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This book is a very different type of mathematics textbook. Because of this, users new to it, and its companion books that form the Discovering the Art of Mathematics library1, need context for the book's purpose and what it will ask of those that use it. This preface sets this context, addressing first the Explorers (students), then both Explorers and Guides (teachers) and finishing with important information for the Guides.
This work has been supported by the National Science Foundation under awards NSF0836943 and NSF1229515. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reect the views of the National Science Foundation.
These materials are also based on work supported by Project PRIME which was made possible by a generous gift from Mr. Harry Lucas. These materials were originally developed for use in the course Mathematical Explorations at Westfield State. They are now part of the curriculum library for Discovering the Art of Mathematics, all of whose volumes are freely available at http://artofmathematics.org.
The Department of Mathematics at Westeld State has provided fertile ground and enthusiastic support for pedagogical innovation, scholarship on teaching and learning in mathematics, and classroom experimentation. Under the direction of Chairs Karin Vorwerk and previously John Judge, and the organization of Deb Samwell, the department continues to be deeply committed
to student learning - including in its many required core courses. We are thankful to all departmental faculty, current and past, for their support of Discovering the Art of Mathematics. We are grateful for the support of many at our institution over many years. In particular we thank those in Academic Affairs, our one library faculty, Robert Martin, William Lopes, Louann D'Angelo, Elizabeth Preston, and Andrew Bonacci.
We are grateful for the support our our Advisory Board: Jere Confrey, David Farmer, Sandra Laursen, Carmel Schettino, Deborah Schifter and Dorothy Wallace. We thank Lee Zia for his guidance and encouragement.
Student interns Jillian Bates, Chelsea Baker, Joey Grabowski, Jen Roy, Rachel Cloutier, Zach Lancto, Alyssa Danilow, and Barbara Mulvaney helped with many tasks that supported the project.
Thanks to Sage Fox Consulting, Ken Rath and Emily DeHaro-Otero in particular, for their work on assessment and evaluation.
A great debt of gratitude to Tom Hoogendyk who designed and keeps running our Discovering the Art of Mathematics website.
Acknowledgements
Preface
Notes to the Explorer
Navigating This Book
Directions for the Guides
Chapter Dependencies
A little Introduction
Mathematics and Dancing
Symmetry in Mathematics and Dance
Moving in Symmetry in the Plane
Switching between Symmetries
The Mirror
Same limbs...
Switching between two kinds of Symmetry
Line Dancing
Switching between three kinds of Symmetry
Glide Reflections
Symmetry Choreograhy
Further Investigation
Dance in Symmetry in a Line
More Symmetries of the Plane
Planar Symmetries and Groups
Frieze Patterns
Frieze Patterns and Feet
Are there exactly 7 Frieze Patterns?
Further Investigations
Wallpaper Patterns
Non-Euclidean Symmetries
Salsa Rueda
Learning the Basic Dance
Salsa Rueda – Da Me and Da Me Dos
Further Investigations and Connections
The Space of Partner Salsa Dancing
Counting Positions
Salsa Dance Moves
Further Investigations
Contra Dancing
Contra Dancing and Permutations
Further Investigations
Permutation Connections
Contra Dancing and Groups
Further Investigation: Contra Dancing and Groups
Other Connections
Maypole Dancing
Which Ribbon Pattern is Created by the Dance?
Which Dance Arrangement Leads to this Ribbon Pattern?
Further Investigations
Connections
Bibliography
Index