Thursday, April 28, 2011

Synonyms in mathematics and string theory: originally called octaves, now known as octonions or Cayley numbers

Octonions build an eight-dimensional number system— the largest of the four sets of numbers of normed devision algebra [1,2]. Baez and Huerla (both with first name John) describe how they were discovered by John Graves (another John!), who called them octaves [3]. Graves communicated his findings to the Irish mathematician William Rowan Hamilton. Although Hamilton was not interested in these numbers, he reported them at the Irish Royal Society. Without any written publication, Arthur Cayley, one of the Invariant Twins [4], rediscovered the octonions and beat Graves to publication. Thus, octonions are also known as Cayley numbers—and not as Graves numbers

Octonions are the elements in a Cayley algebra [5]. Like the multiplication of quaternions, the multiplication of octonions is noncommutative: order matters. Multiplication of octonions is not even associative. But they have two “very good properties” [2]: every nonzero octonion has a multiplicative inverse, and two nonzero octonions never multiply together to give zero. John Graves amateur interest (he was a lawyer) in algebra and his imagination of numbers with supernatural properties remains striking, especially, when considering their growing importance in explaining the universe and modeling the matter and forces therein.  

Keywords:  multidimensional systems, vector space, supersymmetry, spinors

References and further reading
[1] John C. Baez: The Octonions [].
[2] Timothy Gowers (Editor): The Princeton Companion to Mathematics. Princeton University Press, Princeton, New Jersey, 2008; pp. 275 to 279.
[3] John C. Baez and John Huerta: The Strangest Numbers in String Theory. Sci. Am. May 2011, 304 (5), pp. 60-65 [].
[4] Chapter 20 “Invariant Twins” in Men of Mathematics by E. T. Bell. Simon & Schuster, New York,  1937.
[5]  Wolfram MathWorld: Octonion [].

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