To Henry Baker: I point out that just as 1/complex performs an inversion in the unit circle, 1/quaternion and 1/octonion do also in the unit sphere in 4 and 8 dimensions respectively -- up to a conjugation i.e. mirror-reflection in all 3 cases. All of the multiplications in these domains multiply "norms" (sum of squares), i.e. |ab| = |a|*|b|; they are bilinear in dimensions 4 and 8, i.e. c*(a+b)=ca+cb and (a+b)*c=ac+bc; noncommutative in dimensions>=4 (ab .noteq. ba); and non-associative in dimensions>=8 (ab*c noteq a*bc). There also happen to be particularly nice lattices in 2, 4, and 8 dimensions. You favored the square lattice in 2D, but the equilateral triangle lattice is nicer. I can see why you did not want it since 1/2 + i*sqrt(3)/2 is "irrational," but this was a somewhat arbitrary judgment depending on favoring 0+i over that. If we instead favored the eq.triangle lattice ("Eisenstein integers") then the "Gaussian integers" square lattice would be the ones which were "irrational." The Eisenstein integers are just as good as the Gaussians internally speaking, i.e. they are multiplicatively closed, enjoy unique factorization theorem, etc. Your geometric way of looking at things means you can use other lattices essentially as easily as the simple cubic integer lattice. In 4 dimensions, a very nice lattice is the "Hurwitz quaternions" which have all coordinates=integer/2 and sum is integer. Each point has 24 equally-nearest neighbors. It is closed under quaternionic multiplication. In 8 dimensions the nice lattice is the "integral octonions" arising from the E8 lattice. Each point has 240 equally-nearest neighbors. It is closed under octonionic multiplication. Those both enjoy unique factorization theorems albeit the precise forms of these theorems is not immediately obvious and the noncommut and/or nonassoc makes your brain hurt. Given these multiplicative-closure properties, approximating a real octonion x (say) as a "ratio" p/q of octonions, meaning p * inverse(q) = p * conjugate(q) / |q|^2, kind of makes more sense as a goal, i.e. x * q = p and if x were an "integral octonion" (meaning on that lattice, which is not the simple cubic 8D lattice) then x*q would also be. But note q*x and x*q are in general unequal. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)