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|Type=[[:Category:Near Miss|Near Miss]] |
|Type=[[:Category:Near Miss|Near Miss]] |
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|Triangles=|Squares=|Pentagons=|Rhombic=|Rods=60+90|Spheres=32|Author=[[User:Amafirlian|Amafirlian]] 00:58, 9 August 2007 (UTC)}} |
|Triangles=|Squares=|Pentagons=|Rhombic=|Rods=60+90|Spheres=32|Author=[[User:Amafirlian|Amafirlian]] 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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− | The model is heavy (about 1kg) and strong: it's quite suitable for playing the game of ''Geomag Catch''. |
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− | {{clr}} |
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== Building Instructions == |
== Building Instructions == |
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− | # Build the lower |
+ | # Build the lower halve of the rhombic triacontahedron. |

− | # Start putting support rods in (3 connected rods) |
+ | # Start putting support rods in (3 connected rods). |

− | #* 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. |
+ | #* 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. |

− | #* Count edges: support rods always span four edges of the triacontahedron. |
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#* Also keep a look out for the five intersecting tetrahedra that develop. |
#* Also keep a look out for the five intersecting tetrahedra that develop. |
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# Complete the upper halve of the rhombic triacontahedron. |
# Complete the upper halve of the rhombic triacontahedron. |
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# Put in the remaining support rods. |
# Put in the remaining support rods. |
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− | * |
+ | * This construction method also provides the easiest way of building the [[Five Intersecting Tetrahedra|five intersecting tetrahedra]]! |

{{clr}} |
{{clr}} |
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== Different Views == |
== Different Views == |
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− | <gallery |
+ | <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> |
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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> |
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</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 "root-two-phi 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. |
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== 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] |
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* [[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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[[Category:Polyhedron]] |
[[Category:Polyhedron]] |
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− | [[Category:Zonohedron]] |
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[[Category:Near Miss]] |
[[Category:Near Miss]] |
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[[Category:Intersecting Structure]] |
[[Category:Intersecting Structure]] |
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− | [[Category: |
+ | [[Category:Support Structure]] |

− | [[Category: |
+ | [[Category: 5-Star]] |

+ | |||

⚫ |