complex numbers - If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go:

• 1-tuple: Real numbers.



• 2-tuple: Complex numbers.


• 4-tuple: Quaternions.


• 8-tuple: Octonions.

The Wikipedia link describes this doubling process:

In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one.

But if these are just vectors in the end, I wonder if there are vectors in the odd dimensions like 3, 5, etc., or other non-power-of-two dimensions like 10, 12, etc.. This way there would be a potentially more general construction describing the vector, and the power-of-two case would be a special case, sort of thing.