Truncated dodecahedron

Truncated dodecahedron

(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 32, E = 90, V = 60 (χ = 2)
Faces by sides20{3}+12{10}
Conway notationtD
Schläfli symbolst{5,3}
t0,1{5,3}
Wythoff symbol2 3 | 5
Coxeter diagram
Symmetry groupIh, H3, [5,3], (*532), order 120
Rotation groupI, [5,3]+, (532), order 60
Dihedral Angle10-10:116.57
3-10:142.62
ReferencesU26, C29, W10
PropertiesSemiregular convex

Colored faces

3.10.10
(Vertex figure)

Triakis icosahedron
(dual polyhedron)

Net

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:

A = 5 \left(\sqrt{3}+6\sqrt{5+2\sqrt{5}}\right) a^2 \approx 100.99076a^2
V = \frac{5}{12} \left(99+47\sqrt{5}\right) a^3 \approx 85.0396646a^3

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.

Orthogonal projections
Centered by Vertex Edge
3-10
Edge
10-10
Face
Triangle
Face
Decagon
Image
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
image

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.

Schlegel diagrams are similar, with a perspective projection and straight edges.

Orthographic projection Stereographic projections

Decagon-centered

Triangle-centered

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:

Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={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.

Dimensional family of truncated spherical polyhedra and tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
 
*832
[8,3]...
 
*32
[,3]
 
[12i,3] [9i,3] [6i,3] [3i,3]
Figures
Schläfli t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{,3} t{12i,3} t{9i,3} t{6i,3} t{3i,3}
Coxeter
Dual
figures
Triakis
figures

V3.4.4

V3.6.6

V3.8.8

V3.10.10

V3.12.12

V3.14.14

V3.16.16

V3.∞.∞
Coxeter

Truncated dodecahedral graph

Truncated dodecahedral graph

5-fold symmetry schlegel diagram
Vertices 60
Edges 90
Automorphisms 120
Chromatic number 2
Properties Cubic, Hamiltonian, regular, zero-symmetric

In the mathematical field of graph theory, a truncated dodecahedral graph is 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


See also


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