Light crossing a boundary bends by a single rule: n₁·sinθ₁ = n₂·sinθ₂. Into a denser medium it bends toward the normal; into a rarer one, away — and past one angle it cannot cross at all and turns entirely back. Down the center, data flows: the two indices and an angle go in, the engine solves for the refracted ray, and the crossing comes out. The blue team builds and defends it; the red team tries to break it.
source Snell derived it 1621 (unpublished); first in print R. Descartes, La Dioptrique (1637); earlier still Ibn Sahl, On Burning Mirrors and Lenses (984). AMBER — no single stable primary link; see the documented lineage. Rendered, not quoted.
Angles are measured from the normal (perpendicular to the boundary). The law is a single conserved quantity across the interface:
n₁·sinθ₁ = n₂·sinθ₂. Solve it: θ₂ = arcsin( n₁·sinθ₁ / n₂ ). If that ratio exceeds 1 there is no real angle — the ray cannot leave, and all of it reflects. That threshold is the critical angle θᶜ = arcsin(n₂/n₁), and it exists only when n₁ > n₂.
Live values for the current setting:
| quantity | value |
|---|
Snell fixes the direction a boundary turns a ray; it does not say what is travelling. What bends is a wave — the field carried by the wave equation uᵗᵗ = c²uᵅᵅ.
Refraction is what happens when that wave meets a slower region (c = c₀/n): the wavefronts pile up, and to stay continuous across the seam the crest angle must obey exactly n₁sinθ₁ = n₂sinθ₂. One sphere carries the wave; this one bends it. Each sphere is the next one's premise.
The blue team's live check: re-run the solver over a fixed battery of crossings and confirm the sine relation round-trips and the critical angle is arcsin(n₂/n₁). If red tampers, this badge is where it shows.
Three numbers define the crossing: the index of the medium the ray starts in (n₁), the index it enters (n₂), and the angle of incidence θ₁ measured from the normal. A larger index means slower light and a denser medium.
| medium | n (approx) |
|---|---|
| vacuum | 1.000 |
| air | 1.0003 |
| water | 1.333 |
| crown glass | 1.52 |
| diamond | 2.417 |
Feed these into the panel below and it solves for the refracted ray — or reports that none exists.
Move any slider — the refracted angle and the critical angle are solved from the sine relation on the spot, never looked up.
What the machine produces, proven: for every crossing either a single refracted angle θ₂ that satisfies n₁sinθ₁ = n₂sinθ₂ to 1e-12 and round-trips back to θ₁, or — when n₁ > n₂ and θ₁ exceeds the critical angle — total internal reflection, no real solution. At normal incidence (θ₁=0) the ray does not bend.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
So "the ray goes here" is only half the crossing; the energy split, the phase, and the colour are all outside this one frame — which is why the panel reports geometry, not brightness.
"Snell discovered the law." Cut. Ibn Sahl had it in 984; Snell (1621) never published; Descartes printed it first (1637). In France it is the Snell–Descartes law — the machine cites all three.
"Light always bends toward the normal." Cut. Only into a denser medium (n₂>n₁). Into a rarer one it bends away, and past the critical angle it does not cross at all — both computed live.
"Total internal reflection is perfectly total." Kept, corrected. Geometrically yes, but an evanescent field leaks into the second medium; across a thin gap it tunnels (frustrated TIR). Snell gives the ray, not that field.
The red team's move: replace the sine relation with a linear one — n₁θ₁ = n₂θ₂ — a plausible-looking forgery. Refraction angles and the critical angle both drift. The blue team's witness (window 7) is watching.
Swap sinθ for θ and the round-trip n₂sinθ₂ no longer returns n₁sinθ₁ — the witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.