Truncated dodecahedron
Truncated dodecahedron | |
---|---|
(Click here for rotating model) | |
Type | Archimedean solid Uniform polyhedron |
Elements | F = 32, E = 90, V = 60 (χ = 2) |
Faces by sides | 20{3}+12{10} |
Conway notation | tD |
Schläfli symbols | t{5,3} |
t0,1{5,3} | |
Wythoff symbol | 2 3 | 5 |
Coxeter diagram |
|
Symmetry group | Ih, H3, [5,3], (*532), order 120 |
Rotation group | I, [5,3]+, (532), order 60 |
Dihedral angle | 10-10: 116.57° 3-10: 142.62° |
References | U26, C29, W10 |
Properties | Semiregular convex
|
Colored faces |
3.10.10 (Vertex figure) |
Triakis icosahedron (dual polyhedron) |
Net |
In
triangular
faces, 60 vertices and 90 edges.
Geometric relations
This polyhedron can be formed from a regular dodecahedron by truncating (cutting off) the corners so the pentagon faces become decagons and the corners become triangles.
It is used in 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
Cartesian coordinates for the vertices of a truncated dodecahedron with edge length 2φ − 2, centered at the origin,[1]
are all even permutations of:
- (0, ±1/φ, ±(2 + φ))
- (±1/φ, ±φ, ±2φ)
- (±φ, ±2, ±(φ + 1))
where φ = 1 + √5/2 is the golden ratio.
Orthogonal projections
The truncated dodecahedron has five special
Coxeter planes
.
Centered by | Vertex | Edge 3-3 |
Edge 10-10 |
Face Triangle |
Face Decagon |
---|---|---|---|---|---|
Solid | |||||
Wireframe | |||||
Projective symmetry |
[2] | [2] | [2] | [6] | [10] |
Dual |
Spherical tilings and Schlegel diagrams
The truncated dodecahedron can also be represented as a
spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal
, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
perspective projection
and straight edges.
Orthographic projection | Stereographic projections | |
---|---|---|
Decagon-centered |
Triangle-centered | |
Vertex arrangement
It shares its
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:
Family of uniform icosahedral polyhedra | |||||||
---|---|---|---|---|---|---|---|
Symmetry: [5,3], (*532) | [5,3]+, (532) | ||||||
{5,3} | t{5,3} | r{5,3} | t{3,5} | {3,5} | rr{5,3} | tr{5,3} | sr{5,3} |
Duals to uniform polyhedra | |||||||
V5.5.5 | V3.10.10 | V3.5.3.5 | V5.6.6 | V3.3.3.3.3 | V3.4.5.4 | V4.6.10 | V3.3.3.3.5 |
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.
*n32 symmetry mutation of truncated spherical tilings: t{n,3} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry *n32 [n,3] |
Spherical | Euclid. | Compact hyperb. | Paraco. | |||||||
*232 [2,3] |
*332 [3,3] |
*432 [4,3] |
*532 [5,3] |
*632 [6,3] |
*732 [7,3] |
*832 [8,3]... |
*∞32 [∞,3] | ||||
Truncated figures |
|||||||||||
Symbol | t{2,3} | t{3,3} | t{4,3} | t{5,3} | t{6,3} | t{7,3} | t{8,3} | t{∞,3} | |||
Triakis figures |
|||||||||||
Config.
|
V3.4.4
|
V3.6.6 | V3.8.8 | V3.10.10 | V3.12.12
|
V3.14.14 | V3.16.16 | V3.∞.∞ |
Truncated dodecahedral graph
Truncated dodecahedral graph | |
---|---|
Hamiltonian, regular, zero-symmetric | |
Table of graphs and parameters |
In the
graph of vertices and edges of the truncated dodecahedron, one of the Archimedean solids. It has 60 vertices and 90 edges, and is a cubic Archimedean graph.[2]
Circular |
Notes
- ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
- ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269
References
- ISBN 0-486-23729-X. (Section 3-9)
- Cromwell, P. (1997). Polyhedra. United Kingdom: Cambridge. pp. 79–86 Archimedean solids. ISBN 0-521-55432-2.
External links
- Weisstein, Eric W., "Truncated dodecahedron" ("Archimedean solid") at MathWorld.
- Klitzing, Richard. "3D convex uniform polyhedra o3x5x - tid".
- Editable printable net of a truncated dodecahedron with interactive 3D view
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra