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Type=[[:Category:Near MissNear Miss]] 

Type=[[:Category:Near MissNear Miss]] 

Triangles=Squares=Pentagons=Rhombic=Rods=60+90Spheres=32Author=[[User:AmafirlianAmafirlian]] 00:58, 9 August 2007 (UTC)}} 

Triangles=Squares=Pentagons=Rhombic=Rods=60+90Spheres=32Author=[[User:AmafirlianAmafirlian]] 00:58, 9 August 2007 (UTC)}} 
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The rhombic triacontahedron is made from thirty golden rhombii, with the sharp ends meeting at twelve valency five nodes, and the shallow ends meeting in twenty valency three nodes. 





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It is possible to build a rhombic triancontahedron "naked" with Geomag bars, but it is very unstable: the friction between the joins is just enough to keep the model afloat (TODO: picture). This model has an internal support structure that renders the solid rigid.

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This support structure is amazingly close. It consists of [[Five Intersecting Tetrahedrafive intersecting tetrahedra]], connected to the spheres in the rhombic triacontahedron with valency 3. The size of the edges of the tetrahedra is 3 rods. 
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This support structure is amazingly close (the error is less than 1%). It consists of [[Five Intersecting Tetrahedrafive intersecting tetrahedra]], connected to the spheres in the rhombic triacontahedron with valency 3 (there are twenty of them). The length of the tetrahedral edges is 3 rods. 

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The model is heavy (about 1kg) and strong: it's quite suitable for playing the game of ''Geomag Catch''. 

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{{clr}} 






== Building Instructions == 

== Building Instructions == 




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# Build the lower half of the rhombic triacontahedron. Put a few diagonals into the rhombii using a different color. As the model takes shape discard these. 
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# Build the lower halve of the rhombic triacontahedron. 
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# Start putting support rods in (3 connected rods). Once you have a triangle of internal support, immediately turn this into a tetrahedron. 
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# Start putting support rods in (3 connected rods). 
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#* Basically it suffices to connect one side to a sphere of valency 3, and see to which sphere of valency 3 the other side leads you. 
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#* Basically it suffices to connect one side to a sphere of valency 3, and see to which sphere of valency 3 the other side leads you. 
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#* Count edges: support rods always span four edges of the triacontahedron. 


#* Also keep a look out for the five intersecting tetrahedra that develop. 

#* Also keep a look out for the five intersecting tetrahedra that develop. 

# Complete the upper halve of the rhombic triacontahedron. 

# Complete the upper halve of the rhombic triacontahedron. 

# Put in the remaining support rods. 

# Put in the remaining support rods. 




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* As an unexpected bonus, this construction method also provides the easiest way of building the [[Five Intersecting Tetrahedrafive intersecting tetrahedra]]! 
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* This construction method also provides the easiest way of building the [[Five Intersecting Tetrahedrafive intersecting tetrahedra]]! 





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== Different Views == 

== Different Views == 
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<gallery captionalign="left"> 
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<gallery> 

Image:Rhombic triacontahedron near miss.jpg<small>View on a sphere with valency 5</small> 

Image:Rhombic triacontahedron near miss.jpg<small>View on a sphere with valency 5</small> 

Image:Rhombic triacontahedron near miss b.jpg<small>View on a sphere with valency 5</small> 

Image:Rhombic triacontahedron near miss b.jpg<small>View on a sphere with valency 5</small> 
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Image:Rhombic triacontahedron near miss d.jpg<small>Closeup of the support structure after removal of a sphere with valency 5</small> 

Image:Rhombic triacontahedron near miss d.jpg<small>Closeup of the support structure after removal of a sphere with valency 5</small> 

</gallery> 

</gallery> 
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== Calculations == 

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''This model is indeed a very near miss.'' 

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The support struts may be seen as "step three diagonals" of the dodecahedron formed by the points of the intersecting tetrahedra. 

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This strut is itself the diagonal of a square whose sides are formed by the inscribed pentagram on the dodecahedral faces, which have length the golden ratio, phi (=1.618approx) [this is a "roottwophi bar" see ???] 

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So, the support strut is sqrt(2) times phi times the length of the dodecahedral edge. 

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The dodecahedral edge is the short diagonal of the golden rhombus of the 

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rhombic triacontahedron: this length is 4/sqrt(10+2*sqrt(5)) (1.051 approx). 

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It's the same as the inverse of the circumradius of the icosahedron, see [http://mathworld.wolfram.com/Icosahedron.html Mathworld] 

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Multiplying all these together gives us the '''geometric''' ratio 

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between the rhombic triacontahedron edge length and the support strut. 

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* Geometric ratio = 2.406 

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* Actual geometric distance = 93.45mm 

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* Length of strut = 3 bars + 0.61mm 

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We know that once the error gets down to around half a millimetre, the structure can usually absorb it. 

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A very small smile is permitted. 






== Related Links == 

== Related Links == 




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* [http://www.geomagmasters.com Geomag Structures and Projects Site] 

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* [http://www.geomagmasters.com/Shapes.htm Glow in the Dark Rhombic Triacontahedron] 


* [[Wikipedia:Rhombic triacontahedron]] 

* [[Wikipedia:Rhombic triacontahedron]] 




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== Other Versions == 





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* [http://www.flickr.com/photos/karlhorton/sets/72157601630325142/ flickr set by Karl Horton] 

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Voted: [[User:Karl HortonKarl Horton]] 09:10, 9 August 2007 (UTC) 


[[Category:Polyhedron]] 

[[Category:Polyhedron]] 
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[[Category:Zonohedron]] 


[[Category:Near Miss]] 

[[Category:Near Miss]] 

[[Category:Intersecting Structure]] 

[[Category:Intersecting Structure]] 
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[[Category:Reinforcement Structure]] 
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[[Category:Support Structure]] 
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[[Category: *****]] 
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[[Category: 5Star]] 

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Voted: [[User:Karl HortonKarl Horton]] 09:10, 9 August 2007 (UTC) 