Editing Kepler–Poinsot polyhedron (section)

=== Non-convexity ===
These figures have [[pentagram]]s (star pentagons) as faces or vertex figures. The [[small stellated dodecahedron|small]] and [[great stellated dodecahedron]] have [[star polygon|nonconvex regular]] [[pentagram]] faces. The [[great dodecahedron]] and [[great icosahedron]] have [[convex polygon|convex]] polygonal faces, but pentagrammic [[vertex figure]]s.

In all cases, two faces can intersect along a line that is not an edge of either face, so that part of each face passes through the interior of the figure. Such lines of intersection are not part of the polyhedral structure and are sometimes called false edges. Likewise where three such lines intersect at a point that is not a corner of any face, these points are false vertices. The images below show spheres at the true vertices, and blue rods along the true edges.

For example, the [[small stellated dodecahedron]] has 12 [[pentagram]] faces with the central [[pentagon]]al part hidden inside the solid. The visible parts of each face comprise five [[isosceles triangle]]s which touch at five points around the pentagon. We could treat these triangles as 60 separate faces to obtain a new, irregular polyhedron which looks outwardly identical. Each edge would now be divided into three shorter edges (of two different kinds), and the 20 false vertices would become true ones, so that we have a total of 32 vertices (again of two kinds). The hidden inner pentagons are no longer part of the polyhedral surface, and can disappear. Now [[Planar graph#Euler's formula|Euler's formula]] holds: 60&nbsp;&minus;&nbsp;90&nbsp;+&nbsp;32&nbsp;=&nbsp;2. However, this polyhedron is no longer the one described by the [[Schläfli symbol]] {5/2,&nbsp;5}, and so can not be a Kepler–Poinsot solid even though it still looks like one from outside.