Trailing zero: Difference between revisions

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The [[dihedral angle]]s for the [[edge-transitive]] polyhedra are:
Ed is what individuals call me and my wife doesn't like it at all. For many years he's been residing in Alaska and he doesn't plan on altering it. My working day job is an information officer but I've currently applied for another one. To play lacross is the factor I love most of all.<br><br>Feel free to visit my web-site; online psychic reading ([http://www.seekavideo.com/playlist/2199/video/ just click the following page])
{| class="wikitable"
|- align="center"
! Picture
! Name
! [[Schläfli symbol|Schläfli<BR>symbol]]
! [[Vertex configuration|Vertex/Face<BR>configuration]]
! exact dihedral angle<BR>(radians)
! approximate<BR>dihedral angle<BR>(degrees)
|-align="center"
! colspan=6 | [[Platonic solid]]s (regular convex)
|- align="center"
| [[Image:Tetrahedron.png|30px]]
| align="left" | [[Tetrahedron]]
| {3,3}
| (3.3.3)
| arccos(1/3)
| 70.53°
|- align="center"
| [[Image:Hexahedron.png|30px]]
| align="left" | [[Hexahedron]] or [[Cube (geometry)|Cube]]
| {4,3}
| (4.4.4)
| π/2
| 90°
|- align="center"
| [[Image:Octahedron.png|30px]]
| align="left" | [[Octahedron]]
| {3,4}
| (3.3.3.3)
| π &minus; arccos(1/3)
| 109.47°
|- align="center"
| [[Image:Dodecahedron.png|30px]]
| align="left" | [[Dodecahedron]]
| {5,3}
| (5.5.5)
| π &minus; arctan(2)
| 116.56°
|- align="center"
| [[Image:Icosahedron.png|30px]]
| align="left" | [[Icosahedron]]
| {3,5}
| (3.3.3.3.3)
| π &minus; arccos(&radic;5/3)
| 138.19°
|-align="center"
! colspan=6 | [[Kepler-Poinsot solid]]s (regular nonconvex)
|- align="center"
| [[Image:Small stellated dodecahedron.png|30px]]
| align="left" |[[Small stellated dodecahedron]]||{5/2,5}
| (5/2.5/2.5/2.5/2.5/2)
| π &minus; arctan(2)
| 116.56°
|-  align="center"
| [[Image:Great dodecahedron.png|30px]]
| align="left" |[[Great dodecahedron]]||{5,5/2}
| (5.5.5.5.5)/2
| arctan(2)
| 63.435°
|-  align="center"
| [[Image:Great stellated dodecahedron.png|30px]]
| align="left" |[[Great stellated dodecahedron]]||{5/2,3}
| (5/2.5/2.5/2)
| arctan(2)
| 63.435°
|-  align="center"
| [[Image:Great icosahedron.png|30px]]
| align="left" |[[Great icosahedron]]||{3,5/2}
| (3.3.3.3.3)/2
| arcsin(2/3)
| 41.810°
|- align="center"
! colspan=6 | [[Quasiregular polyhedron|Quasiregular polyhedra]] ([[Rectification (geometry)|Rectified regular]])
|- align="center"
| [[Image:Uniform polyhedron-33-t1.png|30px]]
| align="left" | [[Tetratetrahedron]]
| r{3,3}
| (3.3.3.3)
| <math> \pi - \arccos{\left( \frac{1}{3} \right)} </math>
| 109.47°
|- align="center"
| [[Image:Cuboctahedron.png|30px]]
| align="left" | [[Cuboctahedron]]
| r{3,4}
| (3.4.3.4)
| <math> \pi - \arccos{\left( \frac{1}{\sqrt{3}} \right)} </math>
| 125.264°
|- align="center"
| [[Image:Icosidodecahedron.png|30px]]
| align="left" | [[Icosidodecahedron]]
| r{3,5}
| (3.5.3.5)
| <math> \pi - \arccos{ \left( \sqrt{ \frac{ (5 + 2\sqrt 5)}{15} } \right) } </math>
| 142.623°
|- align="center"
| [[Image:Dodecadodecahedron.png|30px]]
| align="left" | [[Dodecadodecahedron]]
| r{5/2,5}
| (5.5/2.5.5/2)
| π  arctan(2)
| 116.56°
|- align="center"
| [[Image:Great icosidodecahedron.png|30px]]
| align="left" | [[Great icosidodecahedron]]
| r{5/2,3}
| (3.5/2.3.5/2)
|
|
|- align="center"
! colspan=6 | Ditrigonal polyhedra
|- align="center"
| [[Image:Small ditrigonal icosidodecahedron.png|30px]]
| align="left" | [[Small ditrigonal icosidodecahedron]]
| a{5,3}
| (3.5/2.3.5/2.3.5/2)
|
|
|- align="center"
| [[Image:Ditrigonal dodecadodecahedron.png|30px]]
| align="left" | [[Ditrigonal dodecadodecahedron]]
| b{5,5/2}
| (5.5/3.5.5/3.5.5/3)
|
|
|- align="center"
| [[Image:Great ditrigonal icosidodecahedron.png|30px]]
| align="left" | [[Great ditrigonal icosidodecahedron]]
| c{3,5/2}
| (3.5.3.5.3.5)/2
|
|
|- align="center"
! colspan=6 | [[Hemipolyhedron|Hemipolyhedra]]
|- align="center"
| [[Image:Tetrahemihexahedron.png|30px]]
| align="left" | [[Tetrahemihexahedron]]
| o{3,3}
| (3.4.3/2.4)
|
| 54.73°
|- align="center"
| [[Image:Cubohemioctahedron.png|30px]]
| align="left" | [[Cubohemioctahedron]]
| o{3,4}
| (4.6.4/3.6)
|
| 54.73°
|- align="center"
| [[Image:Octahemioctahedron.png|30px]]
| align="left" | [[Octahemioctahedron]]
| o{4,3}
| (3.6.3/2.6)
|
| 70.53°
|- align="center"
| [[Image:Small dodecahemidodecahedron.png|30px]]
| align="left" | [[Small dodecahemidodecahedron]]
| o{3,5}
| (5.10.5/4.10)
|
| 26.063°
|- align="center"
| [[Image:Small icosihemidodecahedron.png|30px]]
| align="left" | [[Small icosihemidodecahedron]]
| o{5,3}
| (3.10.3/2.10)
|
| 116.56°
|- align="center"
| [[Image:Great dodecahemicosahedron.png|30px]]
| align="left" | [[Great dodecahemicosahedron]]
| o{5/2,5}
| (5.6.5/4.6)
|
|
|- align="center"
| [[Image:Small dodecahemicosahedron.png|30px]]
| align="left" | [[Small dodecahemicosahedron]]
| o{5,5/2}
| (5/2.6.5/3.6)
|
|
|- align="center"
| [[Image:Great icosihemidodecahedron.png|30px]]
| align="left" | [[Great icosihemidodecahedron]]
| o{5/2,3}
| (3.10/3.3/2.10/3)
|
|
|- align="center"
| [[Image:Great dodecahemidodecahedron.png|30px]]
| align="left" | [[Great dodecahemidodecahedron]]
| o{3,5/2}
| (5/2.10/3.5/3.10/3)
|
|
|- align="center"
! colspan=6 | [[Polyhedron#Quasi-regular duals|Quasiregular dual solids]]
|- align="center"
| [[Image:Hexahedron.png|30px]]
| align="left" | [[Cube|Rhombic hexahedron]]<BR>(Dual of tetratetrahedron)
| -
| V(3.3.3.3)
| π &minus; π/2
| 90°
|- align="center"
| [[Image:Rhombic dodecahedron.png|30px]]
| align="left" | [[Rhombic dodecahedron]]<BR>(Dual of cuboctahedron)
| -
| V(3.4.3.4)
| π &minus; π/3
| 120°
|- align="center"
| [[Image:Rhombic triacontahedron.png|30px]]
| align="left" | [[Rhombic triacontahedron]]<BR>(Dual of icosidodecahedron)
| -
| V(3.5.3.5)
| π &minus; π/5
| 144°
|- align="center"
| [[Image:DU36 medial rhombic triacontahedron.png|30px]]
| align="left" | [[Medial rhombic triacontahedron]]<BR>(Dual of dodecadodecahedron)
| -
| V(5.5/2.5.5/2)
| π  π/3
| 120°
|- align="center"
| [[Image:DU54 great rhombic triacontahedron.png|30px]]
| align="left" | [[Great rhombic triacontahedron]]<BR>(Dual of great icosidodecahedron)
| -
| V(3.5/2.3.5/2)
| π  π/(5/2)
| 72°
|- align="center"
! colspan=6 | Duals of the ditrigonal polyhedra
|- align="center"
| [[Image:DU30 small triambic icosahedron.png|30px]]
| align="left" | [[Small triambic icosahedron]]<BR>(Dual of small ditrigonal icosidodecahedron)
| -
| V(3.5/2.3.5/2.3.5/2)
|
|
|- align="center"
| [[Image:DU41 medial triambic icosahedron.png|30px]]
| align="left" | [[Medial triambic icosahedron]]<BR>(Dual of ditrigonal dodecadodecahedron)
| -
| V(5.5/3.5.5/3.5.5/3)
|
|
|- align="center"
| [[Image:DU47 great triambic icosahedron.png|30px]]
| align="left" | [[Great triambic icosahedron]]<BR>(Dual of great ditrigonal icosidodecahedron)
| -
| V(3.5.3.5.3.5)/2
|
|
|- align="center"
! colspan=6 | [[Hemipolyhedron#Duals of the hemipolyhedra|Duals of the hemipolyhedra]]
|- align="center"
| [[Image:Tetrahemihexacron.png|30px]]
| align="left" | [[Tetrahemihexacron]]<BR>(Dual of tetrahemihexahedron)
| -
| V(3.4.3/2.4)
| π  π/2
| 90°
|- align="center"
| [[Image:Hexahemioctacron.png|30px]]
| align="left" | [[Hexahemioctacron]]<BR>(Dual of cubohemioctahedron)
| -
| V(4.6.4/3.6)
| π  π/3
| 120°
|- align="center"
| [[Image:Hexahemioctacron.png|30px]]
| align="left" | [[Octahemioctacron]]<BR>(Dual of octahemioctahedron)
| -
| V(3.6.3/2.6)
| π  π/3
| 120°
|- align="center"
| [[Image:Small dodecahemidodecacron.png|30px]]
| align="left" | [[Small dodecahemidodecacron]]<BR>(Dual of small dodecahemidodecacron)
| -
| V(5.10.5/4.10)
| π  π/5
| 144°
|- align="center"
| [[Image:Small dodecahemidodecacron.png|30px]]
| align="left" | [[Small icosihemidodecacron]]<BR>(Dual of small icosihemidodecacron)
| -
| V(3.10.3/2.10)
| π  π/5
| 144°
|- align="center"
| [[Image:Small dodecahemicosacron.png|30px]]
| align="left" | [[Great dodecahemicosacron]]<BR>(Dual of great dodecahemicosahedron)
| -
| V(5.6.5/4.6)
| π  π/3
| 120°
|- align="center"
| [[Image:Small dodecahemicosacron.png|30px]]
| align="left" | [[Small dodecahemicosacron]]<BR>(Dual of small dodecahemicosahedron)
| -
| V(5/2.6.5/3.6)
| π  π/3
| 120°
|- align="center"
| [[Image:Great dodecahemidodecacron.png|30px]]
| align="left" | [[Great icosihemidodecacron]]<BR>(Dual of great icosihemidodecacron)
| -
| V(3.10/3.3/2.10/3)
| π  π/(5/2)
| 72°
|- align="center"
| [[Image:Great dodecahemidodecacron.png|30px]]
| align="left" | [[Great dodecahemidodecacron]]<BR>(Dual of great dodecahemidodecacron)
| -
| V(5/2.10/3.5/3.10/3)
| π  π/(5/2)
| 72°
|}
 
== References ==
* [[Coxeter]], ''Regular Polytopes'' (1963), Macmillian Company
** ''Regular Polytopes'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (Table I: Regular Polytopes, (i) The nine regular polyhedra {p,q} in ordinary space)
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-7 to 3-9)
* {{MathWorld |title=Uniform Polyhedron |id=UniformPolyhedron}}
 
[[Category:Polyhedra]]

Revision as of 22:14, 26 February 2014

Ed is what individuals call me and my wife doesn't like it at all. For many years he's been residing in Alaska and he doesn't plan on altering it. My working day job is an information officer but I've currently applied for another one. To play lacross is the factor I love most of all.

Feel free to visit my web-site; online psychic reading (just click the following page)

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