page_title: Octonion
section_title: Fano plane mnemonic
hierarchical_section_title: Octonion / Definition / Fano plane mnemonic
context_page_description: In mathematics, the octonions are a normed division algebra over the real numbers, meaning it is a hypercomplex number system; Octonions are usually represented by the capital letter O, using boldface O or blackboard bold. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
  Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions.
context_section_description: A convenient mnemonic for remembering the products of unit octonions is given by the diagram, which represents the multiplication table of Cayley and Graves. This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the Fano plane. The lines are directional. The seven points correspond to the seven standard basis elements of Im(O) (see definition below). Each pair of distinct points lies on a unique line and each line runs through exactly three points.
  Let (a, b, c) be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by
  ab = c and ba = −c
  together with cyclic permutations. These rules together with
  1 is the multiplicative identity,
   for each point in the diagram
  completely defines the multiplicative structure of the octonions. Each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.
caption: A mnemonic for the products of the unit octonions.[8]