Summary
There is a simple combinatorial anomaly, making possible some special linear algebra and thereby some special geometry, that occurs only in dimensions 1, 2, 4, and 8. The consequences are wide ranging and in particular lead to the existence of the complex numbers, the quaternions and the octonions. This article explains why the anomaly exists only in these dimensions using elementary linear algebra.
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Acknowledgments
My thanks to Douglass L. Grant and his December 2018 article in this Magazine, “Proving Euler’s Four-Square Lemma Using Linear Algebra,” where he uses the algebraic identity derived from the n = 4 case to obtain the corresponding number-theoretic fact about the sum of four squares [Citation2]. It was this article that led me down the path that resulted in these notes.
Special thanks to Roger Vogeler, professor of mathematics at Central Connecticut State University. He has helped to greatly improve the both the accuracy and the readability of these notes.
Of course, the book On Quaternions and Octonions by John H. Conway and Derek A. Smith has been an invaluable resource [Citation1]. I was greatly saddend to hear of Conway’s passing in April of 2020. His insights and expository style will be greatly missed.
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Franklin R. Gould
Frank Gould graduated from Berea Colege in 1965 with a degree in physics. He subsequently earned an MA in physics at Syracuse University. After working for many years at Aetna as a programmer and performance analyst, he returned to graduate school at Wesleyan University, earning a PhD in topological groups in 2009. He also enjoys all kinds of puzzles and hiking mountain trails.