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An octonion form an 8-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the quaternions. . . . . . .
Communtative: No.
Associative: No.
(a²+b²+c²+d²+e²+f²+g²+h²)(m²+n²+o²+p²+q²+r²+s²+t²) =
(am-bn-co-dp-eq-fr-gs-ht)² +
(bm+an+do-cp+fq-er-hs+gt)² +
(cm-dn+ao+bp+gq+hr-es-ft)² +
(dm+cn-bo+ap+hq-gr+fs-et)² +
(em-fn-go-hp+aq+br+cs+dt)² +
(fm+en-ho+gp-bq+ar-ds+ct)² +
(gm+hn+eo-fp-cq+dr+as-bt)² +
(hm-gn+fo+ep-dq-cr+bs+at)²
The identity follows from the fact that the norm of the product of two octonions is the product of the norms, in a way similar to the quaternion identity and the complex product identity (see Sums of Squares).
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a*b
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b.1 | b.e1 | b.e2 | b.e3 | b.e4 | b.e5 | b.e6 | b.e7 |
| a.1 | 1 | e1 | e2 | e3 | e4 | e5 | e6 | e7 |
| a.e1 | e1 | -1 | e4 | e7 | -e2 | e6 | -e5 | -e3 |
| a.e2 | e2 | -e4 | -1 | e5 | e1 | -e3 | e7 | -e6 |
| a.e3 | e3 | -e7 | -e5 | -1 | e6 | e2 | -e4 | e1 |
| a.e4 | e4 | e2 | -e1 | -e6 | -1 | e7 | e3 | -e5 |
| a.e5 | e5 | -e6 | e3 | -e2 | -e7 | -1 | e1 | e4 |
| a.e6 | e6 | e5 | -e7 | e4 | -e3 | -e1 | -1 | e2 |
| a.e7 | e7 | e3 | e6 | -e1 | e5 | -e4 | -e2 | -1 |
fix this table.
. . . . . . explain how fano plane works for multiplication. Add the fano plane to the geometry glossary.
Wikipedia: Octonion
Mathworld: Octonion and Degen's Eight-Square Identity
Euclidean Space: octonion
Fact Archive: Octonion
Tony Smith: http://www.valdostamuseum.org/hamsmith/3x3OctCnf.html
. . . . . . relate this to 3x3 matrices
Complex, Quaternion, Octonion, and Sedenion numbers are n-tuples of real numbers, where n=2,4,8,16, respectively.
Sums of Squares - Every positive integer can be expressed as the sum of four squares; Primes of the form 4k+1 (and their products using the Complex Product Identity) can be expressed as the sum of two squares; Primes of the form 4k+3 (and their products using the Quaternion Identity) can be expressed as the sum of four squares.
The webmaster and author of this Math Help site is Graeme McRae.