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
126.96.36.199 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 rcurve is relatively large (e.g., rcurve >> Ln). For areas containing smaller radii of curvature (rcurve > Ln), 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 domes539 and the polio virus are symmetrical spherical arrangements of alternating triangles.384 (Many viruses are regular icosahedra.)
Last updated on 17 February 2003