A catalog of all 32 Cayley-Dickson-like doubling prod- ucts on ordered pairs (a,b) · (c,d) for which (1,0) is the left and right identity and for which x·x � = x � ·x = kxk 2 given the conju- gate (a,b) � = (a � ,−b). Only eight of these...
moreA catalog of all 32 Cayley-Dickson-like doubling prod- ucts on ordered pairs (a,b) · (c,d) for which (1,0) is the left and right identity and for which x·x � = x � ·x = kxk 2 given the conju- gate (a,b) � = (a � ,−b). Only eight of these are true Cayley-Dickson doubling products, since 24 of them do not satisfy the quaternion properties. Each of the eight Cayley-Dickson products has a dis- tinctive representation in the Fano Plane. 1. Product of finite sequences of real numbers In this catalog, Cayley-Dickson-like products are considered to be products of finite sequences of real numbers. This may seem odd since Cayley-Dickson products are normally considered to be products de- fined on ordered pairs. In order to define a Cayley-Dickson-like product of any two finite sequences of real numbers, every finite sequence x will be identified with an infinite sequence of the form x = x0,x1,x2,··· ,xn−1,0,0,0,··· and the ordered pair of any two sequences x and y (whether finite or infinite)...
