Take the seven-circle Seed of Life and keep going: place the compass at each new intersection and draw another circle of the same radius. The result is a lattice in which every circle passes through the centres of six neighbours and every centre lies on six circles. Mathematically it is the overlapping circles grid, and it is the visual form of the closest possible packing of equal circles in the plane, the same arrangement bees reach in honeycomb and oranges reach in a crate.
The conventional depiction stops at nineteen complete circles and trims the ragged partial arcs with two concentric bounding rings, but that is a presentational choice rather than a limit in the geometry. Grids of 1, 7, 19, 37, 61, 91 and 127 circles are all standard, each ring adding six more around the last. Left to run, the pattern covers the plane forever.
Nearly every figure of elementary plane geometry can be read out of it. Pairs of circles give the vesica piscis. The centres give equilateral triangles, regular hexagons and the hexagram. Selecting thirteen of the circles gives the figure modern writers call the Fruit of Life, and joining those thirteen centres gives Metatron's Cube.
That richness is real, and it is the honest reason the pattern has held attention for nearly three thousand years. It is also, less romantically, a consequence of the symmetry rather than evidence of anything hidden: a lattice with six-fold symmetry will necessarily contain every shape that six-fold symmetry admits. Finding hexagons in it is not a discovery. It is the definition.