Figure 6. Gauss Circle Problem As Figure 6 shows, Gauss proved that the number of integer coordinates located N(r) in a circle of radius have the following relationship: r = + N(r) r2 E(r), where |E(r)| 2r. We define 1, since we do not count the center as a neighbor of n = N(r) – itself. For all neighbors, we determine the angle between the top of the neighbor’s pole and the pole at the current point If any angle is greater than the critical slope angle, then Pij. move one unit height of sand toward one of the directions where the angle is greatest. Then repeat the process until no angle is greater than the critical slope angle. An alternative method would be to equally distribute the amount among all poles where the angle is greatest. If a sufficient number of neighbors are used, then the result will be a pile shaped roughly like a cone. Table 1 shows the number of neighbors and the resulting pile shape for a range of possible For example, if is 4, then is 46 and the shape has 20 edges. r. r n Table 1. Radius vs. pile shape r Base Shape r n 4 Square 8 Square
Figure 6. Gauss Circle Problem As Figure 6 shows, Gauss proved that the number of integer coordinates N(r) located in a circle of radius r have the following relationship: N(r) = r2 + E(r), where |E(r)| 2r. We define n = N(r) – 1, since we do not count the center as a neighbor of itself. For all neighbors, we determine the angle between the top of the neighbor’s pole and the pole at the current point Pij. If any angle is greater than the critical slope angle, then move one unit height of sand toward one of the directions where the angle is greatest. Then repeat the process until no angle is greater than the critical slope angle. An alternative method would be to equally distribute the amount among all poles where the angle is greatest. If a sufficient number of neighbors are used, then the result will be a pile shaped roughly like a cone. Table 1 shows the number of neighbors and the resulting pile shape for a range of possible r. For example, if r is 4, then n is 46 and the shape has 20 edges. Table 1. Radius r vs. pile shape r n Base Shape 4 Square 8 Square
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