# Excavated dodecahedron

Excavated dodecahedron :
| (see 3D model) ||
| 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, self-dual polyhedron |
In geometry, the excavated dodecahedron is a star polyhedron that looks like a dodecahedron with concave pentagonal 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.

## 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.) Each face is a self-intersecting 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 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.

## Related polyhedra

| A pentakis dodecahedron (left) with inverted pyramids (right) has the same surface as the excavated dodecahedron. | The faces of the e. d. (left) are part of the faces of the great icosahedron (right). Extending the short edges of a hexagon until they meet gives the triangle that contains it. Replacing each self-intersecting hexagon with a convex one gives a figure containing the edges of the compound of five cubes (middle). But this is not really a polyhedron, because each of these edges belongs to only one face. |
| The great dodecahedron (left) is an excavated icosahedron. It also has 60 visible triangles. But unlike the e. d. (right) it has convex faces and thus no inner edges. | The faces of the e. d. (left) are part of the faces of the great icosahedron (right). Extending the short edges of a hexagon until they meet gives the triangle that contains it. Replacing each self-intersecting hexagon with a convex one gives a figure containing the edges of the compound of five cubes (middle). But this is not really a polyhedron, because each of these edges belongs to only one face. |