The Saturday Morning Math Group is a UT sponsored outreach program aimed at junior high and high school students, their teachers, and their parents. The program is free and no pre-registration is required. All regular SMMG meetings are held on the ground floor of R. L. Moore Hall, the mathematics bu…
Farey fractions are just ordinary old fractions, but ordered in a particular way. We will see that a Farey fraction has a very special relationship with its neighbors. In particular, we will see that adding Farey fractions is more fun than adding regular fractions!
In this talk we explore the surprising knots we find by cutting up different types of mobius bands. By cutting in creative ways we see that each band secretly contains a different knot. We get these distinct knots through a series of twists and snips. In the end we discover that indeed these knots are special. These are torus knots! In the end we talk about how we can make these torus knots by stretching and gluing donut shaped figures.
On March 30, 2014, Aaron Fenyes introduced us to an unusual kind of computer: one that thinks in strings of text instead of stacks of numbers. To get an idea of what these machines could do, we built some ourselves, using ordinary desktop computers to design and test them. We made machines that churned out everything from text art to arithmetic, and held a competition to find the machine that could run the longest for its size.
Combinatorial games are games played by two people who alternate turns. The games are characterized by the fact that no element of chance is involved in the games, and the players have full knowledge of the game positions at all times. When both players have the same set of allowable moves, the games are called impartial games. The game of Nim is a classic example of an impartial game. In this talk, we look at some examples of impartial games and show different ways to represent the games with the goal of determining winning strategies.
We explore the ideas of continuously changing the shape of a space, the key idea of topology. By gluing together simple pieces we can get new shapes representing manifolds. We used clay manipulatives to try to build manifolds like the sphere surface of a ping pong ball or the torus surface of a shower curtain ring with different numbers of balls of clay that have been stretched but not broken. At the end we tested out ways to calculate the magic number (Euler characteristic) for the different surfaces.
On January 19, 2014, to kick off a new season of UT Austin's Sunday Math Circle, we spent an afternoon playing with the shape our group is named for. Graduate student Alice Mark gave us a taste of the deep and intricate math hidden in a simple-sounding task: packing coins, oranges, and other circular objects into boxes. We started out working in small groups, trying to pack as many pennies as we could onto a quarter-size sheet of paper. The two most popular strategies both arranged the pennies in a regular pattern: a square grid, like a checkerboard, or a triangular grid, like a honeycomb. After comparing some popular strategies, we examined our packings mathematically, calculating the fraction of the paper we'd managed to cover with pennies and counting pennies' nearest neighbors. Then we started thinking about idealized square and triangle packings; we counted nearest neighbors and found the fraction each packing would cover of an infinite plane. (Of course, we had to figure out what it means to "cover a fraction of an infinite plane" first...) When we got tired of pennies on planes, we moved up into the next dimension. We played with different ways of stacking oranges, which turned out to be a lot more complicated than the penny arrangements we'd been investigating.
Dr. Dan Freed led a discussion and practice session on problem solving. Students learned about strategies like generalizing or specializing a problem, finding a simpler problem, varying or augmenting a problem, or working backwards in order to find a solution. All these techniques are useful across a wide variety of problems, and students practiced them by working together on a number of problems, like figuring out how many matches were needed for a tennis tournament of 50 people, or finding the volume of a small pond
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