THE THIN LENS

One equation ties where an object sits to where its image forms, and how big it looks. Slide the object across the focal point and the same law hands you a real inverted image, then a virtual magnified one — the refraction of Snell’s law shaped into a curve of glass. Rendered, not quoted.

source E. Halley, “An Instance of the Excellence of the Modern Algebra, in the Resolution of the Problem of Finding the Foci of Optick Glasses Universally,” Phil. Trans. R. Soc. 17 (1693) 960–969; paraxial form after C. F. Gauss, Dioptrische Untersuchungen (1841). rstl.1693.0074 — AMBER: pre-1800 print, no guaranteed-stable open copy

Blue Team · builds & defends

The Model

A thin lens has negligible thickness, so both refractions collapse onto one plane at distance f. Every object distance d_o maps to exactly one image distance d_i:

1/f = 1/dₒ + 1/dᵢ m = −dᵢ/dₒ

Sign convention: d_i>0 = real image on the far side; d_i<0 = virtual image on the near side. m negative ⇒ inverted; |m| is the size ratio h_i/h_o. Converging lens f>0, diverging lens f<0.

The Lineage

Object → image in one equation. A lens is Snell refraction at two curved surfaces, added up. This sphere folds the-snell-law (n₁ sin θ₁ = n₂ sin θ₂, one ray bending at one interface) into a whole surface of glass: sum the bend of the front and back faces in the paraxial limit and Snell becomes the lensmaker’s 1/f, which becomes this imaging law.

Neighbour: the-snell-law → the-thin-lens → (grating / diffraction spheres downstream).

The Witness

A live re-run of selfcheck() against the pure engine below — the same function proven at boot. It confirms the law holds and flips red the instant the Tamper (window 6) corrupts the sign.

checking…

Re-checks: residual < 1e-9, di(∞)=f, 2f→2f at m=−1, virtual magnifier, diverging reducer.

The Machine

Data In in ↓

Focal length and object distance — and the object height. Everything else is derived.

focal length f = 10.0 cm object distance dₒ = 30.0 cm object height hₒ = 4.0 cm

refract

The Panel LIT

image distance dᵢ—

magnification m—

image height hᵢ—

residual |Δ|—

—

image

Data Out out ↓

The proven result: the pair (d_i, m) that satisfies the imaging law exactly, with the image classified real/virtual and upright/inverted from the signs alone.

engine LIT

Red Team · attacks & breaks

The Adversary wall

“The thin-lens law is only ever an approximation — so any exact assertion is a lie.”

Honest wall. The paraxial law is exact only for small angles and zero thickness. Real lenses show spherical aberration, coma, chromatic aberration (f depends on λ), and field curvature. This engine is exact inside its own definition: given the paraxial model, the algebra is provably correct to 1e-9. It never claims a real glass lens obeys it perfectly — only that the equation does.

The Graveyard

~~“A lens always makes things bigger.”~~ Only inside f (virtual magnifier). Beyond 2f a converging lens shrinks; a diverging lens always shrinks.
~~“A virtual image can be caught on a screen.”~~ d_i<0 means the rays only appear to diverge from a point — no screen catches it, only the eye.
~~“Put the object at the focal point to get a sharp image.”~~ At d_o=f the rays leave parallel: d_i→∞, no image forms at all.
~~“Bigger f means more magnification.”~~ m depends on where the object sits relative to f, not on f alone.

The Tamper

The disclosed planted void. Flip the imaging law to the wrong sign — 1/f = 1/dₒ − 1/dᵢ — and the object at 2f no longer images at 2f. The Witness (window 7) catches it live and goes red.

state: clean