Kaleidoscopes

The rule

A kaleidoscope draws only a small part of the image. You pick a number of sectors, n, and cut the disc like a pie: each sector spans 360° / n. You draw in just one, then copy the motif by rotation into all n sectors:

copies = n

With a mirror, each sector is itself cut in two by an axis, and the motif is reflected too: you draw in a half-sector of 180° / n, and the motif is replayed in the n sectors, once the right way round, once flipped:

copies = 2n

This is the real kaleidoscope, the one with mirrors. All the rest is symmetry: the figure never holds more ideas than the small piece you started with.

What changes from one algorithm to the next

Four of them use the same symmetry around a centre, the dihedral group; the fifth takes it away from the centre to tile the whole plane. What also changes is what gets drawn in the fundamental domain, and whether we draw strokes or fold pixels.

Stained-Glass Shards

Stained glass. Curved shards are drawn in the fundamental domain alone, then replayed on every copy. An outline, like the lead of a stained-glass window, can surround them or vanish.

Frost Ribbons

Filaments follow a polar noise field, confined to the fundamental domain, then replayed on the copies: each ribbon drawn becomes 2n.

Dihedral Mandala

Concentric rings of geometric motifs. Each motif is drawn once in the fundamental domain, then replayed on the 2n copies; some rings also carry a thin guide circle.

Kaleidoscopic Fold

The optical kaleidoscope, with no drawn stroke. Each pixel is folded into the fundamental domain before sampling a continuous field (fractal noise and radial waves): the symmetry costs only an angle fold.

p4m Tiling

The symmetry leaves the centre and tiles the whole plane (the p4m wallpaper group). Each pixel is folded into the fundamental triangle of a square cell, by horizontal, vertical and diagonal mirrors, then samples a continuous field.