Author

Michael Chang

Date of Award

2017

First Advisor

Aaron Williams

Second Advisor

Harold Hastings

Abstract

A Gray code is an ordered sequence of all the possibilities of a certain type of combinatorial object, where each successive object differs from the previous one by a relationship known as a closeness condition. For example, the binary reflected Gray code an ordering of all the binary strings of length n with the closeness condition that adjacent strings differ in only one digit. For example, when n = 2 the order is 00, 01, 11, 10 where the underlined bit changes to create the next string. In contrast the standard “numerical” (or “lexicographic”) order of these strings for n = 2 is 00, 01, 10, 11 and it does not satisfy the closeness condition since two bits change between 01 and 10. Gray codes are used in many fields including engineering, computer science, and game theory since the closeness condition between successive objects can lead to greater efficiency in certain contexts. This thesis explores the potential applications of Gray codes not in science and engineering, but rather in music and visual art. We will find that Gray codes provide very interesting sequences of numbers and “anti-patterns” that lend themselves well to these representations. Furthermore, these representations can help us develop a deeper understanding and appreciation of the Gray code and how they are created.

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