Excavated dodecahedron
In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pyramids in place of its faces. Its exterior surface represents the Ef_{1}g_{1} stellation of the icosahedron. It appears in Magnus Wenninger's book Polyhedron Models as model 28, the third stellation of icosahedron.
Excavated dodecahedron  

Type  Stellation 
Index  W_{28}, 26/59 
Elements (As a star polyhedron)  F = 20, E = 60 V = 20 (χ = −20) 
Faces  Star hexagon 
Vertex figure  Concave hexagon 
Stellation diagram  
Symmetry group  icosahedral (I_{h}) 
Dual polyhedron  self 
Properties  noble polyhedron, vertex transitive, selfdual polyhedron 
Description
All 20 vertices and 30 of its 60 edges belong to its dodecahedral hull. The 30 other internal edges are longer and belong to a great stellated dodecahedron. (Each contains one of the 30 edges of the icosahedral core.) There are 20 faces corresponding to the 20 vertices. Each face is a selfintersecting hexagon with alternating long and short edges and 60° angles. The equilateral triangles touching a short edge are part of the face. (The smaller one between the long edges is a face of the icosahedral core.)
Core  Long edges  Faces  Hull  Cut 

Icosahedron 
G. s. dodecahedron 
Dodecahedron 
one hexagonal face in blue 
Faceting of the dodecahedron
It has the same external form as a certain facetting of the dodecahedron having 20 selfintersecting hexagons as faces. The nonconvex hexagon face can be broken up into four equilateral triangles, three of which are the same size. A true excavated dodecahedron has the three congruent equilateral triangles as true faces of the polyhedron, while the interior equilateral triangle is not present.
The 20 vertices of the convex hull match the vertex arrangement of the dodecahedron.
The faceting is a noble polyhedron. With six sixsided faces around each vertex, it is topologically equivalent to a quotient space of the hyperbolic order6 hexagonal tiling, {6,6} and is an abstract type {6,6}_{6}. It is one of ten abstract regular polyhedra of index two with vertices on one orbit.[1][2]
Related polyhedra



References
 Regular Polyhedra of Index Two, I Anthony M. Cutler, Egon Schulte, 2010
 Regular Polyhedra of Index Two, II Beitrage zur Algebra und Geometrie 52(2):357387 · November 2010, Table 3, p.27
 H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808, 3.6 6.2 Stellating the Platonic solids, pp.96104
Notable stellations of the icosahedron  
Regular  Uniform duals  Regular compounds  Regular star  Others  
(Convex) icosahedron  Small triambic icosahedron  Medial triambic icosahedron  Great triambic icosahedron  Compound of five octahedra  Compound of five tetrahedra  Compound of ten tetrahedra  Great icosahedron  Excavated dodecahedron  Final stellation 

The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry. 