Excavated dodecahedron

Excavated dodecahedron
Type	Stellation
Index	W28, 26/59
Elements (As a star polyhedron)	F = 20, E = 60 V = 20 (χ = −20)
Faces	Star hexagon
Vertex figure	Concave hexagon
Symmetry group	icosahedral (Ih)
Dual polyhedron	self

Animation of an excavated dodecahedron (click to view)

In geometry, the excavated dodecahedron is a star polyhedron having 60 equilateral triangular faces. Its exterior surface represents the Ef₁g₁ stellation of the icosahedron. It appears in Magnus Wenninger's book Polyhedron Models as model 28, the third stellation of icosahedron.

As a stellation

Stellation diagram	Stellation	Core	Convex hull
		Icosahedron	Dodecahedron

As a faceting

It has the same external form as a certain facetting of the dodecahedron having 20 self-intersecting hexagons as faces. This form is a noble polyhedron.

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

• 
  One of the star hexagon faces highlighted.
• 
  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.

Related polyhedra

• 
  It is related to the pentakis dodecahedron, but has inverted pyramids.

It is topologically equivalent to a quotient space of the hyperbolic order-6 hexagonal tiling and can be topologically deformed into this tiling by making the hexagons regular. As such, it is a regular polyhedron of index two:

References

• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp.96-104

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.