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Octonion

The set of octonions, also sometimes called Cayley numbers and denoted O, consists of the elements in a Cayley algebra. A typical octonion is of the form

a+bi_0+ci_1+di_2+ei_3+fi_4+gi_5+hi_6,

where each of the triples (i_0,i_1,i_3), (i_1,i_2,i_4), (i_2,i_3,i_5), (i_3,i_4,i_6), (i_4,i_5,i_0), (i_5,i_6,i_1), (i_6,i_0,i_2) behaves like the quaternions (i,j,k). Octonions are not associative. They have been used in the study of eight-dimensional space, in which a general rotation can be written as

x^'->((((((xc_1)c_2)c_3)c_4)c_5)c_6)c_7.

See also

Cayley Number, Degen's Eight-Square Identity, Quaternion

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References

Baez, J. C. "The Octonions." Bull. Amer. Math. Soc. 39, 145-205, 2002.Conway, J. H. and Guy, R. K. "Cayley Numbers." In The Book of Numbers. New York: Springer-Verlag, pp. 234-235, 1996.Conway, J. and Smith, D. On Quaternions and Octonions. Wellesley, MA: A K Peters, 2001.Okubo, S. Introduction to Octonion and Other Non-Associative Algebras in Physics. New York: Cambridge University Press, 1995.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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Octonion

Cite this as:

Weisstein, Eric W. "Octonion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Octonion.html

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