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Truncated dodecahedron
Truncateddodecahedron.jpg
(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 CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png
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
Polyhedron truncated 12 max.png
Colored faces
Truncated dodecahedron vertfig.png
3.10.10
(Vertex figure)
Polyhedron truncated 12 dual max.png
Triakis icosahedron
(dual polyhedron)
Polyhedron truncated 12 net.svg
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.

Contents

Geometric relationsEdit

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 volumeEdit

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

 

Cartesian coordinatesEdit

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 projectionsEdit

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
Solid      
Wireframe          
Projective
symmetry
[2] [2] [2] [6] [10]
Dual          

Spherical tilings and Schlegel diagramsEdit

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 arrangementEdit

Related polyhedra and tilingsEdit

It is part of a truncation process between a dodecahedron and icosahedron:

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.

Truncated dodecahedral graphEdit

Truncated dodecahedral graph
 
5-fold symmetry schlegel diagram
Vertices60
Edges90
Automorphisms120
Chromatic number2
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

See alsoEdit

NotesEdit

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

ReferencesEdit

  • 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 linksEdit