Grids of equal circles at uniform spacings.
Version of Thursday 26 September 2019.
Dave Barber's other pages.

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