Optics 015 · Ray worlds, boundaries, and natural optics

Unilluminable Room

A three-dimensional mirrored billiard chamber separates direct visibility, exact specular reachability, and sampled diffuse scattering. Dense angular launch sets, repeated finite-wall intersections, reachable-floor point clouds, and a persistent dark-set diagnostic all consume the same room geometry.

Sampled reachable fraction

f_{\mathrm{reach}}

$0.50$

Finite-order dark samples

N_{\mathrm{dark}}

$50\%$

Direction-norm residual

\varepsilon_{\lVert\hat{\mathbf k}\rVert}

$0.00\pi$

Model regime $\text{valid model regime}$

The computation samples finitely many launch angles, reflections, and floor points. It is a reproducible numerical diagnostic of a hidden set, not a mathematical proof that a region remains dark after infinitely many reflections; diffraction, wall roughness, partial absorption, and full radiative transfer are omitted.

Physics tutorial

Audit the physics of Unilluminable Room

BackgroundA mirrored room is a dynamical system. At every wall hit the outgoing direction follows $\hat{\mathbf k}_{\mathrm r}=\hat{\mathbf k}_{\mathrm i}-2(\hat{\mathbf k}_{\mathrm i}\!\cdot\!\mathbf n)\mathbf n$ . Concavity matters because the next mathematical ray–line crossing may lie outside the finite wall segment or beyond an earlier wall.

Why it mattersCan a mirrored room keep a finite region dark from one point source after many exact reflections?

Start with the essentials

Focus question: Can a mirrored room keep a finite region dark from one point source after many exact reflections?
One-sentence intuition: A numerical dark set after a finite reflection order is $\mathcal D_N=\Omega\setminus\bigcup_{m=0}^{N}\mathcal R_m$ . It is useful evidence only when launch angle, wall intersections, coverage tolerance, and reflection order are reported; it is not automatically a theorem about infinitely many bounces.

Core mathematical model

Specular reflection map

\hat{\mathbf k}_{\mathrm r}=\hat{\mathbf k}_{\mathrm i}-2(\hat{\mathbf k}_{\mathrm i}\!\cdot\!\mathbf n)\mathbf n

The unit wall normal reverses only the incident normal component, preserving speed and equal incidence and reflection angles.

First valid wall event

t_*=\min\{t>0:\mathbf r_0+t\hat{\mathbf k}\in\partial\Omega\}

The minimum positive finite-segment intersection prevents a ray from tunneling through a nearer mirror to reach a farther construction line.

Finite-order reachability audit

f_{\mathrm{reach}}=\frac{N_{\mathrm{sample}}-N_{\mathrm{dark}}}{N_{\mathrm{sample}}}

The fraction depends on angular sampling, reflection order, floor sampling, and distance tolerance, so all four belong to the numerical claim.

Common difficulties

Mistaking a red patch for a proof

Typical misconceptionIf a sampled floor region stays red after several reflections, no specular trajectory can ever enter it.

Better mental modelA finite simulation establishes only failure to find a path within its declared launch grid and order. A rigorous hidden-set result needs an analytic construction or convergence argument beyond this apparatus.

Run the experiment

01
Scene 1: Concave visibility audit
Move the point source across the concave chamber. Follow several cyan rays to their first wall and confirm that none crosses an opaque boundary segment.
What to observe: Source motion changes which concave pockets are directly connected, but every legal path remains a sequence of finite intersections.
02
Scene 2: Hidden-set billiard
Enter the hidden-set billiard and compare the dense ray family with green reachable and red dark floor samples after fourteen reflections.
What to observe: The dark sample count can persist under much denser reflection history without being promoted to an infinite-order theorem.
03
Scene 3: Specular versus diffuse walls
Increase the diffuse fraction and identify the explicit gold secondary branches that add new outgoing directions at sampled wall hits.
What to observe: Diffuse scattering fills the floor progressively because it enlarges phase-space support; changing only the red opacity would not reproduce the new trajectories.