This report presents a selection of grids of overlapping circles, where all the circles are the same size, and the center-to-center distance from a circle to its nearest neighbor is a constant. Wikipedia has much more to say about this.
Section 1. In figure 1a is a grid of equal circles arranged triangularly; hardly anything could be simpler. The center-to-center distance (CCD) from each circle to its nearest neighbor is equal to 1 diameter, as indicated by the number 1.0000 in the lower left corner.
In figure 1b the circles' CCD is now half a diameter (indicated by 0.5000), leading to considerable overlap, and yielding the traditional "flower of life" pattern. Now, 61 circles fit in the same overall space as 37 did before.
figure 1a 19 circles | figure 1b 61 circles |
Alternatively, figure 1b can be constructed as the grid of figure 1a succesively overlaid with the 14-circle configurations in green of figures 1c-d-e. Each of the added circles has its center on the point of tangency between two of the original circles. By the same token, each of the original circles has its center on the point of tangency between two of the added circles. Within any one of the three added groups, the CCD is 1 diameter, as with figure 1a.
figure 1c figure 1a plus 14 circles | figure 1d figure 1a plus 14 circles | figure 1e figure 1a plus 14 circles |
This overlay-with-shift technique is quite valuable in creating interesting patterns. As another example, figures 1f-g show similar overlays, in blue, but the centers of the added circles are now at the midpoints of the voids between the original circles. This is substantively different from the placement of the overlays of figures 1c-d-e. In figure 1h only the edges of the circles in this combination are drawn, revealing the effect.
figure 1f figure 1a plus 12 circles | figure 1g figure 1a plus 12 circles | figure 1h 43 circles |
Section 2. In figure 2a, the CCD is √(3/4) ≈ 0.8660 diameters, yielding areas where two circles overlap and points where exactly three arcs intersect. Figure 2b spaces the circles at half that distance, resulting in areas where as many as six circles overlap.
figure 2a 19 circles | figure 2b 61 circles |
Figure 2b could also have been produced by overlaying three offset copies of figure 2a onto itself. Employed would be the same procedure as used in producing figure 1b from 1a-c-d-e.
Figures 2c-d-e are analogous to figures 1f-g-h. The result is much like figure 1b, the "flower of life".
figure 2c figure 2a plus 12 circles | figure 2d figure 2a plus 12 circles | figure 2e 43 circles |
Section 3. In figure 3a, the distance between centers is √(1/2) ≈ 0.7071 diameters, yielding areas where two or three circles overlap. There are no points where three arcs intersect. Where two arcs intersect, they are perpendicular.
The presence of 90-degree angles in a 60-degree grid may explain why it is difficult to find compounds that have significant numbers of tangencies or multiple-arc intersections.
figure 3a 19 circles |
If figure 1h is redrawn with a CCD of √(3/2) ≈ 1.2247, an equivalent grid is created. This underscores the point that many grids have multiple schemes of construction.
figure 3b 43 circles |
Section 4. In figure 4a, the distance between centers is √(1/7) ≈ 0.3780 diameters. Figure 4b spaces the circles at half that distance. Because of the extensive overlapping, the circles are for clarity drawn much larger than in the previous sections.
figure 4a 37 circles |
figure 4b 61 circles |
Section 5. In figure 5a, the distance between centers is √(3/28) ≈ 0.3273 diameters.
figure 5a 37 circles |
Section 6. In figure 6a, the distance between centers is √(3/52) ≈ 0.2402 diameters.
figure 6a 37 circles |