# Cluster 6

Mathematics

Subject to revisions until January 1 2017

**Instructors:**Lawrence Marx,Abigail Thompson, Nelson Max, Bradley Ballinger**Prerequisites:**Algebra II**Typical Field Trips:**Exploratorium, California Academy of Science- This is a
option only.**FIRST CHOICE**

## Introduction

This cluster is designed to introduce students with a strong interest in mathematics to several different advanced topics. Many of these topics would ordinarily only be seen at the advanced undergraduate level, but all lend themselves to an introductory course at the high school level. No prior experience in any of these topics is expected, but enthusiasm for and interest in mathematics is essential.

## Core Course (4 Weeks)

### Combinatorics

Enumerative combinatorics provides a way to count in complicated mathematical settings, and we will learn several important counting techniques in this course. In addition, we will learn some related ideas in elementary probability and number theory. The following is an example of a combinatorics problem that we will encounter in our four-week journey through this interesting area of mathematics:

If a teacher returns a test to her class of 10 students at random, what is the probability that no student gets his or her own test?

## Supplementary Courses

### Symmetry (1 Week)

One can analyse images and shapes in terms of their symmetrically repeating patterns. A symmetry of a geometric figure is a motion which does not change the figure's appearance. These motions form a group in the mathematical sense, because if one such motion is followed by another, the combined motion also does not change the figure's appearance. We will study point groups of motions which keep a figure's center fixed. For example, the symmetry group of a square has 8 motions: the rotations of 0, 90, 180, and 270 degrees about the center (the first of these does not actually move anything), and the reflections in the horizontal line, the vertical line, and the two diagonal lines through the center. We will also look at point groups for 3D figures, for example, of a soccer ball sewn from twelve black pentagons and twenty white hexagons, or a baseball cover sewn from two leather pieces along a curved line.

We will study frieze groups for repeating patterns along a strip, "wallpaper" groups for patterns repeating along two directions in a plane, and space groups for repeating patterns in space, which appear in crystals. Along the way, we will study regular and semi-regular polyhedra (3D shapes bounded by flat regular polygon faces), kaleidoscopic patterns generated by mirrors, and tilings which fill a plane by regular polygons, or by identically shaped tiles.

### Introduction to Graph Theory (1.5 Weeks)

A graph is a collection of points, called vertices, connected by edges. Using graphs to describe information has applications from scheduling tournaments to cryptography. We’ll look some of the at the basics of graph theory, including graph coloring problems, Euler’s formula for graphs on surfaces, and graph planarity. Klein bottles will make a guest appearance when we look at graphs on non-orientable surfaces.

### Knots, Links and the Topology of Space (1.5 Weeks)

From knotted strands of DNA, to tangled necklaces, to the basis of String Theory, knots and links arise in many different applications. We’ll look at the mathematical theory of knots, and learn what knot invariants can tell us about them. We’ll connect knot theory to the study of all 3-dimensional spaces. For example, we'll examine whether or not our universe is flat. We thought the earth was flat for a long time... what about the 3-dimensional universe we live in? If we send a rocket off into space programmed to go "straight", will it eventually come back to where it started, like what happens if you go “west” long enough starting at a point on the equator? We'll look at some of the possibilities by studying what is known about the geometry and topology of space. What possibilities are there for 3-dimensional spaces? Can you come to the “edge” of the universe?