Can we extend the real numbers by using hexagonal coordinates on a plane?

Solution 1:

It depends on what context you want to think of these as.

If you want to look at $\mathbb{Z}[\zeta_3]$, this is a structure that is studied. It's studied own it's own, and is known as the Eisenstein Integers, and as an example of $\mathbb{Z}[\zeta_p]$ which has a lot of interesting properties as a collection of rings. These are called Cyclotomic Fields.

If you want to look at it as $\mathbb{R}[\zeta_3]$, then it is the case that it is the same field as $\mathbb{R}[\zeta_4]=\mathbb{C}$. The expression as $\mathbb{C}$ is usually considered better because $\mathbb{R}[i]$ has an orthogonal basis. In circumstances where a hexagonal grid is relevant, it is used.

Solution 2:

This coordinate system and its use in hexagonal fields are relevant in representing hexagonal automata, circle packing and crystalline structures (e.g ice and snowflakes)

Mostly with optimization and in representing hexagonal data.

Most simulation software I encounter apply trigonometric transformations directly to each hexagonal cell which is a floating point nightmare so this method using a cubic coordinate system would yield simpler algorithms with its simpler but effective structure.

I know Im 6 years late and this should have been posted as a comment, but I made my account just now so feel free to convert this "answer" into a comment

The only issue I see with the canonical form is the breaking of continuous linearity, which is still the advantage of the r+q+s=0 form, making the latter much easier to program and project to 3d euclidean space