Multiply exact quaternions and octonions

The exact-algebras plugin is written in RiX and keeps every component as an exact Integer or Rational.

Runnable RiXRun cell

.Plugin.Load("exact-algebras"); i := .exactAlgebras.Quaternion(0, 1, 0, 0); j := .exactAlgebras.Quaternion(0, 0, 1, 0); k := i * j; {: .exactAlgebras.Components(k), .exactAlgebras.NormSquared(k) };

Conjugation and inverse

The same recursive Cayley–Dickson implementation handles octonions. Division uses right division, so a / b means a * b.Inverse().

Runnable RiXRun cell

.Plugin.Load("exact-algebras"); o := .exactAlgebras.Octonion(1, 2, 3, 4, 5, 6, 7, 8); inverse := .exactAlgebras.Inverse(o); {: .exactAlgebras.NormSquared(o), .exactAlgebras.Components(o * inverse), .exactAlgebras.Components(.exactAlgebras.Conjugate(o)) };

Quaternion and octonion dimensions are not mixed implicitly. Exact rational scalars, however, promote naturally into the real component.

Runnable RiXRun cell

.Plugin.Load("exact-algebras"); q := .exactAlgebras.Quaternion(1, 2); .exactAlgebras.Components(q + 1/2);