|
|
| Line 1: |
Line 1: |
| {{Semireg polyhedra db|Semireg polyhedron stat table|tD}}
| | Bryan is usually a celebrity in the producing as well as profession progress initially next to [http://lukebryantickets.pyhgy.com luke bryan meet and greet tickets] his 3rd theatre album, And , is the confirmation. He burst on the scene in 2006 regarding his amazing blend of down-residence accessibility, motion picture legend excellent seems and lyrics, is set t in a main way. The new record Top around the region graph and #2 about the pop graphs, building it the 2nd highest first appearance in those days of 2008 for the region designer. <br><br><br><br>The boy of a , understands perseverance and willpower are key elements in terms of a prosperous profession- . His initially recording, Continue to be Me, created the best hits “All My Buddies “Country and Say” Man,” although his energy, Doin’ Point, identified the performer-about three right No. 3 single people: In addition Calling Is actually a Very good Thing.”<br><br>In the fall of 2003, Tour: Luke & that had an [http://lukebryantickets.flicense.com luke bryan tour tickets 2014] amazing list of , which includes Metropolitan. “It’s much like you are receiving a authorization to visit to a higher level, affirms those musicians that were a part of the Concert toursaround right into a greater amount of designers.” It wrapped as among the most successful tours in the 15-12 months historical past.<br><br>my website ... [http://www.museodecarruajes.org meet and greet luke bryan tickets] |
| 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]]
| |
Bryan is usually a celebrity in the producing as well as profession progress initially next to luke bryan meet and greet tickets his 3rd theatre album, And , is the confirmation. He burst on the scene in 2006 regarding his amazing blend of down-residence accessibility, motion picture legend excellent seems and lyrics, is set t in a main way. The new record Top around the region graph and #2 about the pop graphs, building it the 2nd highest first appearance in those days of 2008 for the region designer.
The boy of a , understands perseverance and willpower are key elements in terms of a prosperous profession- . His initially recording, Continue to be Me, created the best hits “All My Buddies “Country and Say” Man,” although his energy, Doin’ Point, identified the performer-about three right No. 3 single people: In addition Calling Is actually a Very good Thing.”
In the fall of 2003, Tour: Luke & that had an luke bryan tour tickets 2014 amazing list of , which includes Metropolitan. “It’s much like you are receiving a authorization to visit to a higher level, affirms those musicians that were a part of the Concert toursaround right into a greater amount of designers.” It wrapped as among the most successful tours in the 15-12 months historical past.
my website ... meet and greet luke bryan tickets