The Infinity Cell: Standing Inside a Reflection Group

Reflections in two mirrors meeting at a dihedral angle $\theta$ compose to a rotation by $2\theta$ about their shared edge. The reflected images around that edge close up seamlessly if and only if

$\theta = \frac{\pi}{k}, \quad k \in \mathbb{Z}.$

Among the Platonic solids, exactly one passes this test. The cube’s dihedral angle is $90° = \pi/2$: four mirror-image rooms fit around every edge, and the reflection group generated by its six face planes tiles Euclidean space as the cubic honeycomb $\{4,3,4\}$. Inside a mirrored cube, deep reflections form a perfect lattice — straight lines stay straight forever, to arbitrary depth. Switch the visualization to the cube and look: the interior is indistinguishable from an infinite crystalline corridor of rooms.

The dodecahedron’s dihedral angle is $\arccos(-1/\sqrt{5}) \approx 116.565°$. Three copies around an edge cover $349.70°$, leaving an angular defect of about $10.31°$ per circuit. The recursion never closes. Every reflection chain accumulates that defect, and the deeper you look, the more the accumulated error curls the image — that is exactly the kaleidoscopic swirl you see far inside the physical piece. It is not an imperfection of the build; it is the geometry of $\mathbb{E}^3$ refusing to be tiled by dodecahedra.

The other solids fail more dramatically: the tetrahedron misses by $7.36°$, the octahedron by $31.59°$, the icosahedron by $83.62°$. The dodecahedron’s near-miss is the aesthetic sweet spot — close enough to closing that the corridors read as almost periodic, wrong enough that they shear into spirals at depth.

One more structure worth seeing: the dodecahedron’s 30 edges fall into 15 parallel pairs whose directions group into 5 mutually orthogonal triples — the edge-level trace of the five inscribed cubes, on which the rotation group acts as $A_5$. The Five Frames pattern colors each edge by its triple.

The defect can be repaired — not in Euclidean space, but by bending space itself.

Shrink the cell in $S^3$. In spherical space, dihedral angles grow with size. Inflate the dodecahedron until $\theta = 120° = 2\pi/3$ and the mirrors generate $\{5,3,3\}$ — the 120-cell. Standing inside the mirrored cell, you would see its reflections close up exactly: 120 dodecahedral rooms receding along great-circle geodesics. The infinity mirror becomes a finite, perfect object — you are inside the 120-cell.

Grow the cell in $H^3$. In hyperbolic space, angles shrink with size. Grow the dodecahedron until $\theta = 90°$ and the mirrors generate the order-4 dodecahedral honeycomb $\{5,3,4\}$ — infinitely many right-angled dodecahedral rooms tiling hyperbolic space.

The physical art piece is the Euclidean shadow of these two exact objects, wedged between a spherical polytope and a hyperbolic honeycomb, belonging to neither.

The obvious implementation — mirror the scene recursively with render-to-texture — costs $(\text{faces})^{\text{depth}}$ passes and dies immediately. Convexity gives a better way: a convex cell is an intersection of half-spaces $\mathbf{n}_i \cdot \mathbf{x} \le d_i$, so from any interior point the nearest boundary along a ray is a handful of plane tests. Each pixel runs the whole recursion analytically: find the nearest face, classify the hit (LED strip → emit and stop; frame → absorb; mirror → attenuate and reflect), and iterate to a bounce cap. That loop is the infinite mirror room. The CPU uploads a camera matrix and a clock; everything else — thousands of virtual LEDs at fifty reflection depths — is computed per-pixel, per-frame, in one fullscreen pass.

Depth falloff comes from three multiplicative losses, matching the physics of real half-mirrors: reflectivity per bounce, a slightly cyan mirror tint, and colored haze per unit path length. Most rays terminate early on a strip or frame, which is what keeps the average cost far below the bounce cap.

Built with TypeScript + raw WebGL2, zero dependencies. Single-pass analytic raytrace, HDR bloom via Kawase blur, ACES tonemap. Geometry invariants and the cube’s mirror-lattice exactness are verified numerically in CI.