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{{Semireg polyhedra db|Semireg polyhedron stat table|tD}} | |||
In [[geometry]], the '''truncated dodecahedron''' is an [[Archimedean solid]]. It has 12 regular [[decagon]]al faces, 20 regular [[triangular]] faces, 60 vertices and 90 edges. | |||
__TOC__ | |||
==Geometric relations== | |||
This [[polyhedron]] can be formed from a [[dodecahedron]] by [[Truncation (geometry)|truncating]] (cutting off) the corners so the [[pentagon]] faces become [[decagon]]s and the corners become [[triangle]]s. | |||
It is used in the [[cell-transitive]] hyperbolic space-filling tessellation, the [[Bitruncation#Self-dual .7Bp,q,p.7D polychora/honeycombs|bitruncated icosahedral honeycomb]]. | |||
==Area and volume== | |||
The area ''A'' and the [[volume]] ''V'' of a truncated dodecahedron of edge length ''a'' are: | |||
:<math>A = 5 \left(\sqrt{3}+6\sqrt{5+2\sqrt{5}}\right) a^2 \approx 100.99076a^2</math> | |||
:<math>V = \frac{5}{12} \left(99+47\sqrt{5}\right) a^3 \approx 85.0396646a^3</math> | |||
==Cartesian coordinates== | |||
The following [[Cartesian coordinates]] define the vertices of a [[Truncation (geometry)|truncated]] [[dodecahedron]] with edge length 2(τ−1), centered at the origin:<ref>{{mathworld |title=Icosahedral group |urlname=IcosahedralGroup}}</ref> | |||
:(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, ±τ<sup>2</sup>) | |||
:(±τ<sup>2</sup>, ±τ, ±2) | |||
:(±2, ±τ<sup>2</sup>, ±τ) | |||
where τ = (1 + √5) / 2 is the [[golden ratio]] (also written φ). | |||
==Orthogonal projections== | |||
The ''truncated dodecahedron'' has five special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A<sub>2</sub> and H<sub>2</sub> [[Coxeter plane]]s. | |||
{|class=wikitable | |||
|+ Orthogonal projections | |||
|- | |||
!Centered by | |||
!Vertex | |||
!Edge<br>3-10 | |||
!Edge<br>10-10 | |||
!Face<br>Triangle | |||
!Face<br>Decagon | |||
|- | |||
!Image | |||
|[[File:Dodecahedron_t01_v.png|120px]] | |||
|[[File:Dodecahedron_t01_e3x.png|120px]] | |||
|[[File:Dodecahedron_t01_exx.png|120px]] | |||
|[[File:Dodecahedron_t01_A2.png|120px]] | |||
|[[File:Dodecahedron_t01_H3.png|120px]] | |||
|- align=center | |||
!Projective<br>symmetry | |||
|[2] | |||
|[2] | |||
|[2] | |||
|[6] | |||
|[10] | |||
|} | |||
== Vertex arrangement== | |||
It shares its [[vertex arrangement]] with three [[nonconvex uniform polyhedra]]: | |||
{|class="wikitable" width="400" style="vertical-align:top;text-align:center" | |||
|[[Image:Truncated dodecahedron.png|100px]]<br>Truncated dodecahedron | |||
|[[Image:Great icosicosidodecahedron.png|100px]]<br>[[Great icosicosidodecahedron]] | |||
|[[Image:Great ditrigonal dodecicosidodecahedron.png|100px]]<br>[[Great ditrigonal dodecicosidodecahedron]] | |||
|[[Image:Great dodecicosahedron.png|100px]]<br>[[Great dodecicosahedron]] | |||
|} | |||
== Related polyhedra and tilings == | |||
It is part of a truncation process between a dodecahedron and icosahedron: | |||
{{Icosahedral truncations}} | |||
This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2n.2n), and [n,3] [[Coxeter group]] symmetry. | |||
{{Truncated figure1 table}} | |||
==See also== | |||
*[[:Image:Truncateddodecahedron.gif|Spinning truncated dodecahedron]] | |||
*[[Icosahedron]] | |||
*[[Icosidodecahedron]] | |||
*[[Truncated icosahedron]] | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9) | |||
*{{cite book|author=Cromwell, P.|year=1997|title=Polyhedra|location=United Kingdom|publisher=Cambridge|pages=79-86 ''Archimedean solids''|isbn=0-521-55432-2}} | |||
==External links== | |||
*{{mathworld2 | urlname = TruncatedDodecahedron| title = Truncated dodecahedron | urlname2 = ArchimedeanSolid | title2 = Archimedean solid}} | |||
*{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|o3x5x - tid}} | |||
*[http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=1e9V3YL5nW2MMkIcdn0TdMHHhXMiuoCQGqz2g3IjH7orIJ5iBy9LQ80CKQP1GAP9MmtklgzVBcF5ZfK9LsPLcjTfCVtbQWJrpIJTarRzJGitPNEnHrk3rNm5pr6Gzui1P5MD7RwSrFu6TKzjy5qQl5PYokM9mcFWcoPivzjQxlRGa1eVpVmZl5Uv2nXTaX5RSgc2N5B3daPbsAUEsCGxrnbgMLCKvMvztIjl44GGTstwl3pC589OwhVUTHvkTzg6b4dpshGHQn4ajtxQA8chKkqzW1wKBsKuMpbqE4oCXbIi2sfEgppN1tcDBWVOJUXQfPiEglU1jtQi7fUj5xDW2PpZtdwQDmwpC3Lk&name=Truncated+Dodecahedron#applet Editable printable net of a truncated dodecahedron with interactive 3D view] | |||
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra] | |||
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra | |||
{{Archimedean solids}} | |||
{{Polyhedron navigator}} | |||
[[Category:Uniform polyhedra]] | |||
[[Category:Archimedean solids]] |
Revision as of 13:04, 21 November 2013
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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:
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.
Centered by | Vertex | Edge 3-10 |
Edge 10-10 |
Face Triangle |
Face Decagon |
---|---|---|---|---|---|
Image | |||||
Projective symmetry |
[2] | [2] | [2] | [6] | [10] |
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: Template:Icosahedral truncations
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.
Template:Truncated figure1 table
See also
Notes
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References
- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
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External links
- Template:Mathworld2
- Template:KlitzingPolytopes
- Editable printable net of a truncated dodecahedron with interactive 3D view
- The Uniform Polyhedra
- Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Template:Archimedean solids Template:Polyhedron navigator
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