The original version of the essay below was first published to geometryresearch at the Swarthmore Math Forum in April of 1998. Beyond Flatland:
by Kirby Urner 

Agglomerating equiradiused spheres outwardly from a central sphere in concentric cuboctahedral layers gives an isotropic Barlow packing more commonly known as the facecentered cubic or fcc. The packing subsumes the tetrahedral arrangement used to stack cannon balls, called the 'brass monkey' by some in USA civil war days, and commonly used for fruitstacking in grocery stores. The nuclear sphere, plus layers out to one of the cuboctahedron's 8 trianglar facets, defines a tetrahedral packing. 

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 By the same token, a tetrahedral packing, with an apexial entry point and fallpattern with 3 choices per peg, to the base corners of a tet, defines "Pascal's Tetrahedron" of countable pathways, again with a kind of peakshaped statistical outcome. 1 1 1 1 1 1 1 2 2 3 3 4 4 1 2 1 3 6 3 6 12 6 1 3 3 1 4 12 12 4 1 4 6 4 1 Pascal's Tetrahedron also describes the structure of diamond crystals, where the carbon atoms arrange in stacked tetrahedra. Another feature of the fcc pegboard is the fact that any four pegs not in the same plane define a tetrahedron of whole number volume, relative to a unit tetrahedron defined by four closepacked fcc spheres. This is true for skew as well as regular tetrahedra. If the spheres have unit radius, then each sphere center is distance 2 from 12 surrounding centers, but this 2 also = 1 interval or 1 diameter, so the unit volume tetrahedron is likewise a model of 1x1x1 or 1^3, measuring in intervals. 


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PART II: The Octet Truss 

The most isotropic Barlow packing of 12 spheres around every 1 in a cuboctahedral conformation, serves as a basis for explorations in crystallography. Considered as a skeletal arrangement of edges, all length 2 and interconnecting adjacent unit radius sphere centers, we get a spaceframe known to engineers as the octettruss. Alexander Graham Bell was among the first to study the octettruss circa 1907 for its relevance to large scale structures, such as towers and giant kites. Fuller made use of the same spaceframe in his geometric investigations, naming it the isotropic vector matrix. 

The four trusses may be paired to give two sets of vertices arranged in a cubic pattern, ala XYZ, with one set having vertices at the centers of the other's cubes. This pattern is known as bodycentered cubic or bcc in crystallography. As Kepler once discovered, spheres packed in an fcc lattice may expand to become rhombic dodecahedra, thereby filling all the interspheric voids. An octet truss is a skeletal spaceframe with all edges perpendicular to rhombic facets. The spheres inscribed in the rhombic dodecas "kiss" at these face centers. The rhombic dodecahedron's 14 vertices occupy the centers of the 8 tetrahedral and 6 octahedral voids surrounding any fcc sphere. Its volume is 6 relative to the tetrahedron's, thereby providing the beginnings of our concentric hierarchy: Shape Volume   Tetrahedron 1 Cube 3 Octahedron 4 Rh Dodecahedron 6 Cuboctahedron 20 Table 1: Concentric Hierarchy 
The volume 4 octahedron inscribes as the long diagonals of the rhombic dodecahedron and defines the vertices of a fourth octet truss paired with the first i.e. the one defined by the rhombic dodecahedron centers. 
The above concentric hierarchy shapes may all be fractured into "common denominator" modules, a minimal set of which consists of the A and the B mods, irregular tetrahedra with left and right mirrored versions (or insideout versions), both of volume 1/24. Left and right A mods, plus a B mod of either hand, make a minimum tetrahedral spacefiller or MITE of volume 1/8. All MITEs (MInimum TEtrahedra) are outwardly identical and interchangeable, regardless of the internal B's handedness. Eight MITEs make a Coupler of volume 1, another spacefiller. Our more complete hierarchy now looks like this: 

Shape Volume   A module 1/24 B module 1/24 MITE 1/8 Coupler 1 Tetrahedron 1 Cube 3 Octahedron 4 Rh Dodecahedron 6 Cuboctahedron 20 Table 2: Concentric Hierarchy References

PART III: The Jitterbug 

The 20volumed cuboctahedron embeds in the fcc lattice and is defined by 12 unitradius spheres packed around a nuclear sphere. Considered as a wireframe with flexible joints, it twistcontracts in either a clockwise or counterclockwise direction by bringing pairs of vertices along the diagonals of its square faces closer together. When all edges are length 2, the resulting icosahedron's volume is about 18.51 relative to the cuboctahedron's of 20. 
The jitterbug transformation may be computerized using STRUCK, a Java application for building structures using edges which push or pull exponentially when forced away from their predefined restlengths. STRUCK also permits sets of edges to smoothly alter their restlengths through time, as in this case of the jitterbug, where the diagonals of the cuboctahedron's six square faces go from 2 x root(2) to 2 (and onward to zero at octaphase). STRUCK optionally writes successive animation frames in POV format for raytracing and moviemaking purposes. 




Some fivefold symmetric shapes, the 30faceted rhombic triacontahedron for example, may be tightly shrinkwrapped around the unitradius fcc sphere, as is Kepler's fourfold symmetric rhombic dodecahedron. An interesting fact about the shrinkwrapped rhombic triacontahedron is how closely it misses having a volume of precisely 5. By shrinking its unit radius by a hair (~0.0005) we can make each of its 120 Tmodules have a volume of precisely 5/120 or 1/24, the same as the A and B modules discussed above  a useful mnemonic for fitting this fivefold symmetric shape into our growing concentric hierarchy. The Tmod is also the principal Koski mod, which he recursively disassembles into smaller and smaller phiscaled versions of itself, down to his arbitrarily small "remainder tets". 

Shape Volume   A module 1/24 B module 1/24 T module 1/24 MITE 1/8 Coupler 1 Tetrahedron 1 Cube 3 Octahedron 4 Rh Triacontahedron 5 Rh Dodecahedron 6 Icosahedron 18.51... Cuboctahedron 20 Table 3: Concentric Hierarchy With the above concentric hierarchy in mind, a geometry student can look at an octet truss and superimpose an easily memorable system of scaled shapes. Both four and fivefold symmetric members are represented, with a bridging transformation, and are concentric and hierarchically arranged. Many of the shapes are also dual pairs, which pairs may then be combined to give additional shapes. For example, the cube and octahedron are duals, and combine to give the rhombic dodecahedron. Given the streamlining effects of merging fcc packing with the octet truss and a concentric hierarchy of easytoremember volumes, the essentials of this curriculum are likely to gravitate down to lower grade levels, such that all of the above will be in some form accessible to an average 14 year old. Using TV, the internet, and film, it should be possible to communicate this primary level information quickly to a fairly large and global audience. We hope to have this job completed or well underway by the end of 1998. References:
Oregon Curriculum Network 