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Make 3D Solid Shapes - Truncated Dodecahedron / Усеченный додекаэдр
Make 3D Solid Shapes - Truncated Dodecahedron / Усеченный додекаэдр
Published: 2013/09/24
Channel: ART KOSEKOMA
Solid Shapes And Their Nets: Truncated Dodecahedron / Усічений додекаедр / Усеченный додекаэдр
Solid Shapes And Their Nets: Truncated Dodecahedron / Усічений додекаедр / Усеченный додекаэдр
Published: 2014/10/26
Channel: ART KOSEKOMA +
Giant Truncated Dodecahedron Sphere (Zen Magnets)
Giant Truncated Dodecahedron Sphere (Zen Magnets)
Published: 2013/09/12
Channel: Mathnetism
Truncated Dodecahedron
Truncated Dodecahedron
Published: 2011/12/31
Channel: James Olsen
Modular Papercraft:  Truncated dodecahedron for cutting machine
Modular Papercraft: Truncated dodecahedron for cutting machine
Published: 2016/07/05
Channel: blackfoldworkshop
TUTORIAL Giant Truncated Dodecahedron Sphere Decagonal Subunit (Zen Magnets)
TUTORIAL Giant Truncated Dodecahedron Sphere Decagonal Subunit (Zen Magnets)
Published: 2013/09/17
Channel: Mathnetism
Paper Archimedean Solids
Paper Archimedean Solids
Published: 2016/04/16
Channel: paper polyhedra
031 How to Build a Dodecahedron
031 How to Build a Dodecahedron
Published: 2012/09/18
Channel: ToolTutor
VeryPuzzle Void Truncated Icosidodecahedron (VTI) & Scramble
VeryPuzzle Void Truncated Icosidodecahedron (VTI) & Scramble
Published: 2016/04/24
Channel: RPKVids
Truncated Icosahedron Video Tutorial SolidWorks
Truncated Icosahedron Video Tutorial SolidWorks
Published: 2013/11/05
Channel: CAD Gill
Vertex-Truncation Process Between Dodecahedron And Icosahedron / Усечение додекаэдра
Vertex-Truncation Process Between Dodecahedron And Icosahedron / Усечение додекаэдра
Published: 2013/03/24
Channel: ART KOSEKOMA
Unusual Polyhedral Dice
Unusual Polyhedral Dice
Published: 2015/12/10
Channel: mathsgear
Превращение додекаэдра в усечённый додекаэдр, conversion of Truncated dodecahedron
Превращение додекаэдра в усечённый додекаэдр, conversion of Truncated dodecahedron
Published: 2012/04/10
Channel: Mnogogranniki Ru
Tetrahedrally Truncated Dodecahedron Clock
Tetrahedrally Truncated Dodecahedron Clock
Published: 2015/06/22
Channel: Sharjeel Aziz
Blender 3D - Dodecahedron
Blender 3D - Dodecahedron
Published: 2015/03/06
Channel: ASopticom
211mm Truncated dodecahedron
211mm Truncated dodecahedron
Published: 2015/03/01
Channel: RoboSprout
Polyhedra With Articulated Faces : Truncated Dodecahedron
Polyhedra With Articulated Faces : Truncated Dodecahedron
Published: 2015/06/08
Channel: Laboratoire de robotique de l'université Laval / Laval University Robotics Laboratory
NeoCube Truncated dodecahedron
NeoCube Truncated dodecahedron
Published: 2011/01/24
Channel: fastandfaster3
Vertex-Truncation Process Between Dodecahedron And Icosahedron / Усечение додекаэдра
Vertex-Truncation Process Between Dodecahedron And Icosahedron / Усечение додекаэдра
Published: 2014/11/19
Channel: ART KOSEKOMA +
Truncated Dodecahedron 截角十二面體
Truncated Dodecahedron 截角十二面體
Published: 2013/05/21
Channel: Helen Yu
Folding a Stellated Truncated Dodecahedron from Thirty Pieces of Paper
Folding a Stellated Truncated Dodecahedron from Thirty Pieces of Paper
Published: 2013/01/22
Channel: Zander B
Truncated dodecahedron from 120 Buckyballs + ASMR
Truncated dodecahedron from 120 Buckyballs + ASMR
Published: 2017/03/18
Channel: The Little Viridian Fighter
Truncated Dodecahedron to Great Rhombicosidodecahedron
Truncated Dodecahedron to Great Rhombicosidodecahedron
Published: 2012/01/18
Channel: wolframmathematica
Truncated Dodecahedron to Great Rhombicosidodecahedron
Truncated Dodecahedron to Great Rhombicosidodecahedron
Published: 2009/07/10
Channel: wolframmathematica
Converging Tetrahedrons
Converging Tetrahedrons
Published: 2013/12/21
Channel: scopeman3
Vertex-Truncation Process Between Dodecahedron And Icosahedron / Усечение додекаэдра
Vertex-Truncation Process Between Dodecahedron And Icosahedron / Усечение додекаэдра
Published: 2016/05/27
Channel: KOSEKOMA MATHART
3d Tutorial | Nested Dodecahedron | 3dsmax
3d Tutorial | Nested Dodecahedron | 3dsmax
Published: 2015/07/08
Channel: John Malcolm
Truncated dodecahedron
Truncated dodecahedron
Published: 2015/02/27
Channel: RoboSprout
Truncated Malice
Truncated Malice
Published: 2012/06/02
Channel: Indigobassnotes
20Row Truncated Dodecahedron
20Row Truncated Dodecahedron
Published: 2013/02/03
Channel: Dimitri Tishchenko
Truncation of Dodecahedron
Truncation of Dodecahedron
Published: 2012/10/02
Channel: takashiyoshinorw
Flower truncated dodecahedron
Flower truncated dodecahedron
Published: 2011/12/02
Channel: Edo Timmermans
Construction of the Snub Dodecahedron
Construction of the Snub Dodecahedron
Published: 2016/09/13
Channel: Adam Montney
Open cage truncated dodecahedron of octahedra 2
Open cage truncated dodecahedron of octahedra 2
Published: 2010/04/18
Channel: 88625652
Octagon Stack Truncated Octahedron Dodecahedron (Zen Magnets)
Octagon Stack Truncated Octahedron Dodecahedron (Zen Magnets)
Published: 2013/05/19
Channel: Mathnetism
Truncated Hyper-Dodecahedron in rotation
Truncated Hyper-Dodecahedron in rotation
Published: 2016/06/02
Channel: David Diamondheart
Handmade Rubik
Handmade Rubik's cube type puzzle modification "3x3x4 Truncated rhombo-hexagonal dodecahedron v1"
Published: 2010/04/09
Channel: RFLo9Puzzles
Origami Snub Dodecahedron
Origami Snub Dodecahedron
Published: 2011/04/18
Channel: Lina S.
3D Modeling Polyhedra in Fusion 360: Part 2 - Icosahedron and Dodecahedron
3D Modeling Polyhedra in Fusion 360: Part 2 - Icosahedron and Dodecahedron
Published: 2017/02/03
Channel: Dave Tishmack
hyperbolic truncated dodecahedron2
hyperbolic truncated dodecahedron2
Published: 2010/12/10
Channel: Kazuaha
Polyhedra With Articulated Faces : Truncated Icosahedron
Polyhedra With Articulated Faces : Truncated Icosahedron
Published: 2015/06/08
Channel: Laboratoire de robotique de l'université Laval / Laval University Robotics Laboratory
Dissection of a Rhombic 210-hedron into a Combination of Icosahedron and Truncated Dodecahedron
Dissection of a Rhombic 210-hedron into a Combination of Icosahedron and Truncated Dodecahedron
Published: 2009/07/08
Channel: wolframmathematica
Review : Truncated Megaminx
Review : Truncated Megaminx
Published: 2012/08/03
Channel: Prin Kijviwattanakarn
Rhombic Dodecahedron Puzzle Review
Rhombic Dodecahedron Puzzle Review
Published: 2013/02/08
Channel: ptrsdboy
Rhombic Dodecahedron Truncated Octahedron Tessellation (Zen Magnets)
Rhombic Dodecahedron Truncated Octahedron Tessellation (Zen Magnets)
Published: 2011/08/07
Channel: Dimitri Tishchenko
Math Monday - Light Stick Construction
Math Monday - Light Stick Construction
Published: 2011/11/16
Channel: National Museum of Mathematics
uniform polyhedron and dual_truncates
uniform polyhedron and dual_truncates
Published: 2016/02/01
Channel: 王德祥
Stellated Rhombic Dodecahedron Truncated Octahedron Tessellation (Zen Magnets)
Stellated Rhombic Dodecahedron Truncated Octahedron Tessellation (Zen Magnets)
Published: 2011/09/01
Channel: Dimitri Tishchenko
Rhino Polyhedra
Rhino Polyhedra
Published: 2011/09/03
Channel: Alphonso Peluso
How to make Origami Buckyballs from PHiZZ units
How to make Origami Buckyballs from PHiZZ units
Published: 2008/03/25
Channel: barbabellaatje
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WIKIPEDIA ARTICLE

From Wikipedia, the free encyclopedia
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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
Truncated dodecahedron.png
Colored faces
Truncated dodecahedron vertfig.png
3.10.10
(Vertex figure)
Triakisicosahedron.jpg
Triakis icosahedron
(dual polyhedron)
Truncated dodecahedron flat.png
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[edit]

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[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: 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 Dodecahedron t01 v.png Dodecahedron t01 e3x.png Dodecahedron t01 exx.png Dodecahedron t01 A2.png Dodecahedron t01 H3.png
Projective
symmetry
[2] [2] [2] [6] [10]
Dual
image
Dual dodecahedron t12 v.png Dual dodecahedron t12 e3x.png Dual dodecahedron t12 exx.png Dual dodecahedron t12 A2.png Dual dodecahedron t12 H3.png

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
Uniform tiling 532-t01.png Truncated dodecahedron stereographic projection decagon.png
Decagon-centered
Truncated dodecahedron stereographic projection triangle.png
Triangle-centered
Truncated dodecahedron ortho-color.png Truncated dodecahedron schlegel.png Truncated dodecahedron schlegel-tricenter.png

Vertex arrangement[edit]

It shares its vertex arrangement with three nonconvex uniform polyhedra:

Truncated dodecahedron.png
Truncated dodecahedron
Great icosicosidodecahedron.png
Great icosicosidodecahedron
Great ditrigonal dodecicosidodecahedron.png
Great ditrigonal dodecicosidodecahedron
Great dodecicosahedron.png
Great dodecicosahedron

Related polyhedra and tilings[edit]

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 graph[edit]

Truncated dodecahedral graph
Truncated dodecahedral graph.png
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]

Truncated Dodecahedral Graph.svg
Circular

See also[edit]

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]

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