The Flower of Life is a triangular lattice: circle centers sit at integer combinations of two basis vectors 60° apart, e1 = d·(1,0) and e2 = d·(cos60°, sin60°), kept inside a hexagonal boundary of "ring" order n (the same rule that builds a hex-grid map). That boundary — all axial coordinates (a,b) with |a|, |b|, |a+b| ≤ n — produces exactly 1 + 3n(n+1) circles: 1, 7, 19, 37, 61 for n = 0..4, matching the real historical Flower-of-Life ring counts.
Each circle has radius r; the spacing ratio slider changes d (center distance) relative to r. At ratio 1.0 every circle passes exactly through the centers of its six neighbors — the classical construction. Above 1.0 the circles pull apart; below 1.0 they overlap more deeply. The lens where two circles of radius r intersect is a vesica piscis; its area is computed live from the exact formula 2r²·cos⁻¹(d/2r) − (d/2)·√(4r²−d²) for the current ratio.
- Metatron's Cube — every pair of the visible circle centers connected by a straight line; toggle it to see the platonic-solid projections hidden inside the lattice.
- 60° symmetry error — each generated center is rotated by exactly 60° and the nearest existing lattice point is measured; a true hexagonal lattice returns ≈0 px, confirming the construction is a genuine 6-fold-symmetric packing and not a decorative approximation.