Kaleidoscope Geometric Transformations

Assembling the kaleidoscope...

Presets

Mirrors and group

Second mirror: dihedral group D_N (off = rotations only, C_N)

Object cell, pen and look

Auto light pen (Lissajous); drag in the eyepiece to draw Cutaway: show virtual images Auto-orbit (cutaway)

Live readouts

Graphs

Number of images of a point: 2N (dihedral, orange) and N (rotations, blue); current N marked

Reflections before a ray leaves the mirror pair (2000 random rays)

Beads visible through the wedge over time

Frame time (ms), green line = 16.7 ms

Numerical checks

running...

How to use

Look through the eyepiece: the coloured beads and gems sit in a rotating object cell at the far end; two mirrors meeting at the angle pi/N fill the view with N-fold symmetric copies. Change N, tick or untick the second mirror (dihedral D_N versus pure rotations C_N), spin the cell, and draw with the light pen (it runs by itself; drag to add strokes). Press View for the cutaway: the barrel is cut open and you can see the real wedge, the two mirrors and the beads, plus the virtual images they create. Drag to orbit in the cutaway.

The science

Two plane mirrors meeting along the tube axis at the dihedral angle alpha = pi/N generate, by repeated reflection, a set of 2N images of every object (method of images). The images are the orbit of the object under the dihedral group D_N, generated by the rotation R by 2 pi/N about the axis and the reflection M in one mirror:

R^N = I, M^2 = I, M R M = R^(-1), |D_N| = 2N, x' = R^k M^j x

Mirrors containing the axis keep the axial coordinate, so every image lies in the object-cell plane: the 3D device draws a flat pattern, exactly the 2D picture, but with real beads that have depth and tumble. Without the second mirror (or with two mirrors at 2 pi/N and rotation only) the group is the cyclic group C_N with N images. A light ray bouncing between the mirrors follows, when unfolded, a straight line through copies of the wedge; the number of reflections equals the number of mirror lines that straight line crosses, and the exit point is the unfolded exit point folded back into the wedge. Beads are only drawn where they lie inside the wedge (fragments outside are discarded in the source frame), which is what a physical object cell shows.

Simplifications: perfectly reflecting mirrors (no 4 percent loss per bounce, no darkening of high-order images), beads are emissive glass-like solids lit from behind, not ray-traced refraction, and the eyepiece is orthographic-like only near the axis (a small perspective). The barrel, brass fittings, frosted cell, gems, shards and beads are modelled and baked in Blender 5.2; mirrors, images and pen trail are generated live.

What the checks mean

Group laws multiply the actual 3D rotation and reflection matrices. Distinct images counts unique image positions of a generic point. Billiard unfolding traces random rays with real mirror-plane reflections and compares exit point, exit direction and reflection count with the straight-line unfolding folded into the wedge. Rendered orbit reads the instance matrices of the first bead and compares every image position with the group formula.

Keyboard and accessibility

Space pauses, R resets, C re-runs the checks (while the page has focus).
All controls are native inputs with labels.