Excavated dodecahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Excavated dodecahedron
Excavated dodecahedron.png
Type Stellation
Index W28, 26/59
Elements
(As a star polyhedron)
F = 20, E = 60
V = 20 (χ = −20)
Faces Star hexagon face.png
Star hexagon
Vertex figure Medial triambic icosahedron face.png
Concave hexagon
Symmetry group icosahedral (Ih)
Dual polyhedron self
Stellation diagram Stellation core Convex hull
Third stellation of icosahedron facets.png Icosahedron.png
Icosahedron
Dodecahedron.png
Dodecahedron
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.

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.

Related polyhedra[edit]

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 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:[1]

Uniform tiling 66-t2.png

References[edit]

Notable stellations of the icosahedron
Regular Uniform duals Regular compounds Regular star Others
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
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.