GRATING
d·sinθ = mλ run backwards — measure an angle, recover the light, residual zero.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Thousands of slits where Young had two. Each pair still obeys the same interference arithmetic — but with every slit voting, the broad fringes sharpen into knife-edge lines, and the pattern becomes an instrument.
The condition
The slit spacing is one over the ruling density, and constructive interference demands d sine theta equals m lambda. Orders exist only while m lambda fits inside d — the floor of d over lambda, and not one more. Rule the grating finer than the wavelength itself and only the straight-through m = 0 beam survives.
The inversion
Run it backwards and the grating becomes a spectrometer: measure the first-order angle, multiply d by its sine, and the wavelength is recovered — at the defaults, exactly, residual literally zero. The venturi meter recovered flow from pressure; this instrument recovers color from geometry.
Hydrogen through the instrument
Feed it the Balmer quartet the Bohr page predicts and the four lines fan out at exactly the angles the atom priced, each at its true color. And the orders interleave: second-order violet lands beyond first-order red — the dashed overtake — because twice 410 exceeds 656.
Audit
Audited: the inversion returns the input wavelength bitwise at the defaults and to machine epsilon across the whole plane of rulings and colors; Bohr’s predicted H-alpha, measured back through this grating, returns the same float with residual exactly zero — two modules, one number; and the order cutoff holds as an identity at every checked ruling.