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
Properties	noble polyhedron, vertex transitive, self-dual polyhedron

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

Analogous faceting

It has the same external form as a certain facetting of the dodecahedron having 20 self-intersecting hexagons as faces. The non-convex 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.

• 
  One of the star hexagon faces highlighted.
• 
  Its face as a facet of the dodecahedron.

The faceting is a noble polyhedron. With six six-sided faces around each vertex, it is topologically equivalent to a quotient space of the hyperbolic order-6 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

• 
  It is related to the pentakis dodecahedron, but has inverted pyramids.
• 
  The great dodecahedron is an excavated icosahedron (the icosahedron being the dual of the dodecahedron).

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.