A circular lens can't focus a point to a point — so how close can two stars (or two letters) be and still be seen as two, and why does 20/20 vision stop where it does?
▶ Launch the interactive simulationModel the Airy pattern A(x)=[2J₁(x)/x]² of a circular aperture; bisect the Bessel function J₁ for its first zero x₁; form the incoherent sum of two equal point sources, scan a noisy combined profile at the Rayleigh separation to count peaks and read the central saddle dip, and locate the Sparrow limit from where the central curvature flips sign.
the central disk's angular radius is θ_min = (x₁/π)·λ/D = 1.22·λ/D — the celebrated 1.22 is NOT a fudge but x₁/π, the first zero of J₁ (≈3.8317) divided by π, recovered here from scratch. At Rayleigh separation the two Airy disks merge into a profile that dips to ≈0.735 of the peaks (the 'just resolved' signature); squeeze below the Sparrow limit ≈0.78 θ_min (≈0.95 λ/D) and the dip vanishes into one blob. Made concrete: a 2.3 mm daylight pupil at 550 nm gives θ_min≈1 arcminute — exactly the stroke width on the 20/20 eye-chart line, so 'perfect' vision is set by diffraction, not biology, and bigger apertures (telescopes) see finer. The imaging sequel to the diffraction grating: from resolving two colours to resolving two points.