Excavated dodecahedron
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!bgcolor=#e7dcc3 colspan=2|Excavated dodecahedron | |
| align=center colspan=2|240px (see 3D model) | |
| bgcolor=#e7dcc3|Type | Stellation |
| bgcolor=#e7dcc3|Index | W28, 26/59 |
| bgcolor=#e7dcc3|Elements (As a star polyhedron) | F = 20, E = 60 V = 20 (χ = −20) |
| bgcolor=#e7dcc3|Faces | 80px Star hexagon |
| bgcolor=#e7dcc3|Vertex figure | 80px Concave hexagon |
| bgcolor=#e7dcc3|Stellation diagram | 80px |
| bgcolor=#e7dcc3|Symmetry group | icosahedral (Ih) |
| bgcolor=#e7dcc3|Dual polyhedron | self |
| bgcolor=#e7dcc3|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.) 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.)
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| Core
! Long edges ! Faces ! Hull ! Cut |
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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.
File:AD2.png|One of the star hexagon faces highlighted.
File:Excavated dodecahedron face.png|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.[https://arxiv.org/abs/1005.4911 Regular Polyhedra of Index Two, I] Anthony M. Cutler, Egon Schulte, 2010[https://www.researchgate.net/publication/225386108_Regular_Polyhedra_of_Index_Two_II Regular Polyhedra of Index Two, II] Beitrage zur Algebra und Geometrie 52(2):357-387 · November 2010, Table 3, p.27
Related polyhedra
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| {{multiple image | align = left | width = 200 | image1 = Polyhedron truncated 20 dual big.png | image2 = Concave pentakis dodecahedron.png | footer = A pentakis dodecahedron (left) with inverted pyramids (right) has the same surface as the excavated dodecahedron. }} |rowspan="2"| {{multiple image | align = right | perrow = 3 | width = 200 | image1 = Pyritohedral excavated dodecahedron.png | image2 = Pyritohedral great icosahedron core.png | image3 = Polyhedron great 20 pyritohedral.png | image4 = Pyritohedral excavated dodecahedron, face gray.png | image5 = Pyritohedral great icosahedron core, face gray.png | image6 = Polyhedron great 20 pyritohedral, face gray.png | footer = 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. }} |
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| align = left | width = 200 | image1 = Stell12s cells solid 3 stacked.png | image2 = Stell12g cells solid 3 stacked.png | footer = 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
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- 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
{{Icosahedron stellations}}