Background
In their book Regular Complex Polytopes, Coxeter remarks that the three regular complex polytopes are "remarkably similar" to the tetrahedron, cube and octahedron. (p. 127)
Specifically the fact that there are 3 polyhedra $A$, $B$, and $C$, where
- $A$ is self-dual
- $B$ and $C$ are dual to each other
- $C$ is the rectification of $A$
- $A$ is the halving of $B$
So for the three Platonic solids we have the following diagram:
Observation
After reading this I noticed that this isn't the only set of 3 regular polyhedra with this property. I immediately noticed that:
- $A$ = Halved mucube
- $B$ = Mucube
- $C$ = Muoctahedron
also holds the same relationships.
And trivially $A = B = C =$ Square tiling also holds.
If you begin searching you can find plenty of examples of a more general property:
- $A$ and $D$ are dual to each other
- $B$ and $C$ are dual to each other
- $C$ is the rectification of $D$
- $A$ is the halving of $B$
In the above examples $A$ was equal to $D$.
Question
Given an abstract regular polyhedron of type $\{p,p\}$ does $\delta\eta\delta r\{p,p\} = \{p,p\}$? In other words:
- Is the process of rectification, dual, halving, dual the identity?
- Does the following diagram commute:
where nodes indicate Schläfli types rather than specific polytopes.