Wednesday, May 05, 2010

Fresh Fruit

Fig 1: 14 Rhombic Triacontahedra around One

Depicted above is a 2-Frequency rhombic dodecahedron (shaded), of tetravolume 48, anchoring a bevy of rhombic triacontahedra of tetravolume 7.5 at its 14 vertexes.

In terms of our home base ball packing, the IVM, the uniradius spheres would be smaller than these rhombic triacontahedra. This distance across the rhombic dodecahedron, from face center to opposite face center, would be two sphere diameters (hence 2-Frequency).

Because rhombic dodecahedra fill space, what we see here is the nucleus of a repeating pattern or lattice. The rhombic triacontahedra occupy positions in a body cubic centric packing or BCC, with eight around 1 at the corners of a cube. Note that the triacontahedra touch one another at vertices only, do not have shared edges or faces.

Fig 2: Nuclear Rhombic Triacontahedron

Here is a nuclear rhombic triacontahedron (shaded) surrounded by its eight neighbors at the corners of a cube (shown in outline). The edges of the cube define the short face diagonals of the 2F rhombic dodecahedron depicted in Figure 1.

Recall that the volume 6 rhombic dodecahedron and volume 3 cube both decompose into constituent Mites of volume 1/8. The volume of two Mites is equal to the volume of four K-mods, which are the T-mod shaped tetrahedral wedges dividing the 7.5 RT into 120 constituent members.

The Mite itself dissects into two A-mods and one B-mod, each of volume 1/24 and each equal to the volume of the T-mod, that which divides the RT of tetravolume 5 into 120 wedges.

The volume of the K-mod is 3/2 larger than 1/24, i.e. is 1/16 or half that of a Mite. Four K-mods have the volume of a Syte (two Mites).

David Koski supplied these pictures, developed using Scott Vorthmann's vZome. He has a long-standing interest in rhombic triacontahedra and their dissection into phi-scalable modules.

Fig 3: Duotet Cube of volume 24 (eight touching one)