|Rhombic Triacontahedron Near Miss|
|Author||Amafirlian 00:58, 9 August 2007 (UTC)|
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.
It is possible to build a rhombic triancontahedron "naked" with Geomag bars, but its 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.
This support structure is amazingly close (the error is less than 1%). It consists of five 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.
The model is heavy (about 1kg) and strong: it's quite suitable for playing the game of Geomag Catch.
- 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.
- Start putting support rods in (3 connected rods). Once you have a triangle of internal support, immediately turn this into a tetrahedron.
- 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.
- Also keep a look out for the five intersecting tetrahedra that develop.
- Complete the upper halve of the rhombic triacontahedron.
- Put in the remaining support rods.
- As an unexpected bonus, this construction method also provides the easiest way of building the five intersecting tetrahedra!
This model is indeed a very near miss.
The support struts may be seen as "step three diagonals" of the dodecahedron formed by the points of the intersecting tetrahedra.
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 ???]
So, the support strut is sqrt(2) times phi times the length of the dodecahedral edge.
The dodecahedral edge is the short diagonal of the golden rhombus of the rhombic triacontahedron: this length is 4/sqrt(10+2*sqrt(5)) (1.051 approx).
It's the same as the inverse of the circumradius of the icosahedron, see Mathworld
Multiplying all these together gives us the geometric ratio between the rhombic triacontahedron edge length and the support strut.
- Geometric ratio = 2.406
- Actual geometric distance = 93.45mm
- Length of strut = 3 bars + 0.61mm
We know that once the error gets down to around half a millimetre, the structure can usually absorb it.
A very small smile is permitted.
Voted: --Karl Horton 09:10, 9 August 2007 (UTC)