# MATH 125: Explorations in Mathematics

**The Art and Code of Mathematics - ***The Art and Code of Mathematics* invites you to explore what it means to do mathematics - the ways of thinking like a mathematician and the creative art of crafting mathematics. In this course, we will discuss a variety of simply-stated problems and employ intuitive and clever ideas to build mathematical arguments that lead to their solution. We will also see how mathematics permeates the physical world and its code is ever-present in nature. In-class activities and assignments will encourage students to become independent thinkers and share their ideas within their groups to build strong mathematical arguments. The main areas we will explore and discuss are: number patterns; prime numbers and their properties; the Fibonacci Sequence and the Golden Ratio; counting selections of objects that are ordered or unordered - permutations and combinations; the Pigeonhole Principle; Pascal's Triangle and its mysteries; modeling problems with graphs, traversability of graphs, planar graphs, coloring maps, and their associated graphs.

Course Attributes: Mathematical Perspectives

**The Mathematics of Calendars and Timekeeping** - In this course we explore how mathematics permeates one of the most important features of life: how people manage and control the passage of time, and how mathematics has come to serve as an invaluable tool for timekeeping, from ancient times to the present. We study the fundamental astronomical phenomena that form the foundations of all timekeeping and we survey the diverse and fascinating ways in which people of different cultures and ages have developed mathematical solutions to the problems of accurately reckoning times, dates, and periods for regulating their societies.

Course Attributes: Mathematical Perspectives

**Mathematics and the Creative Imagination** - In *Mathematics and the Creative Imagination,* we explore creativity in Mathematics. The Heritage Illustrated Dictionary defines creativity as *"Characterized by originality and expressiveness, imaginative".* Mathematics is marked by originality, imagination, beauty, and the unexpected! In this course, we explore the following topics, designed to expose students to the originality and creativity inherent in mathematics: Game theory, the four-coloring problem, topology, figurate numbers, number theory, the 4^{th} dimension and beyond, cryptography and secret codes, group theory, and logic. Using historical discussions of how each topic of mathematics arose and a thorough discussion of the creativity involved, each of these topics has been selected in order to facilitate students' understanding of how one actually DOES mathematics, how much creativity is involved, and how imaginative, beautiful, and interesting it truly is. As *creativity* is the topic of this course, students have many opportunities throughout the semester to express their own creativity, mostly in the form of written critiques of a variety of books read throughout the semester (*Alice's Adventures in Wonderland, Flatland, The Man who Loved only Numbers, *etc).

Course Attributes: Mathematical Perspectives

**Mathematics and Music** - This course explores the physics of sound waves, frequency, and pitch, all which are easily modeled by mathematics. It also includes a study of Pythagorean turning, just tuning, and mean-tone, well, and equal tuning. Other mathematical topics occur in rhythm, the measure of time in music theory, permutations in change ringing, and mathematical transformations in musical composition.

**Personal Finance** - In the *Personal Finance* course students will explore the mathematics of banking by examining checking and savings accounts, CDs, credit and debit cards and loans. A study of annuities and the time value of money can be seen in applications of insurance, investments, and real estate. Students will manage a virtual stock account in order to learn about gains/losses, yields/growth rates, risk, and actual/expected returns. The advantages/disadvantages of using collectibles in an individual investment portfolio will be investigated.

Course Attributes: Mathematical Perspectives

**Secret Codes** - In *Secret Codes* we explore cryptology, which comprises both cryptography (the art and science of creating secret messages) and cryptanalysis (the art and science of breaking secret messages, i.e. reading the secret message even when you should not be able to) along with steganography (hiding secret messages). We will start with classical ciphers, many dating back thousands of years: shift, affine, Vigenère, substitution, Hill, one-time pads, Enigma, etc. Then after a brief introduction to number theory, we will start on modern methods: the RSA algorithm, DES, and AES: Rijndael, ElGamal, ethics and privacy. All codes will be placed in historical perspective by exploring the political and military contexts in which they were devised.

Course Attributes: Mathematical Perspectives

**Women in Mathematics** - This course is a mathematical topics course which includes reading and writing biographies of women mathematicians. Topics are drawn from the mathematical fields of the women mathematicians and include a study of the conic sections, string geometry, number theory, modular arithmetic, sequences, other number bases, and group theory. Angle measures of polygons, tessellations, and symmetry are explored in the mathematics of quilting. Transformations, border patterns, finite designs, and wall paper patterns are included in the study of the art and geometry of women in southern Africa.

Course Attributes: Mathematical Perspectives, Gender and Diversity Studies