Given this relationship between convex \(2\)-polytopes and roots of unity in that \(2\)-dimensional algebra, let's try to construct a family of algebras such that if a member algebra is \(n\)-dimensional, it has cyclic subgroups under multiplication whose convex hull is an \(n\)-polytope.
Aside from the aforementioned case with polygons, mathematicians don't usually associate cyclic groups with polytopes. Rather, it is the symmetry groups that commonly comes to mind whenever polytopes are mentioned in the context of abstract algebra. But I'd like to show that there is a very natural notion of dimensionality within finite cyclic groups under any unital algebra's multiplication operation, and that it's not always just the two dimensions of the roots of unity in the complex plane.
Polytopic Groups and Algebras
Let's define a "polytopic group" to be a finite cyclic group whose group operation is the multiplication operation of a certain algebra. The span of the group's elements within the algebra's vector space determines the polytopic group's dimensionality, and its convex hull is a polytope with the same dimensionality.
Let's also define an "\(\bf{n}\)-polytopic algebra" as an \(n\)-dimensional real algebra with \(n\)-dimensional polytopic subgroups. Meaning, the dimensionality of the algebra and its highest-dimensional polytopic subgroup is the same.
We can thus say that \(\Bbb{C}\) is a \(2\)-polytopic algebra and \(\Bbb{R}\) is a \(1\)-polytopic algebra. Unfortunately, \(\Bbb{H}\) (the quaternions) and Cayley-Dickson constructions derived from it are NOT polytopic algebras, but they do have \(2\)-polytopic subalgebras. Also, all unital algebras contain the \(1\)-polytopic subalgebra \(\Bbb{R}\).
The following is my attempt at constructing other polytopic algebras. Note that I will be focusing on algebras over the field of real numbers, but you are free to try this exercise for other fields.
