The Octavian Integers O

This chapter determines the correct way to define “the integral octonions” and investigates the geometry of the very interesting lattice E₈ that they form. It deduces that the ring O = O⁸ of “octavian integers” has the Euclidean property—that we can divide with a remainder smaller than the divisor. The chapter shows that a primitive octavian integer ρ of norm mn has precisely 240 left-hand divisors of norm m and 240 right-hand divisors of norm n, each set geometrically similar to the 240 units of O⁸. More generally, it shows that the set of left-hand divisors of a given octavian integer is geometrically similar to the set of all octavian integers of a certain norm. The chapter describes the sets of all possible left-hand and right-hand divisors of p = ρ₁ of norms m = m₀ and n = m₁.