## 5 Results
### Measures of Performance  
Figure 17 provides the visual results of the four DGGSs based on different methods. Our DGGSs have, on average, the most compact cells and smaller area distortions, except for the Snyder equal-area DGGSs. Since equivalent surface areas for the cells are maintained, [PERSON]'s DGGSs exhibit more shape distortions, especially in the cells along the radial lines that extend from a polyhedron face center to its vertices. We used five statistics of shape and area measurements: the range of the different area (standardized area), the standard deviation of the area, the range of compactness, the standard deviation of compactness, and the mean compactness. Figures 18-20 illustrate the performance of DGGSs by the level of resolution for each of the five response measures. Figure 17: The four polyhedral DGGSs. Each figure shows a 4-fold, level-5 triangle subdivision on polyhedrons. The surfaces are shown in an orthographic projection to accentuate the distortion patterns present in each method. Differences in the range of the area distortion in the different methods are clearly distinguished as the resolution increases (shown in Figures 18a, 19a, 20a). The greatest area distortion is found with the Geo Bisector method, followed by (in descending order) the Dymaxion projection method, SACs, and Snyder. The Snyder projection has an area range of zero, of course, since it is an equal-area mapping method. For the compactness measures, the Snyder method has the highest scores both in range and standard deviation (Figures 18c, 19c, 20c). These results indicate that the Snyder equal-area projection has the most shape distortion. The compactness performance of our method is almost the same as that of the Dymaxion method. As the mean of the compactness increases, it approaches the most compact  
Figure 19: Graphs of the five statistics of the area and ZSC response measurements for each method as a sequence by recursion level (level 1 to level 8). These results are for the icosahedron using the 4-fold density ratio. Figure 20: Graphs of the five statistics of the area and ZSC response measurements for each method as a sequence by recursion level (level 1 to level 5). These results are for the icosahedron using the 9-fold density ratio. shape possible. The results of the experiments for this measure show that the Geo Bisector has the best performance (highest values). The histograms of the distributions of the area bias and compactness values (in Figures 21 and 22) show distinct differences between methods. The Snyder projection histogram for the area has only one bar for the single equal-area value (no bias). Our method has a continuous, unimodal histogram. It also has a better shape distortion performance than the Dymaxion and Geo Bisector methods on the tetrahedron (Figure 21a). For the icosahedron, no matter what fold method is used the histogram for our method has a smoother distribution (Figures 22a and23a). The maps of the spatial variation in the area and ZSC values have interesting patterns (in Figures 24-26). For the 4-fold density ratio, the spatial variation in the area (Figures 24a and 25a) has one of two patterns. The Snyder method result is homogeneous with an equal area. The other three methods created partitions with a decrease in area per cell radially from the center. For compactness, each method has a unique pattern (in Figures 24b and 25b). For the 9-fold density ratio, all methods have patterns of area and compactness variation that are very similar to those for the 4-fold ratio (in Figure 26). Figure 21: The distribution histograms of area and compactness for different methods. These values are for the tetrahedron partitions. Figure 22: The distribution histograms of area and compactness for different methods. These values are for the icosahedron partitions. Figure 23: The distribution histograms of area and compactness for different methods. These values are for the icosahedron partitions.