Overlapping Circles Grid

An overlapping circles grid is an arrangement of congruent circles with centers on a lattice and radii large enough that neighboring circles overlap. In the common triangular form, the centers lie on a triangular lattice and each radius equals the nearest-neighbor distance.

The number of circles in a patch consisting of a central circle and successive complete hexagonal rings is the centered hexagonal number, also called a hex number,

which for , 2, ... gives 1, 7, 19, 37, ... (OEIS A003215). The pattern can be continued indefinitely.

Such triangular grids occur in decorative art from antiquity through the medieval and Renaissance periods (Wolfram 2002, pp. 43 and 873; Katz and Jin 2016).

A related pattern occurs at the Osireion at Abydos, Egypt (Rawles 1997).

The Phoenician ivory carving bears a triangular grid of overlapping circles. It dates from about 900 BC, is presumably from the Mediterranean area, and is now in the British Museum (Wolfram 2002, pp. 43 and 873).

A seven-circle overlapping pattern occurs in a first-century BC floor mosaic from a bathhouse in Herod's palace (Katz and Jin 2016).

A related circle geometric construction appears in Leonardo da Vinci's Codex Atlanticus, folio 309v (Leonardo da Vinci 1478-1519).

Modern usage calls the seven-circle patch the seed of life and the 19-circle patch the flower of life. A different 13-circle selection is called the fruit of life; joining its circle centers by line segments gives Metatron's cube (Melchizedek 1999, pp. 42-43 and 158-160).