Excavated dodecahedron

Excavated dodecahedron
Type	Stellation
Index	W28, 26/59
Elements (As a star polyhedron)	F = 20, E = 60 V = 20 (χ = −10)
Faces	Star hexagon
Symmetry group	icosahedral (Ih)
Dual polyhedron	self
Stellation diagram Stellation core Convex hull Icosahedron Dodecahedron

In geometry, the excavated dodecahedron is a star polyhedron. Its exterior surface represents the Ef₁g₁ stellation of the icosahedron. Magnus Wenninger list it in his Polyhedran models book as model 28, and calls it the third stellation of icosahedron.

It is also a facetting of the dodecahedron. When treated as a self-intersecting polyhedron, it is a noble polyhedron, having 20 star hexagons as faces. Although it is named the excavated dodecahedron, this is not strictly correct: a true excavated dodecahedron would have only the visible triangular faces instead of hexagonal ones.

The 20 vertices of the convex hull match the vertex arrangement of the dodecahedron.

Related polyhedra

• 
  The excavated dodecahedron as the third stellation of icosahedron.
• 
  One of the star hexagon faces highlighted.
• 
  It is related to the pentakis dodecahedron, but has inverted pyramids.
• 
  Its face as a facet of the dodecahedron.

The star hexagon face can be broken up into four equilateral triangles, three of which are the same size. A truly excavated dodecahedron would have the three congruent equilateral triangles as true faces of the polyhedron, while the interior equilateral triangle would be fully inside the polyhedron and would thus not be counted.

It is topologically equivalent to the {6, 6} hyperbolic tiling, by making the hexagons regular. As such, it is a regular polyhedron of index two:

References

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.