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 geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.

3D model of a truncated dodecahedron

Geometric relations edit

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 cell-transitive hyperbolic space-filling tessellation, the bitruncated icosahedral honeycomb.

Area and volume edit

The area A and the volume V of a truncated dodecahedron of edge length a are:

 

Cartesian coordinates edit

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 edit

The truncated dodecahedron has five special orthogonal projections, centered: on a vertex, on two types of edges, and two types of faces. The last two correspond to the A2 and H2 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 edit

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 edit

It shares its vertex arrangement with three nonconvex uniform polyhedra:

 
Truncated dodecahedron
 
Great icosicosidodecahedron
 
Great ditrigonal dodecicosidodecahedron
 
Great dodecicosahedron

Related polyhedra and tilings edit

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 edit

Truncated dodecahedral graph
 
5-fold symmetry Schlegel diagram
Vertices60
Edges90
Automorphisms120
Chromatic number3
Chromatic index3
PropertiesCubic, Hamiltonian, regular, zero-symmetric
Table of graphs and parameters

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

Notes edit

  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 edit

  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. 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 edit