Mirror Maze: How Light Bounces and Bends

The law that runs the whole maze

Every bounce in a mirror maze obeys one rule, known since antiquity: the law of reflection. Light hits a flat (plane) mirror and leaves it at the same angle it arrived, measured from an imaginary line perpendicular to the surface called the normal. The angle of incidence equals the angle of reflection, full stop — it does not matter what colour the light is, how fast it is travelling, or what the mirror is made of, as long as the surface is smooth (specular) rather than rough (diffuse, which scatters light in every direction and is why a piece of paper does not reflect a picture).

angle of incidence (θi)  =  angle of reflection (θr)
                     |  normal
    incoming ray  \  |  /  reflected ray
                   \ | /
    ────────────────●──────────────── mirror surface

A flat mirror simply flips the ray's direction across the normal. In vector form, if d is the incoming direction and n is the unit normal, the reflected direction is r = d − 2(d·n)n. That one line is the entire engine behind every plane-mirror puzzle: trace the beam, find where it hits a surface, compute the new direction with that formula, repeat until the beam leaves the maze or hits the target.

Curved mirrors change the story

A concave mirror (curving inward, like the inside of a spoon) still obeys the same reflection law at every point, but because the surface tilts continuously, nearby parallel rays converge toward a single focal point. That is why concave mirrors can concentrate sunlight into a hot spot, and why they are used in reflecting telescopes and shaving mirrors that magnify. A convex mirror curves outward and does the opposite: parallel rays spread apart as if they came from a point behind the mirror, which is why convex mirrors show a wider field of view — the ones used for shop security and car wing mirrors.

For a mirror shaped like a section of a sphere with radius of curvature R, the focal length is approximately f = R/2 for rays close to the central axis. In a maze puzzle this means a beam that grazes a concave mirror far from its centre bends more sharply than one hitting near the middle — the curvature is not uniform in effect, only in shape.

Refraction: when light changes medium

Mazes that include glass blocks or lenses add a second phenomenon: refraction, the bending of light as it crosses from one transparent medium into another with a different refractive index (n). Light travels slower in glass (n ≈ 1.5) than in air (n ≈ 1.0), and that speed change bends the ray according to Snell's law:

n1 · sin(θ1)  =  n2 · sin(θ2)

n1, n2   refractive indices of the two media
θ1       angle of incidence (from the normal)
θ2       angle of refraction (from the normal)

Push the angle far enough and something else happens: past a critical angle, light cannot leave the denser medium at all and instead reflects internally — total internal reflection. This is the same physics that keeps light bouncing down a fibre-optic cable and traps light inside a diamond, making it sparkle. In a mirror-maze puzzle, a wedge of glass angled steeply enough can act as a perfect mirror without any silvering at all.

Why the puzzle is solvable by hand

Because every interaction — reflection off a plane, reflection off a curve, refraction through glass — is governed by a local, deterministic rule, the whole path of a beam is fixed the instant you place the mirrors. There is no randomness. This is exactly why ray tracing works as a rendering technique in computer graphics: chase one ray backward from the eye, bounce it off every reflective and refractive surface it meets using these same two rules, and you get a physically accurate picture. A mirror maze is ray tracing you can build with your hands.

Frequently asked questions

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