The quaternions, , are a 4-dimensional algebra with basis . To describe the product we could give a multiplication table, but it is easier to remember that:
When we multiply two elements going clockwise around the circle we get the next one: for example, . But when we multiply two going around counterclockwise, we get minus the next one: for example, .
We can use the same sort of picture to remember how to multiply octonions:
This is the Fano plane, a little gadget
with 7 points and 7 lines. The 'lines' are the sides of the triangle,
its altitudes, and the circle containing all the midpoints of the sides.
Each pair of distinct points lies on a unique line. Each line contains
three points, and each of these triples has has a cyclic ordering
shown by the arrows. If and are cyclically ordered
in this way then
This is certainly a neat mnemonic, but is there anything deeper lurking behind it? Yes! The Fano plane is the projective plane over the 2-element field . In other words, it consists of lines through the origin in the vector space . Since every such line contains a single nonzero element, we can also think of the Fano plane as consisting of the seven nonzero elements of . If we think of the origin in as corresponding to , we get the following picture of the octonions:
Note that planes through the origin of this 3-dimensional vector space give subalgebras of isomorphic to the quaternions, lines through the origin give subalgebras isomorphic to the complex numbers, and the origin itself gives a subalgebra isomorphic to the real numbers.
What we really have here is a description of the octonions as a
'twisted group algebra'. Given any group , the group algebra
consists of all finite formal linear combinations of elements
of with real coefficients. This is an associative algebra with
the product coming from that of . We can use any function
© 2001 John Baez