In this paper, we first review local counting methods for perimeter estimation of piecewise smooth binary figures on square, hexagonal, and triangular grids. We verify that better perimeter estimates, using local counting algorithms, can be obtained using hexagonal or triangular grids. We then compare surface area estimates using local counting techniques for binary three-dimensional volumes under the three semi-regular polyhedral tilings: the cubic, truncated octahedral, and rhombic dodecahedral tilings. It is shown that for surfaces of random orientation with a uniform distribution, the expected error of surface area estimates is smaller for the truncated octahedral and rhombic dodecahedral tilings than for the standard cubic or rectangular prism tilings of space. Additional properties of these tessellations are reviewed and potential applications of better surface area estimates are discussed.
Available at: http://works.bepress.com/erik_learned_miller/2/