Draw two circles the same size, each passing through the other’s centre. The almond-shaped overlap is the VESICA PISCIS — ‘the fish’s bladder’ — the seed of Gothic windows, the aureole around Christ, and the compass-and-straightedge start of sacred geometry. Its shape is exact: the pointed oval is taller than it is wide by precisely √3. Slide the circles together.
The vesica piscis is the lens where two equal circles of radius r overlap with their centres exactly r apart (each through the other’s centre). Its height (2r·√3/2 = r√3) to width (r) ratio is precisely √3 ≈ 1.7320508 — the figure from which an equilateral triangle, hexagon, and much of compass-and-straightedge ‘sacred geometry’ are constructed. A fail-loud self-check throws unless the height/width ratio equals √3. ◆ the geometry is exact, node-verified.
LIT: the √3 proportion at the r-apart configuration is exact Euclidean geometry. AMBER: the theological/symbolic weight (the fish, the mandorla, divine union) is the tradition built on a clean construction — the shape is real, the sacredness is the belief.