# Mathematics

3D Visualization of Knots and Braids Janice Pappas
In everyday life, we may encounter an object that appears to be partially occluded, not by other objects, but by parts of the object itself. What does this mean? Visually, one kind of partially occluded object is a knot. Another such object is a braid. Mathematically, a knot is a non-intersecting closed curve, and a braid is the intertwining of a number of “strings” with ends that never turn back on themselves. In this course, we will identify and explore mathematical knots and braids and their properties. We will make physical knots and braids and identify their properties as well. We will determine when unknotting or unbraiding is possible and look at the similarities and differences between mathematical and physical knots and braids. Some of the questions we will address are: Why do certain materials knot more easily than others? What is an effective knot? What is the relation between the number of knot crossings and strength? We are interested in exploring the utility of knots with regard to ropes, wires, paper, and other materials and applied to, for example, boat mooring lines, shoelaces, cables, hangers, surgery sutures, and more. Knot folding techniques will also be studied. Braids will be explored in a similar way. Our goal is to address questions involving mathematical and physical knots and braids. We will also study knots and braids as they are found in a biological or chemical setting. As we delve into concepts concerning knots and braids, we will also learn how to view knot and braid models in virtual reality. We will learn how to use different software packages to devise knots and braids. We will have hands-on time with various materials to create knots and braids, study their properties, and conduct experiments. Students will conduct collaborative research and use software packages to create projects involving knots and/or braids. Projects will be visualized using software packages and at Groundworks in the Duderstadt Center. Using the available resources at Groundworks, students will be able to visualize in 3D the knot projects they create and conduct analyses. A student symposium will be convened in which student projects will be presented.
Prerequisites: having had a science, math or computer science course is helpful, but not necessary. Just bring your enthusiasm for learning about knots, braids and 3D visualization.

Art and Mathematics  –  Martin Strauss
With just a little historical revisionism, we can say that Art has provided inspiration for many fields within Mathematics. Conversely, Mathematics gives techniques for analyzing, appreciating, and even creating Art, as well as the basis for gallery design, digital cameras, and processing of images. In this class we will explore the Mathematics in great works of Art as well as folk art, as a way of studying and illustrating central mathematical concepts in familiar and pleasing material. And we’ll make our own art, by drawing, painting, folding origami papers, and more. Major topics include Projection, Symmetry, Wave Behavior, and Distortion. Projection includes the depiction of three-dimensional objects in two dimensions. What mathematical properties must be lost, and what can be preserved? How does an artwork evoke the feeling of three-dimensional space? We’ll study perspective, depictions of globes by maps, and the role of curvature. Turning to symmetry, we’ll study rotational and reflective symmetry that arise in tiling and other art and math. We’ll study more generalized symmetry like scaling and self-similarity that occurs in fractals as well as every self-portrait, and is central to mathematical concepts of dimension and un very different from the work at coarser scales—it is not self-similar. Describing light as waves and color as wavelength at once explains how mirrors, lenses, and prisms work and explains some uses of light and color in art. Finally, we ask about distorting fabrics and strings, and ask about the roles of cutting, gluing, and of stretching without cutting or gluing. Is a distorted human figure still recognizable, as long as it has the right number of organs and limbs, connected properly? Background in Math and interest in Art suggested. No artistic talent is necessary, though artistically talented students are encouraged to bring art supplies if they are inexpensive and easily transportable.

Fibonacci Numbers  –  Mel Hochster
The Fibonacci numbers are the elements of the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55- where every term is simply the sum of the two preceding terms. This sequence, which was originally proposed as describing the reproduction of rabbits, occurs in astonishingly many contexts in nature. It will be used as a starting point for the exploration of some very substantial mathematical ideas: recursive methods, modular arithmetic, and other ideas from number theory, and even the notion of a limit: the ratios of successive terms (e.g. 13/8, 21/13. 55/34) approach the golden mean, already considered by the ancient Greeks, which yields what may be the most aesthetically pleasing dimensions for a rectangle. As a by-product of our studies, we will be able to explain how people can test certain, very special but immensely large numbers, for being prime. We’ll also consider several games and puzzles whose analysis leads to the same circle of ideas, developing them further and reinforcing the motivations for their study.

Graph Theory  –  Doug Shaw
Ignore your previous knowledge of algebra, geometry, and even arithmetic! Start fresh with a simple concept: Take a collection of points, called vertices, and connect some of them with lines called edges. It doesn’t matter where you draw the vertices or how you draw the lines – all that matters is that two vertices are either related, or not. We call that a “graph” and you’ve taken the first step on the Graph Theory road! Graphs turn up in physics, biology, computer science, communications networks, linguistics, chemistry, sociology, mathematics- you name it! In this course we will discuss properties that graphs may or may not have, hunt for types of graphs that may or may not exist, learn about the silliest theorem in mathematics, and the most depressing theorem in mathematics, learn how to come up with good algorithms, model reality, and construct some mathematical proofs. We will go over fundamental results in the field, and also some results that were only proved in the last year or so! And, of course, we will present plenty of currently unsolved problems for you solve and publish when you get home!

Hex and the 4 Cs  –  Stephen DeBacker