Successor ordinal
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In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.
Geometric relations
This polyhedron can be formed from a dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles.
It is used in the cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.
Area and volume
The area A and the volume V of a truncated dodecahedron of edge length a are:
Cartesian coordinates
The following Cartesian coordinates define the vertices of a truncated dodecahedron with edge length 2(τ−1), centered at the origin:[1]
- (0, ±1/τ, ±(2+τ))
- (±(2+τ), 0, ±1/τ)
- (±1/τ, ±(2+τ), 0)
- (±1/τ, ±τ, ±2τ)
- (±2τ, ±1/τ, ±τ)
- (±τ, ±2τ, ±1/τ)
- (±τ, ±2, ±τ2)
- (±τ2, ±τ, ±2)
- (±2, ±τ2, ±τ)
where τ = (1 + √5) / 2 is the golden ratio (also written φ).
Orthogonal projections
The truncated dodecahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A2 and H2 Coxeter planes.
Centered by | Vertex | Edge 3-10 |
Edge 10-10 |
Face Triangle |
Face Decagon |
---|---|---|---|---|---|
Image | |||||
Projective symmetry |
[2] | [2] | [2] | [6] | [10] |
Vertex arrangement
It shares its vertex arrangement with three nonconvex uniform polyhedra:
Truncated dodecahedron |
Great icosicosidodecahedron |
Great ditrigonal dodecicosidodecahedron |
Great dodecicosahedron |
Related polyhedra and tilings
It is part of a truncation process between a dodecahedron and icosahedron: Template:Icosahedral truncations
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.
Template:Truncated figure1 table
See also
Notes
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References
- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
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External links
- Template:Mathworld2
- Template:KlitzingPolytopes
- Editable printable net of a truncated dodecahedron with interactive 3D view
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Template:Archimedean solids Template:Polyhedron navigator
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