Truncated dodecahedron

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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
3D model of a truncated dodecahedron

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
.

Orthogonal projections
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

  1. ^ Weisstein, Eric W. "Icosahedral group". MathWorld.
  2. ^ Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press, p. 269

References

External links