Sedenion
The
sedenions form a 16-
dimensional algebra over the
reals obtained by applying the
Cayley-Dickson construction to the
octonions.
Like octonions,
multiplication of sedenions is neither
commutative nor
associative.\nBut in contrast to the octonions, the sedenions do not even have the property of being
alternative.\nThey do, however, have the property of being
power-associative.
The sedenions have a multiplicative
identity element 1 and multiplicative inverses, but they are not a
division algebra. This is because they have
zero divisors.
Every sedenion is a real
linear combination of the unit sedenions 1,
e1,
e2,
e3,
e4,
e5,
e6,
e7,
e8,
e9,
e10,
e11,
e12,
e13,
e14 and
e15,\nwhich form a basis of the
vector space of sedenions.\nThe
multiplication table of these unit sedenions looks as follows.
\n\n| × | \n1 | \ne1 | \ne2 | \ne3 | \ne4 | \ne5 | \ne6 | \ne7 | \ne8 | \ne9 | \ne10 | \ne11 | \ne12 | \ne13 | \ne14 | \ne15 | \n
\n |
\n\n| 1 | \n1 | \ne1 | \ne2 | \ne3 | \ne4 | \ne5 | \ne6 | \ne7 | \ne8 | \ne9 | \ne10 | \ne11 | \ne12 | \ne13 | \ne14 | \ne15 | \n
\n\n| e1 | \ne1 | \n-1 | \ne3 | \n-e2 | \ne5 | \n-e4 | \n-e7 | \ne6 | \ne9 | \n-e8 | \n-e11 | \ne10 | \n-e13 | \ne12 | \ne15 | \n-e14 | \n
\n\n| e2 | \ne2 | \n-e3 | \n-1 | \ne1 | \ne6 | \ne7 | \n-e4 | \n-e5 | \ne10 | \ne11 | \n-e8 | \n-e9 | \n-e14 | \n-e15 | \ne12 | \ne13 | \n
\n\n| e3 | \ne3 | \ne2 | \n-e1 | \n-1 | \ne7 | \n-e6 | \ne5 | \n-e4 | \ne11 | \n-e10 | \ne9 | \n-e8 | \n-e15 | \ne14 | \n-e13 | \ne12 | \n
\n\n| e4 | \ne4 | \n-e5 | \n-e6 | \n-e7 | \n-1 | \ne1 | \ne2 | \ne3 | \ne12 | \ne13 | \ne14 | \ne15 | \n-e8 | \n-e9 | \n-e10 | \n-e11 | \n
\n\n| e5 | \ne5 | \ne4 | \n-e7 | \ne6 | \n-e1 | \n-1 | \n-e3 | \ne2 | \ne13 | \n-e12 | \ne15 | \n-e14 | \ne9 | \n-e8 | \ne11 | \n-e10 | \n
\n\n| e6 | \ne6 | \ne7 | \ne4 | \n-e5 | \n-e2 | \ne3 | \n-1 | \n-e1 | \ne14 | \n-e15 | \n-e12 | \ne13 | \ne10 | \n-e11 | \n-e8 | \ne9 | \n
\n\n| e7 | \ne7 | \n-e6 | \ne5 | \ne4 | \n-e3 | \n-e2 | \ne1 | \n-1 | \ne15 | \ne14 | \n-e13 | \n-e12 | \ne11 | \ne10 | \n-e9 | \n-e8 | \n
\n\n| e8 | \ne8 | \n-e9 | \n-e10 | \n-e11 | \n-e12 | \n-e13 | \n-e14 | \n-e15 | \n-1 | \ne1 | \ne2 | \ne3 | \ne4 | \ne5 | \ne6 | \ne7 | \n
\n\n| e9 | \ne9 | \ne8 | \n-e11 | \ne10 | \n-e13 | \ne12 | \ne15 | \n-e14 | \n-e1 | \n-1 | \n-e3 | \ne2 | \n-e5 | \ne4 | \ne7 | \n-e6 | \n
\n\n| e10 | \ne10 | \ne11 | \ne8 | \n-e9 | \n-e14 | \n-e15 | \ne12 | \ne13 | \n-e2 | \ne3 | \n-1 | \n-e1 | \n-e6 | \n-e7 | \ne4 | \ne5 | \n
\n\n| e11 | \ne11 | \n-e10 | \ne9 | \ne8 | \n-e15 | \ne14 | \n-e13 | \ne12 | \n-e3 | \n-e2 | \ne1 | \n-1 | \n-e7 | \ne6 | \n-e5 | \ne4 | \n
\n\n| e12 | \ne12 | \ne13 | \ne14 | \ne15 | \ne8 | \n-e9 | \n-e10 | \n-e11 | \n-e4 | \ne5 | \ne6 | \ne7 | \n-1 | \n-e1 | \n-e2 | \n-e3 | \n
\n\n| e13 | \ne13 | \n-e12 | \ne15 | \n-e14 | \ne9 | \ne8 | \ne11 | \n-e10 | \n-e5 | \n-e4 | \ne7 | \n-e6 | \ne1 | \n-1 | \ne3 | \n-e2 | \n
\n\n| e14 | \ne14 | \n-e15 | \n-e12 | \ne13 | \ne10 | \n-e11 | \ne8 | \ne9 | \n-e6 | \n-e7 | \n-e4 | \ne5 | \ne2 | \n-e3 | \n-1 | \ne1 | \n
\n\n| e15 | \ne15 | \ne14 | \n-e13 | \n-e12 | \ne11 | \ne10 | \n-e9 | \ne8 | \n-e7 | \ne6 | \n-e5 | \n-e4 | \ne3 | \ne2 | \n-e1 | \n-1 | \n
\n
Further reading
- Carmody, Kevin: Circular and Hyperbolic Quaternions, Octonions and Sedenions, Applied Mathematics and Computation 28:47-72 (1988)\n* Carmody, Kevin: Circular and Hyperbolic Quaternions, Octonions and Sedenions - Further results, Applied Mathematics and Computation, 84:27-47 (1997)\n* Imaeda, K., Imaeda, M.: Sedenions: algebra and analysis, Applied Mathematics and Computation, 115:77-88 (2000)
