Grids of mostly equal circles at mostly uniform spacings.
Version of Friday 25 October 2019.
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This report presents a selection of grids of overlapping circles, where the circles are usually the same size, and the center-to-center distance from a circle to its nearest neighbor is usually constant. Wikipedia has much more to say about this.

The first several sections below talk about circles arranged on a triangular grid; after that, rectangular.

Section T1. In figure T1a is a grid of equal circles arranged triangularly. 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 T1b 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 T1a19 circles figure T1b61 circles

Alternatively, figure T1b can be constructed as the grid of figure T1a succesively overlaid with the 14-circle configurations in green of figures T1c-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 T1a.

 figure T1cfigure T1a plus 14 circles figure T1dfigure T1a plus 14 circles figure T1efigure T1a plus 14 circles

This overlay-with-shift technique is quite valuable in creating interesting patterns. As another example, figures T1f-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 T1c-d-e. In figure T1h only the edges of the circles in this T1a-f-g combination are drawn, revealing the effect.

 figure T1ffigure T1a plus 12 circles figure T1gfigure T1a plus 12 circles figure T1h43 circles

Section T2. In figure T2a, the CCD is √(3/4) ≈ 0.8660 diameters, yielding areas where two circles overlap and points where exactly three arcs intersect. Figure T2b spaces the circles at half that distance, resulting in areas where many circles overlap.

 figure T2a19 circles figure T2b61 circles

Figure T2b could also have been produced by overlaying three offset copies of figure T2a onto itself. Employed would be the same procedure as used in producing figure T1b from T1a-c-d-e.

Figures T2c-d-e are analogous to figures T1f-g-h. The result is much like figure T1b, the "flower of life".

 figure T2cfigure T2a plus 12 circles figure T2dfigure T2a plus 12 circles figure T2e43 circles

Section T3. In figure T3a, 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.

 figure T3a19 circles

If figure T1h is redrawn with a CCD of √(3/2) ≈ 1.2247, a grid equivalent to figure T3a is created. This underscores the point that many grids have multiple schemes of construction.

 figure T3b43 circles

Figure T3c, a configuration equivalent to figure T1b, is here obtained by modifying figure T3a as follows: the centers of the circles are unmoved, but the radius is increased to √2 ≈ 1.4142.

Figure T3d is the overlay of T3a and T3c. Note that each small circle is not quite tangent to the large circles near it. The radius of the small circles could be increased by a small amount to address this, but then the intersection of two of their arcs would no longer be exactly perpendicular.

 figure T3c19 circles figure T3d38 circles

When a figure has circles of more than one radius, the number in its lower left-hand corner is omitted, because there would need to be more than one number, and then a question of which number applies to which circles.

In figure T3e, the diameter of each small circle (in red) is 2 sin (15°) ≈ 0.5176 times the diameter of the large.

 figure T3e38 circles

This illustrates how circles of more than one radius can sometimes be combined effectively.

Section T4. In figure T4a, the distance between centers is √(1/7) ≈ 0.3780 diameters. Figure T4b 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 T4a37 circles figure T4b61 circles

Section T5. In figure T5a, the distance between centers is √(3/28) ≈ 0.3273 diameters.

 figure T5a37 circles

Section T6. In figure T6a, the distance between centers is √(3/52) ≈ 0.2402 diameters.

 figure T6a37 circles

Section R1. In figure R1a is a grid of equal circles arranged rectangularly. The center-to-center distance (CCD) from each circle to its nearest neighbor is equal to 1 diameter.

In figure R1b the circles' CCD is now half a diameter (indicated by 0.5000), leading to considerable overlap. Now 63 circles fit in the same overall space as 20 did before.

 figure R1a20 circles figure R1b63 circles

Alternatively, figure R1b can be constructed as the grid of figure R1a succesively overlaid with the configurations in green and blue of figures R1c-d-e. In R1c-d, each of the added circles has its center on the point of tangency between two original circles. In R1e, the center is placed at the midpoint of the void between four original circles. Within any one of the three added groups, the CCD is 1 diameter, as with figure R1a.

 figure R1cfigure R1a plus 16 circles figure R1dfigure R1a plus 15 circles figure R1efigure R1a plus 12 circles

How figures correspond, roughly:

• T1a ↔ R1a
• T1b ↔ R1b
• T1c-d-e ↔ R1c-d
• T1f-g ↔ R1-e

Section R2. In figure R2a, the CCD is √(1/2) ≈ 0.7071 diameters, yielding areas where two circles overlap and points where two pairs of tangent arcs intersect. Figure R2b spaces the circles at half that distance, resulting in areas where many circles overlap; this figure can be broken down into four component grids in the same manner as R1b into R1a-c-d-e. Figure R2c contains all the circles of R2a, but only about half the circles of R2b.

 figure R2a20 circles figure R2b63 circles figure R2c32 circles