Optics 002 · Ray worlds, boundaries, and natural optics

Hall of Mirrors

A labeled mirror gallery constructs single-mirror, corridor, and kaleidoscope image orbits with exact planar reflection. Real and virtual objects share one geometry and scale; color, opacity, physical light paths, and backward construction lines explain what is real.

Ideal image order

N_{\mathrm{ideal}}

$0.50$

Finite-aperture candidates

N_{\mathrm{aperture}}

$50\%$

Reflection residual

\Delta_{\mathrm{refl}}

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Hall of Mirrors

BackgroundA plane mirror image is a geometric reflection of the object across the mirror plane: $\mathbf x^{\prime}=\mathbf x-2\mathbf n\bigl[\mathbf n\cdot(\mathbf x-\mathbf p)\bigr]$ . Repeating that operation creates image orbits for two mirrors; finite mirror panels then decide which backward construction rays can reach their apertures.

Why it mattersWhich virtual images can an observer actually see as mirrors move?

Start with the essentials

Focus question: Which virtual images can an observer actually see as mirrors move?
One-sentence intuition: A plane reflection preserves the complete geometry of the object, so its lateral magnification is $m=+1$ . A virtual image may be translucent or differently colored for explanation, but it must never be drawn smaller than the object.

Core mathematical model

Reflection across a plane

\mathbf x^{\prime}=\mathbf x-2\mathbf n\bigl[\mathbf n\cdot(\mathbf x-\mathbf p)\bigr]

The unit normal and one point on the mirror uniquely define the virtual image position.

Plane-mirror magnification

m=\frac{h_i}{h_o}=+1

The image is upright and the same physical size as the object; its distance behind the plane equals the object distance in front.

Ideal two-mirror image order

N_{\mathrm{ideal}}\approx\frac{2\pi}{\alpha}-1

This angular orbit count is exact only in its stated divisibility and placement cases; the apparatus labels it as ideal.

Common difficulties

Equating a virtual image with emitted light

Typical misconceptionA virtual ghost sends rays outward from its displayed position.

Better mental modelThe dashed backward sight line is a construction. Real light reaches the observer only after reflection from a real mirror panel.

Run the experiment

01
Scene 1: Single virtual image
Move the observer laterally in the single-mirror scene and compare the object and image distances to the mirror plane.
What to observe: The image remains equally far behind the mirror and exactly the same height as the object even though the observer sight line changes.
02
Scene 2: Infinite mirror corridor
Change the corridor angle and follow successive image generations away from the real object.
What to observe: Smaller mirror angle increases ideal orbit order, but finite panels reject construction intersections outside their bounds.
03
Scene 3: Kaleidoscope symmetry
Use the kaleidoscope scene to test rotational closure and observer-aperture candidates separately.
What to observe: A closed angular pattern can contain more ideal images than the finite-aperture candidate count.