Excavated Dodecahedron
Get Excavated Dodecahedron essential facts below. View Videos or join the Excavated Dodecahedron discussion. Add Excavated Dodecahedron to your PopFlock.com topic list for future reference or share this resource on social media.
Excavated Dodecahedron
Excavated dodecahedron
Excavated dodecahedron.png
Type Stellation
Index W28, 26/59
Elements
(As a star polyhedron)
F = 20, E = 60
V = 20 (? = −20)
Faces Star hexagon face.png
Star hexagon
Vertex figure Medial triambic icosahedron face.png
Concave hexagon
Stellation diagram Excavated dodecahedron stellation facets.svg
Symmetry group icosahedral (Ih)
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 Ef1g1 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.) There are 20 faces corresponding to the 20 vertices. 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
Stell12g cells solid 5.png
Icosahedron
Stell12g cells solid 4 stacked.png
G. s. dodecahedron
Stell12g cells solid 3 stacked.png Stell12g cells solid 2 stacked.png
Dodecahedron
Stell12g cells cut 3b solid.png
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.

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]

Uniform tiling 66-t2.png

Related polyhedra

A pentakis dodecahedron (left) with inverted pyramids (right) has the same surface.
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.

References

  1. ^ Regular Polyhedra of Index Two, I Anthony M. Cutler, Egon Schulte, 2010
  2. ^ Regular Polyhedra of Index Two, II Beitrage zur Algebra und Geometrie 52(2):357-387 · November 2010, Table 3, p.27
  • 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



  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Excavated_dodecahedron
 



 



 
Music Scenes