mathbb{C})$ and representations from $F_n$.

Let $P$ be a closed polygon with edges $e_j = v_j v_{j+1}$, $1 \leq j \leq N$ and $v_{N+1} = v_1$ and, at first at least, assume the polygon is convex and has the usual counterclockwise orientation. Erect squares on each edge and connect the nearby corners of the outer edges of the squares. In this way we create a polygon with $2N$ vertices and $2N$ edges: half the edges of course are the edges of the original polygon.

If we talk about the edges only, and identify the directed edge with the complex number it represents in the plane, then the edges of the new figure can be expressed in terms of the edges of the old figure, if you view the (directed) edge as a complex number. If $v$ and $w$ are consecutive edges in the original, then $v$, $i(v - w)$, $w$ are consecutive edges in the new one. Thus the way to think about this is to write the pair $(v, w)$ as a column vector, and multiply it by the matrices

\[ \begin{bmatrix} 1 & 0 \\ i & -1 \end{bmatrix} \qquad \text{and} \qquad \begin{bmatrix} i & -i \\ 0 & 1 \end{bmatrix}. \]

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. By the $6$th or $7$th step, it's pretty incomprehensible, but the $4$th or $5$th iterates can be very nice. Since all the entries of all the matrices have the form $u+iv$, where $u,v \in \{ -1, 0, 1\}$, it also follows that if you start with a figure with lattice point entries, this continues.