**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

**5.2.4.2 Tiling Deforming
Surfaces**

Most physiological surfaces are subject to periodic deformation,
or stretching, along one or both dimensions. Consider the problem of tiling
a surface that is periodically deforming in one direction only. A good example
in human physiology is the large elastic arteries which circumferentially distend
and contract as they absorb ~50% of the stroke volume of each ventricular ejection.
Specifically, diameter oscillations in the pulmonary artery and aorta typically
range from 9%-12%, averaging 11% for men and women under 35 years of age and
6.5% for people over 65.^{521} By
contrast, vessel lengths display almost no variation at all.^{361,521}

In order to tile such a monoaxially deforming surface, individual
nanorobots must have the ability to expand in one dimension only, using their
inflatable bumpers. The nanodevice shape which requires the smallest bumpers
(i.e., minimum volume) to accommodate a given linear expansion is the most space-usage
efficient and is probably preferred. Assuming continuous perimeter bumpers,
compare a series of N square, hexagonal, or triangular prismatic nanorobots
with equal sides of length s, height h, and fully contracted bumpers of minimum
depth b (Fig.
5.3). The center-to-center linear distance between adjacent square units
is L_{0} = s + 2b; between adjacent hexagonal units along the stretch
axis, L_{0} = 3s/2 + 3b/3^{1/2}; and between adjacent triangular
units along the stretch axis, L_{0} = s + 3^{1/2} b.

Now assume the center-to-center distance increases by DL
along the stretch axis. The increase in bumper volume for the squares is DV_{s}
= h DL L_{0} (N-1)/N per square. (For closed
circuits such as complete circular rings of tiles, the (N-1)/N factor drops
out.) The useful interior nanorobot volume (in which nonbumper machinery may
be present) is U_{s} = h s^{2} = h (L_{0} - 2b)^{2}.
For the hexagons, DV_{h} = h DL
L_{0} 3^{1/2} / 3 per hexagon, and U_{h} = h s^{2}
27^{1/2} / 2 where s = 2 ((L_{0}/3^{1/2}) - b) / 3^{1/2}.
For the triangles, DV_{t} = (3^{1/2}
/ 2) h DL (L_{0} + 3^{1/2} b) (N-1)/N
per triangle, and U_{t} = (3^{1/2} / 4) h s^{2} where
s = L_{0} - 3^{1/2} b.

We may now compare the relative volumetric cost of square
(R_{sh}) and triangular (R_{th}) device bumpers to hexagonal
device bumpers using

For any choice of (b/L_{0}) __>__ 0, then R_{sh}
> 1 for all N > 2 and R_{th} > 1 for all N > 0. That is,
bumpers on square or triangular devices inevitably require a larger percentage
change in nanorobot volume to effect a given linear adjustment, hence the hexagonal
prism is always the more space-efficient shape for tasks requiring monoaxial
deformations. (Squares are more efficient than triangles except for 0.43 L_{0
}< b < 0.52 L_{0}.)

Similar results follow from an analysis of tiling a surface
that is simultaneously deforming in both dimensions. Most moving biological
surfaces stretch along both axes at once, often at different rates and phases.
Examples include the skin (which can stretch up to 100% before permanent damage
is observed^{521}), the abdominal
diaphragm, cardiac chamber walls (papillary muscles thicken while shortening
7-20% during an isometric contraction^{362}),
alimentary surfaces, and various bladders, glands and sacs. Tendons typically
stretch 5% (up to 10% in the human psoas tendon^{521})
while contracting slightly in width. Apparently hexagonal prisms are also the
most efficient shape for tessellating nanorobots aggregating on biaxially deforming
surfaces; the intuitive rationale is that the most efficient shape is closest
to a circle, with minimal perimeter length enclosing maximal area. In nature,
epidermal squamous cells are arranged in hexagonal columns and onion skin cells
also form neat arrays of hexagons. Hexagonal territorial partitioning is common
among surface-feeding bird and fish species.^{2029,2030}

Last updated on 17 February 2003