This book has its origins in the research conducted by one of us (Corinne) nearly 30 years ago. She in turn introduced her husband (Tevian) to the octonions more than 20 years ago; they have collaborated on further research in this area ever since. This book reflects Tevian's efforts to understand what Corinne has taught him; although most of the actual writing was done by Tevian, the final product has been shaped every bit as much by Corinne, however indirectly. Rob Wilson has been a part of this collaboration for the last five years, and has shown us that the mathematical structure of the octonions is even richer than we had dreamed, going far beyond the topics presented here. Special mention and thanks are due our students, Jörg Schray, Jason Janesky, Aaron Wangberg, Joshua Kincaid, and Henry Gillow-Wiles, whose published work has been incorporated where appropriate, and to our most recent collaborator, John Huerta.

This book is however dedicated to David Fairlie and Tony Sudbery, Corinne's collaborators and mentors for her early work relating the octonions to string theory. Our ongoing work with the octonions, including this book, is a testament to their vision; they got us started.

This book would not have been possible without the financial support and encouragement of the John Templeton Foundation, for which we are very grateful.

An early draft of the material in Chapter 14 formed the basis for formal and informal seminars at several institutions, including Mount Holyoke and Grinnell Colleges in 2002, and Oregon State University in 2004. Sections 11.5, 12.4, and much of Chapters 13 and 15 are based on the authors' published research, as cited there.

Finally, all figures in this book appeared initially on the authors' website at with a Creative Commons by-nc-nd license, and are used by permission.

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