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The success of the new invariant for ellipses of arbitrary aspect ratio led us to speculate that this new invariant might hold for randomly-centered holes of completely arbitrary shape. To be able to simulate percolation/conduction problems involving such shapes, one is forced to go to a lattice or digital-image approach.
We consider two kinds of shapes. The first are shapes made up of combinations of needles, which are represented in the same way as were the single needles, by removing conducting bonds on a square lattice.
Objects with finite area, like squares or ellipses or more unusual shapes, are represented using digital image techniques , using square lattices of square pixels. Fig. 5 shows a series of circles represented by pixels. The circles were generated by centering a continuum circle, with a diameter D (D = 9,15,21,41) set to an odd number of pixels, on a given pixel. Other pixels were then included in the digital-image representation if their centers fell inside the radius of the continuum circle . It is somewhat surprising, but encouraging, to note that for diameters of 15 pixels or greater, the area of the digitized circle, as determined by the number of pixels included in the circle by the above construction, agrees with the area of the continuum circle, π D2 / 4, to within less than one percent.
Figure 5: Showing a series of circles represented by an increasing number of square pixels arranged on a square lattice. The accuracy of the representation increases with the number of pixels used.
After a digital image is created, say of randomly-centered insulating ellipses oriented in the horizontal and vertical directions, one-pixel-wide electrodes are "glued" on opposite sides, and a conductor network is created with nodes at the center of each pixel. Fig. 6 shows the resulting conductor network superimposed on a piece of the original random image. Conductors with conductance Σij connect nearest-neighbor pixels i and j, which themselves have conductivities σi and σj. The conductance Σij is defined as the series combination of Σi and Σj:
where Σi is the conductance of one half of pixel i. This means that Σi = 2 σi. If pixels i and j are both conductors with conductivity σo, then Σij = σo. If either pixel i or pixel j represent insulating material, then Σij = 0. The last case of interest is if pixel i is on the electrode, and pixel j has finite conductivity σj. The electrodes are considered to have infinite conductivity, which results in the value of Σij = 2 σj.
Figure 6: Showing an example of a digital image with random conducting and insulating regions, along with one-pixel wide electrodes attached to the ends. The network of conductors into which the image is mapped is also shown, superimposed onto the image.