An earlier post included code for multiplying quaternions, octonions, and sedenions. The code was a little clunky, so I refactor it here.

def conj(x): xstar = -x xstar[0] *= -1 return xstar def CayleyDickson(x, y): n = len(x) if n == 1: return x*y m = n // 2 a, b = x[:m], x[m:] c, d = y[:m], y[m:] z = np.zeros(n) z[:m] = CayleyDickson(a, c) - CayleyDickson(conj(d), b) z[m:] = CayleyDickson(d, a) + CayleyDickson(b, conj(c)) return z

The `CayleyDickson`

function implements the Cayley-Dickson construction and can be used to multiply real, complex, quaternion, and octonion numbers. In fact, it can be used to implement multiplication in any real vector space of dimension 2^{n}. The numeric types listed above correspond to *n* = 0, 1, 2, and 3. These are the only normed division algebras over the reals.

When *n* = 4 we get the sedenions, which are not a division algebra because they contain zero divisors, and the code can be used for any larger value of *n* as well. As noted before, the algebraic properties degrade as *n* increases, though I don’t think they get any worse after *n* = 4.

If you wanted to make the code more robust, you could add code to verify that the arguments `x`

and `y`

have the same length, and that their common length is a power of 2. (You could just check that the length is either 1 or even; if it’s not a power of 2 the recursion will eventually produce an odd argument.)

Can you calculate octanion exponents?