**Nanomedicine,
Volume I: Basic Capabilities**

**©
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 p^{2} / 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
arrays^{518} that may be employed
to form near-arbitrary networks^{2843}
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