Last time we looked at well-centered polytopes and I wondered whether a vertex-transitive polytope with well-centered $j$-faces would be completely well-centered. (For $j=0$ or 1 the condition is trivial, since points and edges are always well-centered, so we’d have to require $j=2$ at least.) This seems to be false. But why would I think this?

A completely well-centered polytope is what I’ve called orthodivisible: it can be split into orthoschemes—simplices whose vertices are on a path $P_0, \dotsc, P_d$ with mutually orthogonal edges.

That’s because you can make a path starting at a vertex—let’s call it $P_0$—continuing to the midpoint of an edge—$P_1$—and through the circumcenters of $j$-faces, up to the center of the whole polytope, and all the edges are mutually orthogonal.

They’re mutually orthogonal since the circumsphere $S_F$ of each face $F$ is the intersection of the affine span of $F$ with the circumsphere of the whole polytope $P$, and the normal vector from the circumcenter $c$ of $P$ through any subspace $\operatorname{aff}(F)$ meets it in the center of $S_F$.

Some circumscribable polyhedra with regular 2-faces aren’t well-centered, like the pentagonal pyramid J2:

or a square rotunda:

Although the normal lines through each face’s circumcenter all meet in one point—the polytope’s circumcenter—that’s outside the polytope.

These problems occur because the polytope’s faces don’t “surround” the whole sphere: and so the meeting point of the normal lines through the faces’ circumcenters might not be inside.

A vertex-transitive polytope, though—or indeed a polytope whose symmetries act transitively on all the faces of any particular rank—must surround the sphere in the right way. Even in a vertex-transitive polytope, though, the 2-faces might not be well-centered. Consider the disphenoid frustum from last time, with its affine spans and normal lines:

In combination, though—well-centered 2-faces in a vertex-transitive polytope—it seems like looking within a 3-face at the point where the normal lines through each of its 2-faces’ circumcenters meet, which is the 3-face’s circumcenter, might have to be in the interior.

Yet spidrox puts the kibosh on that.

It’s vertex-transitive, with regular 2-faces, but its square pyramid facets are not well-centered. Notably, they’re barely not well-centered: the circumcenter lies in the boundary.

Could strengthening a bit more—to vertex- and edge-transitive poltyopes—mean that the 3-faces must be well-centered?