**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

**5.2.4.1 Tiling Nondeforming
Surfaces**

If the surface to be covered by a nanorobot aggregate has
constant area, such as the exterior of a bone, tooth, or fingernail, or the
inner surface of a passive sinus or duct, or the skull and meninges, or certain
components of the eyeball (cornea, sclera, etc.), then the question is how to
completely tile a fixed surface using prismatic units of a particular shape,
a familiar problem in spatial geometry.^{519,520}
Orthogonally-arrayed unit circles achieve a packing density of only 78.54%,
while hexagonal packing of unit circles achieves 90.70% coverage. However, as
is well-known, triangles, squares, or hexagons are the only regular polygonal
prismatic solids that can completely tile a plane by themselves, achieving 100%
packing density (Fig.
5.1). There are also an infinite number of irregular planar polygonal tessellations
using a wide variety of tile shapes (Fig.
5.2), although nanorobots with these shapes would suffer extraordinarily
low volume/surface ratios -- an extremely inefficient use of space.

Triangles, rectangles, or regular hexagons (e.g., a roll of
chicken wire) can be used alone to tile a cylindrical surface. The deformation
of nanorobot bumpers into curved wedge segments allows tiling of ellipsoidal
or other randomly curved surfaces provided the minimum radius of curvature r_{curve}
is relatively large (e.g., r_{curve} >> L_{n}). For areas
containing smaller radii of curvature (r_{curve} > L_{n}),
a combination of three regular polygonal prisms may be used to completely tile
any arbitrarily curved surface: hexagonal prisms for flat sections, mixed with
pentagonal prisms to add positive curvature (e.g., convexity, like a sphere)
and heptagonal prisms to add negative curvature (permitting concave surface
deformations). Artificial fullerene and natural radiolarian structures illustrate
this approach (Fig.
2.19).^{522,523,1308}
Triangular prisms can also be used to tile arbitrarily curved surfaces; for
example, some geodesic domes^{539}
and the polio virus are symmetrical spherical arrangements of alternating triangles.^{384}
(Many viruses are regular icosahedra.)

Last updated on 17 February 2003