The endodocehedron, also called the concave pyrohedral dodecahedron, is the concave solid corresponding to the interior void formed when each face of a regular dodecahedron is folded along a diagonal and resulting faces are unfolded to form a cube. It is equilateral and has 12 faces, 30 edges, and 20 vertices.
The endodocehedron with unit edge lengths corresponds to removing six square-base oblique wedges of edge length (where is the golden ratio), height 1/2 and ridge length 1/2 from a cube of edge length .
The endododecahedron can be constructed from 6 pieces as shown above, where each edge has unit length, assembled so that the central edges of adjacent pieces are oppositely oriented.
The endododecahedron of edge length has surface area
and volume
where is again the golden ratio.
The skeleton of the endodecahedron is the dodecahedral graph.