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 Truncated Cuboctahedron of Cubes Truncated Cuboctahedron of Cubes rendered in Geomag Pastelles Type Nested Polyhedra Panels 24 × 72 × Rods 168 × Spheres 96 × Author PolyClare 18:55, 20 January 2010

It's a truncated cuboctahedron on the outside, and a rhombicuboctahedron on the inside, and the two are connected by the cubes.

Using this composition, the numbers of rods, spheres, and panels may be quickly counted up:

• 12 pink cubes: 12 x [ 12 pink rods, 6 pink squares, 4 spheres + 4 spheres / shared by 2 cubes ] = 144 rods & 72 squares & 72 spheres
• 24 blue triangles: 24 x [ 1 blue rod , 1 blue triangle, 0 spheres ] = 24 rods & 24 triangles

Totals given in the summary box.

## Related construction Edit

The construction described above may be created without the blue triangles and rods. This is actually a relatively stable construction when using classic Geomag rods. This is because the direction that the inner cuboctahedron 'wants' to collapse is prevented by the minimum angular limit of the classic Geomag rods. [Thanks to Karl Horton for pointing out the instability of this construction as shown]

This object may be constructed by building a rhombicuboctahedron, and then augmenting some of its square faces (specifically those adjacent to triangles) with cubes. Any square face of the rhombicuboctahedron that is surrounded solely by squares should be left as an open hole (in addition to leaving all the triangular faces as open holes).

## Investigating the instabilityEdit

To experience for myself just how unstable the inner rhombicuboctahedron could be without those last six square panels I decided to build a rhombicuboctahedron from the Geomag Kids rods and panels. And I would build it without the attached cubes, for simplicity and clarity, and to avoid the help that the Geomag minimum angular limit gave. I was expecting that the instability should be magnified because of the the larger dimensions of the Geomag Kids rods.

So first I built the rhombicuboctahedron, only inserting square panels in locations adjacent to triangular faces. As I was building it, it was clear that it was unstable, and wanted to collapse. When I adjusted the faces and gently sat the completed construction on one of its unpanelled square faces, it held together enough to stand on its own. However, when I put it on a triangular face, the thing collapsed into a shape close to a tetrahedron!

 Rhombicuboctahedron with not enough square panels for rigidity Before collapse After collapse
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