Thursday, November 06, 2008

Re: K-Mods etc.

:: algebraic analysis ::
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[ excerpt from Synergeo 44896, typo fixed ]

In review, starting with a Golden Cuboid of edges 1 x phi x 1/phi, we figure out what a module consisting of 1/6th that total volume would be, in tetravolumes, then scale appropriately to gain a rhombic triacontahedron of tetravolume 7.5. We confirm this scale or height is the same we'd get if going off Fuller's calcs for the T-mod height needed for a rhombic triacontahedron of volume 5.

Height = radius of rhombic triacontahedron i.e. from body center to any face center.

Koski proves by construction that his 7.5 rh triac and 6.0 rhombic dodeca have the intersecting meshes that they do, using his phi-scaled modules.

The interesting result here is the concentric hierarchy rhombic triacontahedron of volume 7.5 has a radius of phi/sqrt(2).

Another, related, discovery is the concentric hierarchy cube of volume 2, derived from the pentagonal dodecahedron inscribed as short face diagonals of the above 5.0 volumed rhombic triacontahedron.

In teaching this stuff we could go:

Tetra 1
Coupler 1
Cube 2
Duo Tet 3
Octa 4
Triaconta 5 (t-mods)
Triaconta 5+ (e-mods)
Rhdodeca 6
Triaconta 7.5 (k-mods)
Cubocta 20

t, e and k mods all have the same shape but different scales.

Followup: posting to math-teach @ Math Forum, Nov 29, 2008.