|Grunbaum and Shephard's Classification of Escher-Like Patterns with Applications to Abstract Algebra|
Luke Brian Rawlings
Uploaded by: Pocket Masters
Pockets: Historical Dissertations, 2016 (May) Teachers College Columbia University Ed.D. Dissertations, Gottesman Libraries Archive
Tags: art, Escher, Frieze, pattern
Description/Abstract: This study investigates a link between art and mathematics. It attempts to show that patterns in the Euclidean plane, such as those made popular by the artist M.C. Escher, can function as inspiration for the transmission of mathematical knowledge at the college level. The study is broken up into two parts. The first part revisits a result of Branko Grunbaum and G.C. Shephard, and focuses on how the seven frieze groups associated with infinite strip patterns of the plane can be classified into fifteen pattern types by way of recognizing the relationship between a motif and the symmetry group of the infinite strip to which it belongs. The second part of the study sheds light on rich mathematical concepts associated with monohedral tilings of the plane, otherwise known as tessellations. Concepts found in college mathematics courses that cover group theory, linear algebra, and geometry surface in the work through the language of symmetry. Throughout the work, the study places the mathematics educator as its primary audience, signaling the possible impact that visual inspection of artistic patterns in the plane might have in the learning process. It is asserted that this art is useful as extra examples in several college mathematics courses, among them abstract algebra, linear algebra, geometry, and liberal arts mathematics, where the use of such patterns aids in understanding abstract concepts often encountered in these courses. Over one hundred figures are used to show that learning through visuals is an important part of the work. During the course of the study, a bank of problems forms from both the classification of pattern types and study of tessellations, and is presented in an appendix. The work represents a first form of a handbook which educators might use as a source for examples in a college mathematics classroom.