Octonion

The octonions are a non-associative extension of the quaternions. They were discovered by John T. Graves[?] in 1843, and independently by Arthur Cayley[?], who published the first paper on them in 1845. They are sometimes referred to as Cayley numbers or the Cayley algebra.

The octonions form an 8-dimensional algebra over the real numbers, and can therefore be thought of as octets of real numbers. Every octonion is a real linear combination of the unit octonions 1, e₁, e₂, e₃, e₄, e₅, e₆ and e₇, the multiplication table for which looks as follows.

·	1	e1	e2	e3	e4	e5	e6	e7
1	1	e1	e2	e3	e4	e5	e6	e7
e1	e1	-1	e4	e7	-e2	e6	-e5	-e3
e2	e2	-e4	-1	e5	e1	-e3	e7	-e6
e3	e3	-e7	-e5	-1	e6	e2	-e4	e1
e4	e4	e2	-e1	-e6	-1	e7	e3	-e5
e5	e5	-e6	e3	-e2	-e7	-1	e1	e4
e6	e6	e5	-e7	e4	-e3	-e1	-1	e2
e7	e7	e3	e6	-e1	e5	-e4	-e2	-1

See also Hypercomplex numbers.

External links:

• The Octonions (http://math.ucr.edu/home/baez/Octonions/octonions.html) - an article by John C. Baez
• Octonion Fractals (http://www.stud.ux.his.no/~onar/Octonion.html) - fractals generated using octonion mathematics