© 1999 Robert A. Freitas Jr. All Rights Reserved.

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

5.2.5 Space-Filling Nanodevice Aggregates: Nano-Organs

In specialized structures such as synthetic bones or artificial organs constructed as aggregates of millions or billions of nanorobots, it may be necessary for nanodevices to completely occupy specific three-dimensional volumes. This is the familiar problem of space-filling polyhedra in spatial geometry.

The volumetric packing factor of closely-packed spheres of equal radius is only (3 p² / 64)^1/2 ~ 0.68017;⁵¹⁹ by comparison, protein-interior packing densities are ~60-85%.³²¹¹ The simplest way to completely fill a volume (e.g., packing factor = 1) is to use a prism having a horizontal cross-section of a shape that will tile surfaces, as previously described (Section 5.2.4.1). Multiple planar fully-tiled surfaces can then be stacked vertically to fill the volume. Space-filling forms include triangular, square, and hexagonal prisms (Fig. 5.4). Of these, the hexagonal prism has the highest volume/area ratio and appears to be the most efficient prismatic form.

Only five other shapes can uniformly fill volumes by themselves and thus may be candidates for space-filling nanorobots. The most important of these is the truncated octahedron, similar to the tetrakaidecahedron,¹¹⁰¹ which can fill space all by itself (Fig. 5.5). The truncated octahedron is formed by cutting the 6 corners off a regular octahedron (which has 8 faces, all triangles; Fig. 5.9), making a 14-hedron consisting of 6 small squares and 8 large hexagons, with 24 vertices and 36 edges. A truncated octahedron surrounded by 14 identical polyhedra contacting and matching each of its faces forms a solid unit in space resembling the original unit -- that is, very close to spherical. Such an aggregate with many planar facial contacts permits easy docking, fastening, and transmission of forces in all directions. Weyl,⁵²⁴ following Lord Kelvin,⁵²⁰ long ago recognized that this 14-hedron has the highest volume/area ratio of all the space- filling polyhedra (Table 5.1) and is a close mathematical model of soap bubble froth.

Four other single-species space-filling polyhedra are known. The first is the garnet-shaped rhombic dodecahedron, which has 12 faces, all of which are rhombuses (Fig. 5.6). The volume/area ratio of this 12-hedron is intermediate between the 14-hedron and the hexagonal prism; the shape is found in back-to-back formations in the planar hexagonal honeycomb cell of the bee.⁵¹⁹ The second is the closely related rhombo-hexagonal dodecahedron (Fig. 5.7), which is also self-packing. The third is the non-regular octahedron (8 faces, all triangles; Fig. 5.8), which results from truncating a cube such that all eight vertices are replaced by triangular faces with the new vertices coinciding with the intersection of the face diagonals of the original cube.⁵¹⁹ Hence the equatorial edges are all the same but the vertex edges are half the length of a circumscribed cube's interior diagonal, as distinct from the regular octahedron (Fig. 5.9) with all sides equal, which is not space-filling. The fourth of the rare space-filling solids is the trapezohedra (Fig. 5.10).⁵¹⁸

There are also an infinite number of nonuniform space-filling systems in which two or more kinds of polyhedra are combined.⁵¹⁸ Nanodevices filling space by this means would require at least two distinct nanodevice species of the appropriate shapes. Perhaps the best-known variant is the octet element, a space-filling unit consisting of an ordered combination of regular tetrahedra and regular octahedra. Unique interlocking interfacial or other features can ensure that major subassemblies of the collective structure can only fit together in one correct way as an aid in automatic self-assembly and quality assurance.

Finally, many biological solids are so thoroughly perforated with holes and tubes that they are more properly regarded as porous rather than completely filled solids. These forms are simulated using open packings of polyhedra arrays⁵¹⁸ that may be employed to form near-arbitrary networks²⁸⁴³ of interior voids and channels in artificial organs, as required. Zeolites and other molecular sieves provide excellent models of this configuration.^{382,2468,2469}

Last updated on 17 February 2003