What's a polyhedroid?

Let's try to prove that there can't be hidden faces in any uniform polyhedron P.

All vertices of P are on a sphere (by symmetry). All vertices adjacent to one given vertex are on another sphere (because the edges have the same length), and thus on the intersection of these two spheres which is a circle. So the verf (call it Q, a spheric polygon) has its vertices on a circle.

Points on a circle are convexly independent; none of them is inside the convex polygon formed by the other points. So any vertex of Q is visible from outside, not surrounded and hidden by edges of Q. It follows that any edge of Q has some part (near a vertex) visible from outside.

Returning to 3D, this means that any face of P has some part (near an edge and a vertex) visible from outside.

But why would they refer to hidden faces at all, if they never occur anyway (in 3D)?

I doubt it's based on the existence of higher-dimensional uniform polytopes with hidden parts, because higher dimensions are hard to visualize, and having hidden parts is a visual property.

I think it's based on degenerate polyhedra like

these five. Are those even valid abstract polyhedra? Or are they compounds? In any case, I would say they're degenerate because of the coincident edges, not because of the hidden faces.

In fact some of the 53 accepted polyhedra are degenerate according to my definition:

seside,

sirsid,

gidrid, and

gisdid have coplanar faces. Any others?

Degeneracy can happen in several ways: coincident vertices, collinear edges, coplanar faces, vertices on the wrong edges (impossible for uniforms because a line intersects a sphere in at most 2 points), vertices in the wrong faces (planes), or edges in the wrong faces (planes).

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