I'm attempting to make sense of this. I think it might involve using a fourth spatial dimension.
We have a square in 3 dimensions. (let's say a paper model) so one side is blue and the other side is yellow.
(the colors aren't important, just to visualize)..
so we have 2 square faces (one blue and one yellow), 4 vertices, and 4 edges, satisfying Euler's polyhedra formula ( F+V-E=2)
.. so we can consider it a degenerate polyhedron..
Let's call it a planar square dihedron.
Now, we apply a baseless wedge or "roof" (I like to call it a "cune") to both faces of the square, each perpendicular to each other.
[in the same way that we could take a cube and turn it into a dodecahedron by adding "roofs" on each face, like so:
if we ignore the original edges of the dihedron, and focus just on the new edges formed by the cunes, topologically we now have a figure with 9 edges, 7 vertices, and, it would seem, 4 faces.
It makes a lot more sense if you make a model out of string.
2 of the faces can be thought of either as triangles with one extra vertex between one edge,
or a square with four vertices, 3 which meet with 2 other edges (of other faces) and one of which only has two edges..
and, similarly, the other two faces are either quadrilaterals with one extra vertex, or pentagons with one vertex that doesn't meet with other faces.
The important bit is that we have TWO PENTAGONAL FACES
this is where the fourth dimension comes in..
is it possible to flatten or fold this weird shape in the fourth dimension to get a regualr pentagon? (or a pentagonal dihedron?)