© 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 domes⁵³⁹ and the polio virus are symmetrical spherical arrangements of alternating triangles.³⁸⁴ (Many viruses are regular icosahedra.)

Last updated on 17 February 2003